Evolution Equations and Their Applications in Physical and Life Sciences

evolution equations and their applications in physical and life sciences

PURE

AND APPLIED

MATHEMATICS

A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS EarlJ. Taft Rutgers University NewBrunswick, New Jersey

EDITORIAL M. S. Baouendi University of California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology

Zuhair Nashed University of Delaware Newark, Delaware

BOARD Anil Nerode Cornell University DonaM Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University

S. Kobayashi University of California, Berkeley

David L. Russell Virginia Polytechnic htstitute and State University

Marv#t Marcus University of California, Santa Barbara

Walter Sehempp Universiti# Siegen

W. S. Massey Yale University

Mark Teply University of Wisconsin, Milwaukee

LECTURE NOTES IN PURE AND APPLIED

MATHEMATICS

1. N. Jacobson, ExceptionalLie Algebras 2. L.-A. LindahlandF. Poulsen,ThinSetsin Harmonic Analysis 3. I. Satake,ClassificationTheoryof Semi-Simple AlgebraicGroups 4. F. Hirzebruch et al., DifferentiableManifoldsandQuadratic Forms 5. L Chavel, Riemannian SymmetricSpacesof RankOne 6. R. B. Burckel, Characterization of C(X)Among Its Subalgebras 7. B. R. McDonald et aL, RingTheory 8. Y.-T. Siu, Techniques of Extension on Analytic Objects 9. S. R. Caraduset aL, Calkin AlgebrasandAlgebrasof Operatorson Banach Spaces 10. E. O. Roxinet aL, Differential Games andControlTheory 11. M. OrzechandC. Small, TheBrauerGroupof Commutative Rings 12. S. Thornier,Topology andIts Applications 13. J. M. LopezandK. A. Ross,SldonSets 14. W.W.ComfortandS. Negrepontis,Continuous Pseudometrics 15. K. McKennon and J. M. Robertson,Locally ConvexSpaces 16. M. CarmeliandS. Malin, Representations of the RotationandLorentzGroups 17. G. B. Seligman,RationalMethods in Lie Algebras 18. D. G. deFigueiredo,FunctionalAnalysis 19. L. Cesad et al., NonlinearFunctional AnalysisandDifferential Equations 20. J.J. Sch~ffer, Geometry of Spheresin Normed Spaces 21. K. YanoandM. Kon,Anti-lnvadant Submanifolds 22. W. V. Va~concelo$, TheRingsof DimensionTwo 23. R. E. Chandler,HausdorffCompactiflcations 24. S. P. Franklin andB. V. S. Thomas, Topology 25. S. K. Jain, RingTheory 26. B. R. McDonald andR. A. Morris, RingTheoryII 27. R. B. MuraandA. Rhemtulla,OrderableGroups 28. J. R. Graef,Stability of Dynamical Systems 29. H.-C. Wang,Homogeneous BranchAlgebras 30. E. O. Roxinet al., Differential Games andControlTheoryII 31. R. D. Porter,Introductionto FibreBundles 32. M. Altman,ContractorsandContractorDirectionsTheoryandApplications 33. J. S. Golan,Decomposition andDimension in ModuleCategories 34. G. Fairweather, Finite Element GalerkinMethods for Differential Equations 35. J. D. Sally, Numbers of Generators of Idealsin LocalRings 36. S. S. Miller, Complex Analysis 37. R. Gordon,Representation Theoryof Algebras 38. M. GotoandF. D. Grosshans, Semisimple Lie Algebras 39. A. L Arrudaet al., Mathematical Logic 40. F. VanOystaeyen,Ring Theory 41. F. VanOystaeyen andA. Verschoren, ReflectorsandLocalization 42. MoSatyanarayana, Positively OrderedSemigroups 43. D. L Russell,Mathematics of Finite-Dimensional ControlSystems 44. P.-T. Liu andE. Roxin,Differential Games andControlTheoryIII 45. A. GeramitaandJ. Seberry,Orthogonal Designs 46. J. Cigler, V. Losert, and P. Michor, BanachModulesand Functorson Categoriesof Banach Spaces Economics 47. P.-T, Liu andJ. G. Sutinen,ControlTheoryin Mathematical 48. C. Bymes, Partial Differential EquationsandGeometry 49. G. Klambauer, Problems andPropositionsin Analysis 50. J. Knopfmacher, AnalyticArithmeticof AlgebraicFunctionFields 51. F. VanOystaeyen, Ring Theory 52. B. Kadem, Binary TimeSeries 53. J. Barros-Neto andR. A. Artino, HypoellipticBoundary-Value Problems 54. R. L. Steinberg et al., Nonlinear Partial Differential Equations in Engineering andAppliedScience 55. B. R. McDonald, RingTheoryandAlgebraIII 56. J. S. Golan,Structure Sheaves Overa Noncommutative Ring 57. T. V. Narayana et al., Combinatodcs, Representation TheoryandStatistical Methods in Groups 58. T.A. Burton,Modeling andDifferential Equations in Biology 59. K. H. KimandF. W.Roush,Introductionto Mathematical Consensus Theory

60. J. BanasandK. Goebe/,Measures of Noncompactness in BanachSpaces 61. (3. A. Nielson,DirectIntegralTheory 62. J. E. Smithet al., Ordered Groups 63. J. Cronin,Mathematics of Cell Electrophysiology 64. J. W. Brewer,PowerSedesOverCommutative Rings 65. P. K. Kamthan andM. Gupta,Sequence SpacesandSeries 66. T. G. McLaughlin, Regressive Setsandthe Theoryof Isols et aL, Integral andFunctionalDifferential Equations 67. T. L. Herdman 68. R. Draper,Commutative Algebra 69. W.G. McKay andJ. Patera, Tablesof Dimensions,Indices, andBranchingRulesfor Representationsof SimpleLie Algebras 70. R. L. Devaney andZ. H. Nitecki, ClassicalMechanics andDynamical Systems 71. J. VanGee/,PlacesandValuationsin Noncommutative Ring Theory 72. C. Faith, Injective Modules andInjective QuotientRings Programming with DataPerturbationsI 73. A. Fiacco,Mathematical 74. P. Schultzet al., AlgebraicStructuresandApplications 75. L Bicanet al., Rings,Modules, andPreradicals 76. D. C. KayandM. Breen,ConvexityandRelatedCombinatorial Geometry 77. P. FletcherandI/V. F. Lindgren,Quasi-Uniform Spaces 78. C.-C. Yang,Factodzation Theoryof Meromorphic Functions 79. O. Taussky,Ternary Quadratic Formsand Norms , 80. S. P. SinghandJ. H. Burry,Nonlinear AnalysisandApplications 81. K. B. Hannsgen et al., VolterraandFunctionalDifferential Equations 82. N. L. Johnson et al., Finite Geometries 83. G. I. Zapata,FunctionalAnalysis,Holomorphy, andApproximation Theory Algebra 64. S. GrecoandG. Valla, Commutative 85. A. V. Fiacco,Mathematical Programming with DataPedurbations II 86. J.-B. H#iart-Urruty et al., Optimization 87. A. Figa Talamanca andM. A. Picardello, Harmonic Analysison FreeGroups 88. M. Harada,FactorCategories with Applicationsto Direct Decomposition of Modules 89. V.I. Istr~tescu,Strict Convexity andComplex Strict Convexity 90. V. Lakshmikantham, Trendsin TheoryandPracticeof NonlinearDifferential Equations andJ. B. Srivastava,AlgebraandIts Applications 91. H L. Manocha 92. D. V. Chudnovsky and G. V. Chudnovsky,Classical and QuantumModelsand Arithmetic Problems 93. J. W.Longley,LeastSquaresComputations UsingOrthogonalization Methods 94. L. P. de Alcantara,Mathematical Logic andFormalSystems Functions 95. C. E. Aull, Ringsof Continuous 96. R. Chuaqui,Analysis,Geometry, andProbability 97. L. FuchsandL. Salce,ModulesOverValuationDomains 98. P. FischerandW.R. Smith,Chaos,Fractals, andDynamics 99. W.B. PowellandC. Tsinakis,Ordered AlgebraicStructures 100. G. M. RassiasandT. M. Rassias,Differential Geometry, Calculusof Variations, andTheir Applications 101. R.-E. Hoffmann andK. H. Hofmann, Continuous Lattices andTheir Applications 102. J. H. Lightboume III andS. M. RankinIII, PhysicalMathematics andNonlinearPartial Differential Equations 103. C. A. BakerandL. M. Batten,Finite Geometrics OverCommutative Rings 104. J. W.Breweret al., Linear Systems 105. C. McCroryandT. Shifrin, Geometry andTopology 106. D. W.Kuekeet al., Mathematical Logic andTheoreticalComputer Science 107. B.-L. Lin andS. Simons,NonlinearandConvex Analysis 108. S. J. Lee, OperatorMethods for OptimalControlProblems 109. V. Lakshmikantham, NonlinearAnalysisandApplications MultigddMethods 110. S. F. McCormick, 111. M. C. Tangora,Computers in Algebra 112. D. V. Chudnovsky andG. V. Chudnovsky, SearchTheory 113. D. V. Chudnovsky andR. D. Jenks, Computer Algebra 114. M. C. Tangora,Computers in Geometry andTopology 115. P. Nelsonet al., TransportTheory,InvadantImbedding, andIntegral Equations 116. P. Cldment et al., Semigroup TheoryandApplications 117. J. Vinuesa,Orthogonal Polynomials andTheir Applications et aL, Differential Equations 118. C. M.Dafermos 119. E. O. Roxin, Modem OptimalControl 120. J. C. Diaz, Mathematics for LargeScaleComputing

FunctionalAnalysis 121. P. S. Mi/ojevi~Nonlinear 122. C. Sadosky, AnalysisandPartial Differential Equations 123. R. M. Shortt, GeneralTopology andApplications 124. R. Wong,AsymptoticandComputational Analysis andR. D. Jenks,Computers in Mathematics 125. D. V. Chudnovsky 126. W.D. Walliset al., Combinatorial Designs andApplications 127. S. E/aydi,DifferentialEquations 128. G. Chenet al., DistributedParameter ControlSystems 129. W.N. Ever#t,Inequalities 130. H. G. KaperandM. Garbey,AsymptoticAnalysisandthe Numerical Solution of Partial Differential Equations 131. O. Arinoet al., Mathematical PopulationDynamics 132. S. Coen,Geometry and Complex Variables 133. J. A. Goldstein et al., DifferentialEquations withApplications in Biology,Physics,andEngineering 134. S. J. Andima et al., GeneralTopology andApplications 135. P C/~menteta/., Semigroup TheoryandEvolutionEquations 136. K. Jarosz,FunctionSpaces eta/., p-adicFunctionalAnalysis 137. J. M. Bayod 138. G. A. Anastassiou,Approximation Theory 139. R. S. Rees,Graphs,Matrices,andDesigns 140. G. Abrams eta/., Methods in ModuleTheory 141. G. L. MullenandP. J.-S. Shiue,Finite Fields, CodingTheory,andAdvances in Communications and Computing 142. M. C. JoshiandA. V. Balakdshnan, Mathematical Theoryof Control 143. G. Komatsuand Y. Sakane,ComplexGeometry 144. I. J. Bake~man, Geometric AnalysisandNonlinear Partial Differential Equations andS. Mukai,Einstein MetricsandYang-MillsConnections 145. T. Mabuchi 146. L. FuchsandR. GSbel,AbelianGroups 147. A. D. Pollington andW.Moran,Number Theorywith an Emphasis on the MarkoffSpectrum 148. G. Doreeta/., Differential Equations in Banach Spaces TheoryandDynamicalSystems 149. T. West,Continuum 150. K. D. Bierstedtet al., Functional Analysis Algebra 151. K. G. Fischeret al., Computational 152. K. D. E/worthyet al., Differential Equations, Dynamical Systems, andControlScience RingTheory 153. P.-J. Cahen,et al., Commutative Functions 154. S. C. CooperandW.J. Thron,ContinuedFractionsandOrthogonal 155. P. Clement andG. Lumer,EvolutionEquations,ControlTheory,andBiomathematics 156. M. Gy//enberg andL. Persson,Analysis,Algebra,andComputers in Mathematical Research 157. W.O. Brayet al., Fouder Analysis 158. J. BergenandS. Montgomery, Advances in HopfAlgebras 159. A. R. Magid,Rings, Extensions,andCohomology 160. NoH. Pave/,OptimalControlof Differential Equations 161. M. Ikawa,SpectralandScatteringTheory Methods andStability Theory 162. X. Liu andD. Siege/,Comparison 163. J.-P. Zo/dsio,Boundary ControlandVariation 164. M. K’fi~eket al., Finite Element Methods 165. G. DaPratoandL. Tubaro,Controlof Partial Differential Equations with Applications 166. E. Bel/ico, ProjectiveGeometry 167. M. Costabelet al., Boundary ValueProblems andIntegral Equationsin Nonsmooth Domains 168. G. Ferreyra,G. R. Goldstein,andF. Neubrander, EvolutionEquations 169. SoHuggett,TwistorTheory 170. H. Cooket al., Continua 171. D. F. Anderson andD. E. Dobbs,Zero-Dimensional Commutative Rings 172. K. Jarosz,FunctionSpaces 173. V. Ancona et al., Complex AnalysisandGeometry andApplications 174. E. Casas,Controlof Partial Differential Equations 175. N. Kaltonet al., InteractionBetween FunctionalAnalysis,Harmonic Analysis,andProbability 176. Z. Deng et al., Differential Equations andControlTheory 177. P. Marcelliniet al. Partial DifferentialEquations andApplications Type 178. A. Kartsatos,TheoryandApplicationsof NonlinearOperatorsof AccretiveandMonotone 179. M. Maruyama, Moduliof VectorBundles 180. A. Ursini andP. AglianO,LogicandAlgebra 181. X. H. Caoet a/., Rings,Groups,andAlgebras 182. D. Arnold andR. M. Rangaswamy, Abelian GroupsandModules andA. S. Affa, Matrix-AnalyticMethods in StochasticModels 183. S. R. Chakravarthy

184. J. E. Andersen et al., Geometry andPhysics et al., Commutative RingTheory 185. P.-J. Cahen 186. J. A. Goldsteinet al., StochasticProcesses andFunctionalAnalysis 187. A. Sorbi, Complexity,Logic, andRecursion Theory 188. G. DaPrato andJ.-P. Zol~sio, Partial Differential EquationMethods in Control and Shape Analysis Factorizationin Integral Domains 189. D. D. Anderson, 190. N. L. Johnson, MostlyFinite Geometries Methods of Sturm--Liouville 191. D. HintonandP. W.Schaefer,SpectralTheoryandComputational Problems 192. W.H Schikhofet al., p-adicFunctionalAnalysis 193. S. Sert5z, AlgebraicGeometry 194. G. Caristi andE. Mitidieri, Reaction DiffusionSystems 195. A. V. Fiacco,Mathematical Programming with DataPerturbations 196. M. K’fiSek et al., Finite ElementMethods: Superconvergence, Post-Processing, andA Postedori Estimates 197. S. Caenepeel andA. Verschoren, Rings, HopfAlgebras,andBrauerGroups 198. V. Drensky et al., Methods in RingTheory 199. W.B. JonesandA. Sri Ranga,OrthogonalFunctions,Moment Theory,andContinued Fractions AlgebraicGeometry 200. P. E. Newstead, 201. D. DikranjanandL. Salce, AbelianGroups,Module Theory,andTopology in Computational Mathematics 202. Z. Chenet al., Advances 203. X. CaicedoandC. H. Montenegro, Models,Algebras,andProofs 204. C. Y. Ytldlrlm andS. A. Stepanov, Number TheoryandIts Applications 205. D. E. Dobbs et al., Advances in Commutative Ring Theory 206. F. VanOystaeyen,Commutative Algebraand Algebraic Geometry 207. J. Kakolet al., p-adicFunctional Analysis 208. M. Boulagouaz andJ.-P. "l’ignol, AlgebraandNumber Theory 209. S. Caenepee/ andF. VanOystaeyen,Hopf Algebrasand Quantum Groups 210. F. VanOystaeyen and M. Saorin, Interactions BetweenRing TheoryandRepresentationsof Algebras AlgebraandIts Applications 211. R. Costaeta/., Nonassociative 212. T.-X. He, Wavelet AnalysisandMultiresolutionMethods FunctionSpaces:TheFifth Conference 213. H. HudzikandL. Skt-zypczak, 214. J. Kaflwara et al., Finite or Infinite Dimensional Complex Analysis 215. G. LumerandL. Weis,EvolutionEquationsandTheir Applicationsin PhysicalandLife Sciences Additional Volumesin Preparation

evolution equations and their applications in physical and life sciences proceedings (Karlsruhe),

edited

of the Bad Herrenalb Germany, conference

by

G0nter Lumer University of Mons-Hainaut Mons, Belgium and International SoivayInstitutes for PhysicsandChemistry Brussels, Belgium l_utz Weis Universityof Karlsruhe Karlsruhe, Germany

MARCEL

MARCEL DEKKER,INC. DEKKER

NEWYORK. BASEL

ISBN: 0-8247-9010-3 This bookis printed on acid-free paper. Headquarters MarcelDekker, Inc. 270 Madison Avenue, NewYork, NY10016 tel: 212-696-9000;fax: 212-685-4540 Eastern HemisphereDistribution Marcel Dekker AG Hutgasse4, Posffach 812, CH-4001Basel, Switzerland tel: 41-61-261-8482;fax: 41-61-261-8896 World Wide Web http://www.dekker.com The publisher offers discounts on this book whenordered in bulk quantities. For moreinformation, write to Special Sales/Professional Marketingat the headquartersaddress above.

Copyright © 2001 by Marcel Dekker,Inc. All Rights Reserved. Neither this booknor any part maybe reproduced or transmitted in any form or by any means, dectronic or mechanical, including photocopying,microfilming, and recording, or bYany inform:ation storage and retrieval system, withoutpermissionin writing fromthe publisher. Currentprinting (last digit): 1098765432 I PRINTED IN THE UNITED STATES OF AMERICA

Preface

The sixth International Conferenceon Evolution Equationsand Their Applications in Physical and Life Sciences was held in BadHerrenalb (Karlsruhe), Germany.As the aim of these biannual conferences, it brought together manyof the world’s leading researchers in the mainareas of evolution equations and related fields of application. A particular effort was madeto include in this meeting, along with established scientists, manypromisingyoungresearchers. Manyof the recent developmentsin evolution equations and evolution processes presented in this volumeare related to, or deal directly with, matters of physics, engineering, and life sciences. Among the topics treated here are newdevelopments on: asymptotics in linear and nonlinear systems, maximalregularity for parabolic equations, Gaussian estimates, pseudo-differential operators and boundary value problems, singular problemsinvolving generalized functions, fractional evolution eqtiations, dispersive waves, fully nonlinear problems, chemical reactor theory, disease transport models, superluminal effects in classical physics, blow-upand singular interaction, Feynman integrals for Liouville evolutions, controllability, stochastic analysis of dissipative gradient equations, and vector-valued OmsteinUhlenbeckprocesses. Indeed, four mainorientations clearly emergefromthe lectures and contributions: linear PDEsand semigroups;nonlinear equations; evolution problemsfrom physics, engineering, and mathematicalbiology; stochastic evolutionary processes. Wehave structured the bookby groupingthe contributions accordingto those orientations. This volume, while based on the material presented by the conference participants, is, to be precise, not a mere"proceedings."Abouthalf the contributions contain material further developedafter the conference(with newadditional results) and there are even someimportant contributions from people whowere invited but could not be present at the conference. Theorganizersthank the followinginstitutions for providingthe financial support for the conference: -

DFG(Deutsche Forschungsgemeinschaft) European Union, TMR-Program Ministeriumftir Wissenschaft, Forschungund Kunst Baden-Wtirttemberg Universit~it Karlsruhe, Hochschulvereinigung Karlsruhe 111

iv

Preface

We thank our colleagues and collaborators P. Dufour, A. Fr0hlich, V. Goersmeyer,P. Kunstmann,Ch. Schmoeger,and Z. ~trkalj for assistance and advice during the organization of the meetingand the preparation of the proceedings. In particular, we thank A. Fr/Shlich and M. Schrodt for handling a large part of the technical production. Finally, we thank the contributors, the referees, and Marcel Dekker, Inc., especially Ms.MariaAllegra and Ms.J. Paizzi, for their cooperationin makingthis volumepossible. Giinter Lumer Lutz Weis

Contents

Preface Contributors

oo.

Ill

Semigroupsand Partial Differential Equations Different DomainsInduce Different Heat Semigroupson Co(f1) W. Arendt Gaussian Estimates for SecondOrder Elliptic DivergenceOperators on Lipschitz and CI Domains P. Auscher and Ph. Tchamitchian

o

15

ApproximateSolutions to the Abstract CauchyProblem Boris B~iumer

33

Smart Structures and Super Stability A. V. Balakrishnan

43

Onthe Structure of the Critical Spectrumof Strongly Continuous Semigroups Mark Blake, Simon Brendle, and Rainer Nagel

55

An Operator-Valued Transference Principle and MaximalRegularity on Vector-Valued Lp-Spaces Ph. Clement and Jan Priiss

67

On AnomalousAsymptotics of Heat Kernels A. F. M. ter Elst and Derek W. Robinson

89

On SomeClasses of Differential Operators Generating Analytic Semigroups Angelo Favini, Giskle Ruiz Goldstein, JeromeA. Goldstein, and Silvia Romanelli

105

vl 9.

Contents A Characterization of the GrowthBoundof a Semigroupvia Fourier Multipliers Matthias Hieber

121

10.

Laplace Transform Theory for Logarithmic Regions Peer Christian Kunstmann

125

11.

Exact BoundaryControllability of Maxwell’sEquations in Heterogeneous Media Serge Nicaise

139

A Sufficient Condition for Exponential Dichotomyof Parabolic Evolution Equations Roland Schnaubelt

149

13.

Edge-Degenerate Boundary Value Problems on Cones E. Schrohe and B.-W. Schulze

159

14.

A Theoremon Products of Non-Commuting Sectorial Operators 7~eljko~trkaij

175

15.

The Spectral Radius, Hyperbolic Operators and Lyapunov’sTheorem Vu Quoc Phong

187

16.

A NewApproach to Maximal LfRegularity Lutz Weis

195

12.

Nonlinear Evolution Equations 17.

The Instantaneous Limit of a Reaction-Diffusion System Dieter Bothe

215

18.

A Semigroup Approach to Dispersive Waves Radu C. Cascaval and Jerome A. Goldstein

225

19.

Regularity Properties of Solutions of Fractional Evolution Equations Ph. Clement, G. Gripenberg, and S.-O. Londen

235

20.

Infinite HorizonRiccati Operators in Nonreflexive Spaces WolfgangDesch, Eva FaJangovd, Jaroslav Milota, and Wilhelm Schappacher

247

21.

A Hyperbolic Variant of Simon’s Convergence Theorem A. Haraux

255

22.

Solution of a Quasilinear Parabolic-Elliptic BoundaryValue Problem V. Pluschke

265

Contents

vii

Physical and Life Sciences 277

23.

Singular Cluster Interactions in Few-BodyProblems S. Albeverio and P. Kurasov

24.

Feynmanand Wiener Path Integrals Representations of the Liouville Evolution L Antoniou and O. G. Smolyanov

293

Spectral Characterization of MixingEvolutions in Classical Statistical Physics 1. Antoniou and Z. Suchanecki

301

On Stochastic Schr6dinger Equation as a Dirac Boundary-ValueProblem, and an Inductive Stochastic Limit V. P. Belavkin

311

A Maximum Principle for Fully Nonlinear Parabolic Equations with Time Degeneracy Joachim von Below

329

25.

26.

27.

28.

Dirac Algebra and Foldy-WouthuysenTransform H. O. Cordes

335

29.

OnPerturbations for the ContinuousSpectra of SemigroupGenerators Michael Demuth

347

30.

Mathematical Study of a Coupled System Arising in Magnetohydrodynamics J.-F. Gerbeauand C. Le Bris

31.

A Disease Transport Model K. P. Hadeler, R. lllner, and P. Van DenDriessche

32.

Blow-Upand Hoveringin Parabolic Systems with Singular Interactions: Can We"See" a Hyperfunction? Giinter Lumer

33. Some Asymptotic Problems in Fluid Mechanics ,. N. Masmoudi

355 369

387 395

34.

Lihaits to Causality and Delocalization in Classical Field Theory T. Petrosky and I. Prigogine

405

35.

Remarksto the Blow-upRate of a Degenerate Parabolic Equation Burkhard J. Schmitt and Michael Wiegner

413

Contents

viii ~.

Stability in ChemicalReactor Theory K. Taira and K. Umezu

421

Stochastic Evolution Equations 37.

Banach Space Valued Ornstein-Uhlenbeck Processes Indexed by the Circle Zdzistaw Brze~niak and Jan van Neerven

435

38.

SomeProperties of the KMS-Function Jan A. van Casteren

453

39.

A Generalization of the Bismut-Elworthy Formula Sandra Cerrai

473

40.

Dirichlet Operators for Dissipative Gradient Systems G. Da Prato

483

41.

Generators of Feller Semigroupsas Generators of LP-sub-Markovian Semigroups Niels Jacob

42.

A Note on Stochastic WaveEquations Anna Karczewska and Jerzy Zabczyk

493 501

Contributors

S. Albeverio Institute

for Applied Mathematics, BonnUniversity, Bonn, Germany

I. Antoniou International SolvayInstitutes for Physics and Chemistry,Brussels, Belgium W. Arendt

Department of Mathematics, University of Ulm, Ulm, Germany

P. Auscher UFRSciences-Math6matiques, University of Amiens, Amiens, France A. V. Balakrishnan Electrical Engineering Department, University of California at Los Angeles, Los Angeles, California Boris Biiumer Departmentof Mathematics, University of Nevada, Reno, Reno, Nevada V. P. Belavkin School of Mathematics, NottinghamUniversity, Nottingham, England Mark Blake

St. John’s College, Oxford University, Oxford, England

Dieter Bothe Departmentof Mathematics, University of Paderborn, Paderborn, Germany SimonBrendle Graduate School of Science and Technology, Kobe University, Kobe, Japan Zdzislaw Brzetniak Department of Mathematics, The University of Hull, Hull, England Radu C. Cascaval Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee Sandra Cerrai University of Florence-Dimadefas, Florence, Italy Ph. Cl6mentFaculty of Technical Mathematics and Informatics, Delft University of Technology, Delft, The Netherlands H. O. Cordes Departmentof Mathematics, University of California at Berkeley, Berkeley, California ix

x

Contributors

Giuseppe Da Prato Scuola NormaleSuperiore di Pisa, Pisa, Italy M. DemuthInstitute for Mathematics, Technical University of Clausthal, ClausthalZellerfeld, Germany WolfgangDesch Institute

for Mathematics, University of Graz, Graz, Austria

A. F. M. ter Elst Department of Mathematics and ComputerScience, Eindhoven University of Technology, Eindhoven, The Netherlands Eva Fa’sangov~iDepartment of Mathematical Analysis, Charles University, Prague, Czech Republic Angelo Favini Department of Mathematics, University of Bologna, Bologna, Italy J.-F. Gerbeau CERMICS, Ecole Nationale des Ponts et Chauss6es, Marne-LaVall6e, France GisHe Ruiz Goldstein CERIand Department of Mathematical Sciences, Memphis State University, Memphis,Tennessee Jerome A. Goldstein Department of Mathematics, MemphisState University, Memphis, Tennessee G. Gripenberg Institute Helsinki, Finland

of Mathematics, Helsinki University of Technology,

K.P. Hadeler MathematicsInstitute,

University of Tiibingen, Tiibingen, Germany’

Alain Haraux CNRSLaboratoire d’Analyse Num6rique,Universit6 P. et M. Curie’., Paris, France Matthias Hieber Department of Mathematics, Technical University of Darmstadt, Darmstadt, Germany R. Illner Departmentof Mathematicsand Statistics, British Columbia, Canada

University of Victoria, Victoria,

Niels Jacob Institute for Theoretical Informatics and Mathematics,University of Bundeswehr, Neubiberg, Germany AnnaKarczewskaInstitute Lublin, Poland

of Mathematics, Mafia Curie-Sktodowska University,

Peer Christian KunstmannMathematics Institute Karlsruhe, Germany

I, University of Karisruhe,

xi

Contributors

P. Kurasov Department of Mathematics, Stockholm University, Stockholm, Sweden Claude Le Bris Vall6e, France

CERMICS, Ecole Nationale des Punts et Chauss6es, Marne-La-

S.-O. Londen Institute of Mathematics,Helsinki University of Technology,Helsinki, Finland GfinterLumer Institute of Mathematicsand Informatics, University of MonsHainaut, Mons,Belgium,and International Solvay Institutes for Physics and Chemistry, Brussels, Belgium Nader MasmoudiCEREMADE (UMR7534), University of Paris-Dauphine, France

Paris,

Jaroslav Milota Department of Mathematical Analysis, Charles University, Prague, Czech Republic Rainer Nagel Mathematics Institute,

University of Tiibingen, T0bingen, Germany

Serge Nieaise University of Valenciennes and of Hainaut Cambr~sis, Valenciennes, France T. Petrosky International Solvay Institutes for Physics and Chemistry, Brussels, Belgium V. Pluschke Department of Mathematics and Informatics, University of HalleWittenberg, Halle, Germany I. Prigogine International Solvay Institutes for Physics and Chemistry, Brussels, Belgium Jan Priiss Departmentof Mathematicsand Informatics, University of HalleWittenberg, Halle, Germany DerekW. RobinsonCenter for Mathematics and Its Applications, School of MathematicalSciences, Australian National University, Canberra, Australia Silvia Romanelll Departmentof Mathematics, University of Bari, Bari, Italy WilhelmSchappacher Institute

for Mathematics, University of Graz, Graz, Austria

BurkhardJ. Schmitt Lehrstuhl for Mathematik I, RWTH Aachen, Aachen, Germany

xii

Contributors

Roland Schnauhelt MathematicsInstitute, Germany E. Schrohe Institute

University of T~ibingen, Ti~bingen,

for Mathematics, University of Potsdam, Potsdam, Germany

B.-W. Schulze Institute

for Mathematics, University of Potsdam, Potsdam, Germany

O. G. Smolyanov Department of Mathematics and Mechanics, MoscowState University, Moscow,Russia ~eljko ~trkalj MathematicsInstitute I, University of Karlsruhe, Karlsruhe, Germany Z. SuchaneckiInternational Solvay Institutes for Physics and Chemistry, Brussels, Belgium K. Ta|ra Institute of Mathematics, University of Tsukuba, Tsukuba, Japan Ph. Teham|tchian Faculty of Science and Technology of Saint-J6r6me, University of Aix-MarseilleIII, Marseille, France K. UmeznMaebashi Institute

of Technology, Maebashi, Japan

Jan A. Van Casteren Department of Mathematics and ComputerScience, University of Antwerp (UIA), Antwerp, Belgium P. Van den Dr|essche Department of Mathematics and Statistics, Victoria, Victoria, British Columbia,Canada

University of

Jan van Ncerven Department of Mathematics, Technical University of Delft, Delft, The Netherlands Joachimyon Below Department of Mathematics, Universit6 du Littoral d’Opale, Calais, France Qno¢ Phong Vu Department of Mathematics, Ohio University, Lutz Weis MathematicsInstitute

Athens, Ohio

I, University of Karlsruhe, Karlsruhe, Germany

¯ Michael Wiegner Lehrstuhl I fiir Jerzy Zabczyk Institute Poland

Cfte

Mathematik, RWTH Aachen, Aachen, Germany

of Mathematics, Polish Academyof Sciences, Warsaw,

Different Different

Domains Induce Heat Semigroups on C0(

W. ARENDTUniversit£t Ulm, Angewandte Analysis, [email protected]

D-89069 Ulrn, Germany,

ABSTRACT Let f~l, 122 C ~g be two open, connected sets which are regular in the sense of Wiener. Denote by Ao~ the Laplacian on Co(~j) , j 1, 2. Assume that there exists a non-zero linear mappingU : Co(121) --4 Co(122) such (a)

[U f[ = Ulf[ (f e Co(Q~))

(b)

Uet~X~’ = et~’~:U (t >_ O)

Then it is shown that ~t and f12 are congruent. This result complements [2] where the Laplacian on Lv was considered and U was supposed to be bijective.

.0

INTRODUCTION

Let Ft C ~N be an open set. By A~ we denote the Dirichlet Laplacian on L2 (12) This is a self-adjoint operator generating a contraction semigroup (et’~)t>o If f~ is bounded, then A~ has compact resolvent and hence L2(~) has an orthogonal basis {e~ : n e ~} consisting of eigenvectors of A~ ; i.e.

0
lim A~=~. Now let

fl~

and ~ be two bounded open

sets. We say that ~ and fl~ are isospectral if the operators A~ and A~ have the same sequence of eigenv~lues. It w~s Marc Kac, who asked the following question in his famous paper [12] of 1962: Question (Marc Kac): Let fl~, ~2 be two bounded open, connected sets in which are of cl~s C~ . Assume that ~ and ~2 are isospectral. Does it follow that ~ and ~2 are congruhnt?

2

Arendt

If we have in mind that the sequence of eigenvalues is just the same as the proper fl’equencies of the body, one can indeed reformulate this question by asking "Can you hear the shape of a drum?", which is precisely the title of Kac’s paper. It was knownalready to Kac that the answer is negative if we consider the Laplace Beltrami operator on a compact manifold instead of domains in ~v . In the Euclidean case, a counterexample to Kac’s question was given by Urala~wa [21] if dimension N > 4. For N _> 2 finally, a counterexample was given by Gordon, Webband Wolpert [10] in 1992. Today, very elementary descriptions of examples are given (see Berard [4], Chap~nan [6]). For example, in dimension 2, seven triangles may be put together in the two different ways shown below to produce two polygones which are isospectral but not congruent.

However, it semns that so far no Euclidean counterexample with smooth boundary is known.Thus Kac’s question in the precise form he asked it, is still open. There is another way to look at isospectral sets. Let gt~, ~2 C ~r/g be open and bounded. Then ~1 and ~ are isospectral if and only if there exists a unitary operator U : Le(~,) -~ L~(~e) such eta~=U = Uet’~=~ (t _> O) ~ In fact, we may define U by mapping the orthonormal basis diagonalizing onto the one which diagonalizes A2~=. Let f ~ L~(FI~) . Then the function u(t, x) = (et’x= f)(x) is the unique of the heat equation:

(0.1) A~

Thus (0.1) is equivalent to saying that U maps solutions of (0.2) to solutions. Nowobserve that u(t,x) >_ a. e. if f _ > 0. Moreover, if we thi nk of heat conduction or diffusion as a physical model, then only positive solutions have a physical meaning. So it is natural to consider mappings U which map positive solutions to positive solutions.

Different DomainsInduceDifferent HeatSemigroups By an order isomorphism we unterstand a bijective lineal mappingU : L2 L-" (f~2) satisfying Uf >_0 if and only if f >_ 0

3 (~’~1)

--~ (0.3)

for all f E L~’(f~l) If in (0.1) we replace the unitary operator U by an order isomorphism U , then the following result holds [2, Corollary 3.17]. THEOREM 0.1 Let f~1,122 be two open connected sets which are regular in capacity. If there exists an order isomorphism U : L~(ftl) -~ L2(~2) such that (0.1) holds, then ~ and f~2 are congruent. Having in mind the previous interpretation, we may reformulate Theorem 0.1 by saying that diffusion determines the domain. Werefer to [2] for the proof of Theorem0.1 and to Section 2 - 4 for further explanations, in particular the notion of regularity in capacity. The aim of the present paper is to extend Theorem0.1 in two ways. First of all we will prove that it holds even if we do no longer assume that U is onto. Secondly, we will establish an analogous result where L2 is replaced by a space of continuous function. For our arguments it will be essential that the semigroupgenerated by the Dirichlet Laplacian on Co (f~) is irreducible. Wewill prove this in Section 1 by using that. the semigroupactually consists of classical solutions; i.e. that (0.2) holds. Using classical maximum principle we then obtain irreduciblity.

1

CLASSICAL SOLUTIONS AND STRICT POSITIVITY

OF THE HEAT EQUATION

In this section we show that the heat equation always has classical solution due to interiour elliptic regularity. This is not new (cf. [5, IX 6]), but we use this prove irreducibility with help of the classical strict maximum principle for parabolic equations. This is an alternative much more elementary way in comparison with the use of lower Gaussian bounds (see Davies [7, Theorem3.3.5]). In addition, obtain not only irreducibility in L~ but also in Co(m) which is stronger and will be used in Section 2. Let Ft c/R~ be an open set. Weconsider realizations of the Laplacian on L2 (~) which generate differentiable positive semigroups. Our aim is to show that such a semigroup is automatically strictly positive. The concrete example we will consider later is the Laplacian with Dirichlet boundary conditions. Let T = (T(t))t>_o be a semigroup on a Banach space X with generator A . Then for each k E ~W, the space D(A~) is a Banach space for the norm

tlxlIA~ -- Ilxll +IIAxll+... + IIA~xll ¯ The semigroup is called differentiable In that case one has

(1.1)

if T(t)x ~ D(A) for all t > 0 , x ~ X .

T(.)x ~ ck((o, oc) ; D(Am))

(1.2)

4

Arendt

for all kEtV, mEgg, xeX (see e.g. Pazy[16]). An operator A on L2(f~) is called a realization

of the Laplacian,

Af : A f (in T~(~)’) for all f ¯ D(A) . We now show that a differentiable semigroup T whose generator A is a realization of the Laplacian in L2(f~) governs a classical solution of the heat equation. More precisely, we have the following: THEOREM 1.1 Let T= (T(t))t>_o be a differentiable semigroup L2(f~) whose generator is a realization of the Laplacian. Given f ~ L~(ft) let u(t,x)

= (T(t)f)(x)

(t > O,

Then u ¯ C~((0, c~) x ft) and

ut(t,’~) = ~Xu(t,z)(t > o, x ¯

(1.3)

Weneed the following two results on regularity which are easily proved with help of the Fourier transform (see e.g., [18, Theorem8.12.]). By Hk(f~) we denote the k-th Sobolev Space; i.e. H~(fl) = {f ¯ L2(ft): where a = (al,...

Daf ¯ L2(f~) if I~1

k),

N ,aN) e /N~v denotes a multi-index and lal = ~] o~j its order, j=l

0 ’~’ , Dy= 0~ D~=D~t ..DTv ¯ , local Sobolev spaces are defined lal < k} . Here xve consider, in and D~ as an operator on /9(f2)’ for consistency¯

J=I,’"N, ~W0=Zrv’k){0}={0,1,2,...}. The by H~oc(f~) = {f ¯ L~o~(f~) : D~f e n~o~(E!) if’ the usual way, L~oc(f~) as a subspace of ’D(f~)’ . Weset H°(ft) = L2(f~) H~oc(f~ ) ) = L~oc(f~

LEMMA 1.2 Let k,m ¯ 1No , k > ~y . Then H~oc (l))

C Cm(~2) ¯ In particular,

Here we let C(f~) = C°(gt) be the space of all continuous fl~nctions on ft values in C’ and Ck(f~) is the space of all functions which arc k-times continuously differentiable. LEMMA 1.3 Let u,f ¯ L~o~(fl ) such that Au = f in 79(fl)’. H?o~(a) ¯Moreover, if f e Htoc(f~) , then u e Hk+2/f~) toc

Then u

It is not difficult to see that for k ~ tvV0, H~+c~(f~) = {f ¯ Ht~o~(a): Djf e Htkoc(a) , j = 1,...

,N} .

(1.4)

Proof of Theorem i.I. Let w C f~ be open, bounded such that ~ C f~. Since A is a realization of the Laplacian, it follows from Lemma1.2 that D(A

Different DomainsInduceDifferent Heat Semigroups

5

2k C C2k-P(f~). for all kE~W. Let p> N~ : Then for 2k>p, g~oc(f~) Let wC~ be open and bounded such that c0 C fl. For g : f/-~ ~7, we denote by j(g) the restriction of g to c0. It follows from the closed graph theorem that j defines a bounded operator from D(Ak) into c2k-P(Co) Hence by (1. 2), for ] E L2(f we have joT(.)f ~ cm((o, oc) , c2k-P(Co)) for all k, mE /5/, 2k>p. This implies that u(., .) ~ C’n((0, ec) x w) .

Werecall the classical strict p. 1081].

parabolic maximum principle, see e.g. [8, V § 5, 3.4,

PROPOSITION 1.4 Let T > 0 ¯ Let f~ C ~=gN be open and connected.

Let

u e C2((o,r)x ft) n C([O,r]x fi) suchthat ut(t,x) = Au(t,x) t e (o,r) , Assume that there exist xo ~ I2 , to ~ (O,T] such that u(to,xo) = max u(t, x) ~E(O,’r| Then u is constant. From this we can now deduce that every semigroup generated by a realization the Laplacian is automatically strictly positive wheneverit is positive.

of

THEOREM 1.5 Let 12 C 1RN be open and connected. Assume that T = (T(t))t>_o is a differentiable positive semigroup whose generator A is a realization o] the naplacian. Then T(t)f ~ C~(f~) for all t > 0 , L2(f ~) ; an d if 0 <_ f e Le(fl) , f ~ O, then (T(t)f)(x)

>0

(1.5)

for all x E f~ , t>0. Proof. It follows from Theorem 1.1 that the function u given by u(t,x) (T(t)f)(x) is in C~((0, ec) x f~) and satisfies the heat equation (1.3). that f_>0. Then u(t,x) >_0 for all t>0, x~f~ by hypothesis. Assume that there exists to> 0 , Xo ~ f~ such that u(to,xo) = 0. Let w be open bounded, connected such that D C f~ ¯ The strict maximumprinciple applied to -u shows that u(t, x) = for al l t ~ (0, to ] , x E w.Now a si mple conn ectedness argu ment showsthat u(t,x) = for al l t e (0, to ] , x e f~¯ Sin ce f =li t. r~oU(t,.) in L2(fl) it follows that f = 0. [] REMARK 1.6 (Irreducibility) A positive semigroup T on a Banach lattice is called irreducible if for all f ~ E+ , f ¢ 0 and all qo ~ E~_ , ~o ¢ 0 there exists t > 0 such that (T(t)f , qo) > 0. If in addition T is holomorphic then it is automatically true that (T(t)f , qo) > 0 for all t > 0 (see [15, C-III Theorem 3.2]. Theorem1.5 implies in particular that the semigroup T considcred here is irreducible.

6

Arendt

REMARK 1.7 Of course there do exist realizations of the Laplacian which do not generate a positive semigroup. For example, let A on L’)(0, 1) be given D(A) = {f g2(0,1): f( 0) = -f (1) , f’ (0) = -] Af = f". Then A generates a semigroupwhich is not positive (cf. [1, 3.4], [15, p. 255]). 2INTERTWINING

2

LATTICE

Let f~ C K/N be an open set. L2(f~) ;i.e.

HOMOMORPHISMS ON L

By AT we denote the Dirichlet

Laplacian

on

D(A~) = {f ¯ H01(~): Af ¯ L~(gt)} f = Af. Then AT is a form negative operator which generates a positive contraction semigroup T = (eta~)~>o on Le(f~) By cap(F) in f(HuH~ : u >_1 o n a n ei ghborhood of F} we denote the capacity of a subset F of ~g . Then cap defines an outer measure on ]Rg . An open subset 12 of ~RN is called regular in capacity if cap(B(z, r) \ f~) for all z 6 cOl2 , r > 0. Note that 12 is regular in capacity whenever it is topologically regular, i.e. l~ = f~ ¯ But also the set f~ ---- B(0, 1) \ {(Xl,0): > 0) C K~ is regular in capacity. The aim of this section is to prove the following result which extends Theorem 0.1 mentioned in the introduction. THEOREM 2.1 Let ~1,~’~2 C 1~ N be open, connected and regular in capacity. Assume that there exists a linear operator U : L2(ft~) -> L2(~2) such that U ~ 0 and

(a) lUll Ulfl (f

¯ L 2( n,)) ;

(b) Uet"x~ ’ = e~a~=U (t >_ O) Then f~ is congruent to f~ . Moreprecisely, there exist an isometry "r : ff~.N __~ ~N and a constant c > 0 such that "c(~) = f~t and (U f)(y)

: CI(T(y)) (y

(2.1)

for all f ¯ L:(f~) Here a mapping ~- : ~ ~ ~N is called an isometry if there exist an orthogonal matrix B and a vector b ¯ ~N such that T(y) By+ bfo r all y ¯/ R~: . Tw o open sets 12~ and f~_ are called congruent if there exists an isometry such that ~-(~2~) = f~ . In that case it is easy to see that Uf= fo’r

Different DomainsInduceDifferent Heat Semigroups

7

defines a unitary operator satisfying (a) and (b). Note that the regularity assumption in Theorem 2.1 cannot be omitted. This is made clear in the following remark. R.EMARK2.2 Let ft C ~ be an open set. Then there exists a unique open set fi which is regular in capacity such that (~ D ft and cap(~ \ f~) = 0. This implies in particular that L2(ft) = L2(~) and T =AT(se e [2] for the proofs). Now,if ft is not regular in capacity, then we have gt ¢ f~ , and (a) and {b) satisfied for U the identity operator. Proof of Theorem 2.1. It follows as in [2, (2.16)I that (U f)(y) = { h(y)f(r(y))o for all f e D(fl~), where fl~ C fl~ is component of f12 and h : fl~ ~ (0,~) is Moreover, as in [2, (3.10)] one sees that H~(~) into H~(fl~). Now let 0 < f C~(~) and 9(y)>O for all y~fl~.

that UD(Ftl) C C~(a2)

y eefl~fl’2 \ a (2.2) i open, r : ~ ~ fl~ isometric oneach constant on each component of fl~ U induces a continuous operator from 6 D(fl~) et ~Uf. Then g

Ontheother hand 9 = Uet~ f in H~(fl~). Let k = et~ f. Then H~(fl~). Let kn ~ D(fl~) such that k~ ~ k in H~(fl~). Uk,~ ~ Uk in H~(fl~) and so q.e. after extraction of a suitable subsequence. But Uk = 9 . Hence Uk~9 q.e. Since Uk(y)=O for yeQ~fl~,itfollowsthat 9(y)=O q.e. on f~z ~ fl~ ¯ Since 9 is strictly positive it follows that cap(~ ~ ~) = 0. By [2, Proposition 3.10], it follows that ~ is connected. Thus ~ is an isometry and h eqnal to a constant e > 0. It follows fi’om (2.1) and density that (UI)(~) ~/(~(~))

(2.3) a.e. on ~ for all f e L~(~I) . Note that ~3 := T(~) is an open subset which is regular in capacity. If suffices to showthat cap(~ ~ fiz) = 0 in order to deduce that r(~) = fl~ . Consider the mapping V : L~(~z) ~ L~(~z) given Vg=c-~goT -~. Let W:L2(fi~)~L2(~z) begivenby W=VoU.

Wet~’ = e~7~W (t ~ O) and (W/)(~) for all xe~ and all f~(~). But Wf~H~(fl3). Hence f=0 on q.e. for all f ~ ~(~t) . This implies that cap(~ ~ ~3) = 0, by [2, (3.7)].

3

THE CONGRUENCE PROBLEM WITH RESPECT

Let fl C NNbe an open non-empty set.

TO Co([l)

Weconsider the Dirichlet

Laplacian A0n

on

Co(fl)

{f e C(f~) : for all e > 0 there exists K C f~ compact such that f(x) = 0 whenever x E f~ \ K}

Arendt

8 i.e. the operator Ao~is given by D(Ao~ ) --- {f e C0(12) : Afe Co(f~)} , A0~f =

Here Af is a distribution. Since C0(12) C L.~oc(f~) C 79(12)’ , the definition sense. Note that D(Ao~) ¢: C2(12) since the Laplacian does not satisfy maximal regularity on spaces of continuous functions. It has been investigated in [3] under which conditions Ao~ is generator of a Co -semigroup. The result uses classical notions of Poteutial Theory. DEFINITION3.1 a) Let z E 012 . A barrier is a function w ~ C(12 f’l B) such that Aw <_ 0 in 73(12nB)’ , w(z) = O, w(x) > 0 for x ~ (O7t B)~,{z} where B = B(z,r) is a ball centered at z b) 12 ist regular (in the sense of Wiener) if at each point z ~ 012 there ezists barrier. Every open subset of ~ is regular. In higher dimension, if the boundary of f~ is locally Lipschitz, then gt is regular. But, for example, if N _> 2, then for every x ~ 12, 12 \ {x} is not regular. A bounded open set 12 is regular if and only if the Dirichlet problem is well-posed; i.e. for all ~ e 012 there exists u ~ C((~) such that .ulo ~ = qo and AU--- 0 in/9(12)’ Nowwe describe when Ao~ generates a semigroup (by which we always mean Co-semigroup). The following characterization is proved in [3, Section 3]. THEOREM 3.2

Let ft C ~;~N be open. The following conditions are equivalent.

(i) 12 is regular (in the sense of Wiener). (ii)

~(A~o) ~0

(iii)

A~o generates a holomorphic semigroup. is positive and contracIn that case, the semigroup T(t) = ~’xno generated b y An o rive. Moreover, T is consistent with the semigroup (e~)~o on L~(~); i.e. for

e~I = e~f (~ > 0) It follows from Theorem1.5 that the semigroup (etZX~)t_>ois strictly THEOREM 3.3 Let 12 C O < f E Co(f~) . Then

~N

positiw~:

be open and regular (in the sense of Wiener), Let

(e"Xo"f)(x) for all x e 12, t > 0. Here we use the notation f>_O :¢:~ f(x)>_O f>0 :¢:~ f_>0

forall and

x~12; f~0.

(3.2)

Different DomainsInduceDifferent Heat Semigroups

9

Next we consider two open subsets f~l,~t2 of ~N. A linear C0(~) -~ C0(ft..)) is called a lattice homomorphism

operator

IUfl = UIfI for all f e C0(fl)

U: (3.3)

where Ifl(x) = tf(x)l (x E fll) If U is an order isomorphism (i.e. U is bijective, U >_ 0 and -1 _> 0 ) , then U is clearly a lattice homomorphism(see [19] for more details). Note that every lattice homomorphismU is disjointness preserving, i.e. f . g = 0 implies (U f). (Ug)

(3.4)

for all f,g ~ Co(~1) In fact, if f. g = 0, then inf{[fl ,[gl} = 0. This implies inf{U[fl,Ulgl} Hence IUf] " lUll: (ulfl), (Ulgl) O.

= O..

Nowwe prove the first result on congruence. The space C0(12) is easier to handle than LV-spaces since point evaluations are continuous. PROPOSITION 3.4 Let fll,f~2 C ~.N be open and connected, Co(fl2) be a bounded operator such that

Let U : Co(fh)

(a) f .g = 0 implies (U:). (Ug) = 0 for all f,g e 79(fh) (b) for all y ~ ~ there exists (c)

~UT=U~f for

Then there exist T(~2)

:

~1

such

all

f ~ Co(~) such that (Uf)(y)

f~V(~i)

a constant

c ~ ~{0} and an isometr~ r : ~ ~ ~ satisfying

that

= for all f e Co(~2) . In particular, For the proof we need classical be used in the following form.

e ~ and ~2 are congruent.

regularity properties of the Laplacian. They will

LEMMA3.5 Let ~2 C ~N be open, k e ~V ~ {O} , g ~ C(f~) . If Ag ~ ~(f~) , then g ~ CTM(~) . Proof. Werecall that for a distribution u 6 7)(fl)’ , if Au ~ LP(fl) for p > N, then u e C~(fl) (see e.g. [8, Chapter II, § 3, Proposition 6]). the assertion of the lemma holds for k = 0. Assumethat it holds for some k ~ zNt_){0}. Assume that Age k+~(12). Then g 6 C~(fl) (b y th e ca se k = 0) and ADjg = DjAg ~ C~(fl) Hence Djg ~ C~+x(~) bythe indu ctive hypo thesis k+2(l~). (j=I~...,N). We have shown that g~C [] Proof of Proposition 3.4. Then

1. WeshowthatU’D(~’~I) U/)(fh) C Ck(f~2)

C CC~(~’~2)

Let k e ~WU{O} . (3.5)

I0

Arendt

for k=0. Assume that (3.5) holds for some k. Let f¯:D(~l). Then AUf= UAf ¯ Ck(~) Hence Uf¯ C k+l(i)2) by the lemma. 2. Let y ¯ gt2 . Then ~(f) (Uf)(y) defines a functional ~ ¯ Co(f~t)’ \ {0} . follows fi’om assumption b) that supp ~ is a singleton. Thus, there exist 7(y) ¯ and h(y) ¯ ~ {0} su ch th at ~( f) = h( y)f(T(y)) fo r al l f ¯ C0(f~) . 3. Nowthe proof of [2, Proposition 2.4] shows that h is constant h = c ~ 0 and r : i22 -4 f~l is an isometry. Wedenote its isometric extension to ~N still by r. It remains to show that T(gt2) = i)~ 4. We show that r((~’)2)(~ Take 1 . Let Y o ¯ vqf ~2. Assum e r(yo) ¯ ~. y,~ ¯ f~.~ such that lira y, = Yo ¯ Choose f ¯ /)(f~2) such that f(v(yo)) =: 1 . Then lim (Uf)(y,~)

= lira

Cf(T(y,~))

= C ~t This con tradicts the

fact that

u.f ¯ c0(~). 5. The set V(f~:) is open. Since by 4., 0(r(fl.))) = ~-(0gt2) C 0~1 , it that r([)2) is relatively closed in f~ . Since f~l is connected, we conclude that

fffh) = gh ¯

[]

Condition b) cannot be omitted in Proposition 3.6. In order to see this, it suffices to choose i)~ -- (0, 1) C ~ = ft~ and to take for U the embedding from Co(0, into Co(~) It is surprising that we can omit b) if we strengthen the intertwining condition slightly. For that we will suppose that Aoa~ and Aoa~ are generators. Werecall the following easy description of intertwining operators. PROPOSITION3.6 Let Aj be the generator of a semigroup Tj on a Banach space Ej , j = 1,2 . Let U ¯ ~(E~,E~) . The following are equivalent: (i) UT~(t) : Te(t)U (t (ii)

UD(A~) C D(.4~) and A.zUx : UA~x for all

x ¯ D(A~

Assuming Wiener regularity and the intertwining property we can now show that condition b) of Proposition 3.4 is automatically satisfied. The key argumentis strict positivity in the sense of Theorem3.3. THEOREM 3.7 Let f~,f~2 C ~,N be open, connected and regular (in the sense of Wiener). Let U : Co(fl~) ~ Co(~) bea l at tice hom omorphism, U ~ 0, su ch that Ve’A~o’ = e~h~°~V (t > 0) .

(3.6)

Then there exist a constant c > 0 and an isometry T : j~IV --r IRN satisfying v(l~) = ~ such that

(uf)(~/) = ~f(~-(~/)) (~/¯ for all f ¯ Co(12~) REMARK3.8

By Proposition 3.6, condition (3.6) is equivalent to saying that for

v,f ¯ Co(as) Av = f in D(f~t)’ =~ AUv= Uf in D(f~.~)’

(3.7)

11

Different DomainsInduceDifferent Heat Semigroups which is stronger than condition (C) of Proposition 3.4.

Proof. By the second part of the proof of Proposition 3.4 the operator U is of the form h(y)f(T(y)) (U f)(y) = 0 y e f~2 \

I

where f~ = {y e ft2 : ~ f E Co(ftl) , Vf(y) 0}, h : fl ~ --+ (0, ec) and T:ft~-+f~l are functions. Let 0< fECo(f~i) such that Uf>O. By Theorem 3.3 we have

(ue f)(u) = (e vf)(y) for all y~ ~2,where t >0. This implies that ~t2 =ft~. Thus condition Proposition 3.6 is satisfied and the claim follows.

b) in []

Weconclude this section showing by a counterexample that Theorem 3.7 is not true if we replace Dirichlet boundary conditions by periodic houndary conditions, even if U is an order isomorphism. EXAMPLE 3.9 Consider the Banach lattice E = {f ~ C[-1, 1]: f(-1) = f(1)} with supremum norm and let A be the operator on E given by D(A) = C~[-1,1] : f(~)(-1) = f(~)(1) for k = , Af = f" . T hen A gener at es semigroup T. Let U : E ~ E be given by (Uf)(y)

f(1-y) = f(-1-y)

if 0_
Then UT(t) = T(t)U (t _> 0) even though U is not co~nposition by an isometry. Proof. The operator given Bf = f’ , D(B) = {f ~ EOCI[-1,1] : f’(-1) f~(1)} generates the shift group on E. Hence A = 2 generates a holomorphic semigroup T. One easily sees that UD(A) = D(A) and AUx = UAx (x D(A)) .

4

APPENDIX: REGULARITY IN THE SENSE AND REGULARITY IN CAPACITY

OF WIENER

Let f~ be an open set in /R ~ . We say that ~ is regular in measure (resp., in capacity) if for all z ~ Oft and all r > 0, Ift\B(z,r)l >0 (resp., cap(f~\ B(z,r)) > 0) . Here IFI denotes the Lebesgue measure of a measurable set in h’~ N . It is clear that toplogical regularity (i.e. ~= Ft) implies regnlarity measure, and regularity in measure implies regularity in capacity. EXAMPLE 4.1 Let B = B(0,1) be the euclidean unit ball in e . Then ft = B \ {(x~,0) :0 _< x~ < 1} is regular in capacity but not regular in measure. The reason whythese notions of regularity are introduced in [2] is the following. Consider L2(~t) as a subspace of L:(J~ N) by extending functions by 0 out of

12

Arendt

gt . Then L2(gtl) = L2(Ft2) if and only if [~t~ A f12[ = 0. This in turn, i~nplies that ~ = ~2 if gtl and ft2 are regular in ~neasure. Here fl~,fl~ C ~g arc open sets. If lgtl A ~] = 0, then A~1 = A2~ if and only if cap(12~ A 12,~) = 0, and this in turn implies that f~l = 122 whenever ~1 and ~t2 are regular in capacity (see [2]). REMARK 4.2 The condition "regular in measure" did occur in different context (under different name). It seems to be a crucial condition for smooth approxima,tion in Sobolev spaces (cf. [20]). For example, if f~ C ~2 is open, bounded and star-shaped, then regularity in measure is sufficient for C~(~) being dense I,V~’~(ft) (k ~ ~VV, < p < ~), s ee[20, Theorem B]. I t is als o a necessary condition in special situations (see [20, TheoremC and Example 1]). Next we showthat regularity in the sense of Wiener implies regularity in capacity. Our proof is based on the results of [3]. PROPOSITION 4.3 Let ~ C ff~N be an open set which is regular in the sense of Wiener. Then ~ is regular in capacity. Proo]. Let ~ be open, regular in capacity such that cap((~ \ fl) = 0 (see Proposition 3.18]). Then L2((~)= L2(gt) and A~ = A~. Now assume t, hat l~ # ~. Choose z ~ 0~t~. Let 0 < f ~ Co(f~)~L2(~). Then by [3, (3.3)] (e*A~of) = etA~f. It follows from Theorem 1.5 that c*&gf e C~(~) and (et~ f)(z) > 0 . But etA~ f ~ C~(l~) and etA~ f = et~ f . This contradicts et~f ~ C0(~) ¯ There is a remarkable criterion due to Wiener which describes regularity. that N _> 3. Then f~ is regular in the sense of Wiener if and only if E 2J(N-~) cap (B(z, 2-j) \ a) =

~hat

Assume

(4.1)

for every point z ~ 0~t. Thus Wiener’s criterion is a quantitative version of regularity in capacity. One can also see from Wiener’s criterion that regularity implies regularity in capacity (note howeverthat here N _> 3) . Every open set in ~’~ which satisfies the exterior cone condition (meaning that for each z ~ 0gt there exists a cone in ~3 \ 12 with vertex z) is regular. But there exist cusps which are not regular (see [14, p. 288]). Such a cusp gives an example of an open set in if/3 (or higher dimension) which is regular in capacity but not in the sense of Wiener. In dimension N = 2 the situation is more complicated. In fact, it is knownthat 12 C ~ is regular whenever for each z ~ 0f~ there exists a continuous, injective function f: [0,1] -~ ~ \ ~ such that f(0) = z (see [11, p. 173]). Here is an example of a set in g~2 which is not regular in the sense of Wiener but regular in capacity (and even topologically regular).

Different DomainsInduceDifferent Heat Semigroups

13

PROPOSITION4.4 Let B be the open unit ball in 1~2 and let ft : B \ ( U -~(an,rn) {0})

(4.2)

where a,~=(a,~,0), an>0, lira a,~=0, an+r,~>0 are chosen such that the closed balls -~(an,r,~) are disjoint. Thus ft is open and regular in capacity (and even topologically). However, one can choose r,~ > 0 such that ft is not regular. Weare grateful to Charles Batty for the following probabilistic proof. Some preparation concerning Brownian motion and potential theory is needed (see [17] for example). Let f~ C/~N be an open, bounded set. By {Bt : t >_ 0} we denote the Brownian motion and by P’~ the Wiener measure (x E/RN) . Then regularity can be characterized in terms of Brownian motion in the following way (see [17]). PROPOSITION 4.5

The set ~ is regular if and only if

> 0: e v s e

=0

(4.3)

for all x ~ Oft . REMARK 4.6 a) Condition (4.3) says that Brownian motion starting at Oft has to leave immediately ft with a positive probability (equivalently with probability ~). b) To see the relation with the Dirichlet problem we mention that for f ~ C(Oft) u(x) = E~[f(B~,)]

(x

(4.4)

defines a har~nonic function on ft . Here rfl = inf{t > 0 : Bt ~’ ~} is the first exit time. If (4.3) is satisfied, then !i~ u(z) f( z) for al l z ~ Oft . Thus u is the solution of the Dirichlet problem. We ~nention

that

for

[]

N>_2 and xe/R N\{0},

t>0

P°[B(s) = for so me 0 < s < t] = 0.

(4.5)

Nowwe can prove the proposition. Proof of Proposition ~.5. Wefix a sequence an $ 0 . Let an = (C~n,0) . Let t>0. Then fi~(r) =P°[B(s) ~B(a,~,r) for some s<_t] is decreasing in r>0, and by (4.5), lim,.,0 f,~(r) = 0. This allows us to choose rn > 0 satisfying the requirements of the proposition and such that f~(r~) < 2-~ tbr all n ~ t~ r . Thus P°[B(s) E U B(a,~,r,~)

for some s <_ t] <_ E P°[B(s) ~ B(x,~,rn)

for some

n=l

s _< t] < 1. Consequently, P°[Sr > 0: B(s) e f~ for all s e (0, r)] >_ P°[B(s) ~ for all s E (0, t)] > 0. Thus (4.3) is violated. Of course, the example is also valid in higher dimensions than 2.

14

Arendt

REFERENCES 1. 2. 3.

4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

W. Arendt: Kato’s inequality: a characterisation of gene~utors of positive semigroups. Proc. R. Jr. Acad. 84A (1984) 155- 174. W. Arendt: Does Diffusion Determine the Body? Uhner Seminare Heft 3 (1998) 17- 42. W. Arendt, Ph. B~nilan: Wiener regularity and heat semigroups on .spaces of continuous functions. In: Topics in Nonlinear Analysis. J. Escher, G. Simonett cds., Birkh~iuser, Basel 1999. P. B~rard: Domaines plans isospectraux 5 la Gordon-Webb-Wolpert: une preuve dldmentaire. Afrika Mat. 1 (1993) 135- 146. H. Brezis: Analyse Fonctionnelle. Masson, Paris 1980. S.J. Chapman: Drums that sound the same. Amer. Math. Monthly 102 (1995) 124- 138. E.B. Davies: Heat Kernels and Spectral Theory. Cambridge University Press 1989. R. Dautray, J.-L. Lions: Analyse mathdmatique et calcul numdrique. Masson, Paris 1987. D. Gilbarg, N.S. Trudinger: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1983). C. Gordon, D. Webb, S. Wolpert: Isospectral plane domains on surface,,.~ via Riemannian orbifolds. Invent. Math. 110 (1992) 1 - 22. L.L. Helms: Introduction to Potential theory. Wiley, NewYork 1969. M. Kac: Can one hear the shape of a drum? Amer. Math. Monthly 73 (1966) 1 - 23. O.D. Kellogg: Foundations of Potential Theory. Springer, Berlin 1967. N.S. Landkof: Foundations of ModernPotential Theory. Springer, Berlin 1972. R. Nagel ed.: One-Parameter Semigroups of Positive Operators. Springer LN 1184, Berlin 1986. A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin 1983. S.C. Port, C.J. Stone: Brownian Motion and Classical Potential Theory. Academic Press, NewYork 1978. W. Rudin: Functional Analysis. Mc Graw-Hill, NewYork 1991. H.H. Schaefer: Banach Lattices and Positive Operators. Springer, Berlin (1974). W. Smith, A. Stanoyevitch, D.A. Stegenga: Smooth approximation of Sobolev functions in planar domains. J. London Math. Soc. 49 (1994) 309 - 330. H. Urakawa: Bounded domains which are isospectral but not congruent. Ann. scient, t~c. Norm.Sup. 4~ sSrie, 15 (1982), 441 - 456.

Gaussian Estimates for Second Order Elliptic Divergence Operators on Lipschitz and C1 Domains P. AUSCHER Facult~ de math~matiques et d’Informatique, Universit~ d’Amiens, 33, rue Saint Leu, F-80039 Amiens Cedex 1, and LAMFA,CNRS, UPRES-A6119. E-mail: [email protected] PH. TCHAMITCHIAN Facult6 des Sciences et Techniques de Saint-J6rSme, Universit6 d’Aix-Marseille III, AvenueEscadrille Normandie-Niemen,F-13397 Marseille Cedex 20, and LATP, CNRS, UMR6632. E-mail: [email protected]

ABSTRACT We study the heat kernel of elliptic second order divergence operators defined on Lipschitz or C1 domains subject to Dirichlet or Neumannboundary condition. Our purpose is to obtain gaussian upper bounds and HSlder regularity of these kernels whenwe allow the coefficients to be complex. Weobtain a criterion to decide on whether such estimates hold and apply it in various situations such as uniformly continuous or vmo(fl) coefficients on C1 domains or Lipschitz domains with small Lipschitz constant. Wealso prove an analyticity result for the heat kernels as functions of the coefficients. Although not treated here the strategy works for second order systems subject to Garding inequality.

INTRODUCTION Weconsider second order operators in divergence form on Lipschitz domains with or without lower order terms and subject to Dirichlet or Neumannboundary condition. Weallow complex coefficients. Interior estimates are very well known:see [1], [2], [3], [4], [5] [6], [7] and [8] for real operators and [9], [10] and [11] for complex operators. In our situation, estimates at the boundary are also known. Pointwise gaussian upper bounds hold on arbitrary open sets under Dirichlet boundary condition or on open sets with the extension property under Neumannboundary condition, see [6] and [12]. In the latter reference, other boundary conditions are considered. On Lipschitz domains, 15

16

Auscherand Tchamitcl~iam

boundaryregularity is also available and usually follows from the reflectio~t principle and bilipschitz change of variables. In [13], [14], [15] this is proved for the Green’s and Neumann’sfunctions on bounded Lipschitz domains (that is, for the kernel of the inverse of the operator). As a consequence, such estimates hold for the iheat kernel as well. As we have not found this fact explicitly in the literature, we shall provide an argument (see Sections 3 and 4). Weintroduce a property, called the Gaussian property, which is the conjunction of Gaussian upper bounds together with HSlder regularity in both space variables of the heat kernel. As mentioned,this property is satisfied by real elliptic operators. Nevertheless, it is interesting for manypurposes to see howthis extends to complex ~ operators. For example, we will show that this is a stable property under L (complex) perturbations on the coefficients, hence giving real-analyticity results the heat kernels (of real operators) as functions of the coefficients. The study of the Gaussian property for complex operators can be quite discouraging at first sight on arbitrary Lipschitz domains: constant complexelliptic operators may fail to satisfy good heat kernel bounds because of the counterexample in [16]. This example occurs on a conical domain with large Lipschitz constant. However, for domains with small Lipschitz constant or C~ do~nains then things improve: we will show that constant complex elliptic operators have good heat kernel bounds and so does their L°°- or even BMO-perturbations, or the local analogues, that is coefficients in BUC([~) or in vmo(~). The examples corresponding to BMO or vmo are surprising and cannot be obtained from the reflection principle as this process is not suitable when dealing with BMOnorms on the coefficients. The strategy to obtain heat kernel bounds originates from [10], see also [17], Chapter 1, for a complete argument on R’~, and consists in replacing the Gaussian property I) 3, an equivalent property that is stated in terms of local regularity for weaksolutions of the operator. Here, the definition of a weaksolution incorporates the boundary condition. It is the local nature of the latter property which enables us a reasonable perturbation theory, as is done in Section 4. This article should be regarded as a natural sequel of [10] and [17], so that details will not always be included here. Wealso add that the same strategy can be nsed for second order systems which are elliptic in the sense that they satisfy" a Gardinginequality.

1

NOTATION

For 6 > 0, we denote by A(6) the class of elliptic ellipticity constant 6, that is

matrices in L°°(l~n, M,~(C~))with

and V~e~" ReA(x)~.~>Sl~l

’2,

’*, a.e.

onl~

and .4 is the union of all For open sets f~ in t~ n, we use [[fl[p or [[f[lL,(~) to denote the usual norm the Lebesgue space LP(~l) equipped with Lebesgue measure. The notation H~ (~) stands for the usual Sobolevspace with nor~n(llX;/ll~+ Ilfll~) ’~-,1/’)and H¢] (,[~) is closure of C~"~([~) in ~ (f~). Given A 6,4, coefficients b~.,

GaussianEstimatesfor Elliptic DivergenceOperators

17

of l~’* and a closed subspace V of H1 (it) containing ~ (i t), th e fo J(f,g)=

AVf.Vg÷fb.

Vg÷c.

Vf-~÷df-~

(1)

is regularly accretive on V × V. Denote by L the maximal-accretive operator on L2(it), with largest domain 7)(L) C V, such (Lf,

g)=J(f,g),

f ¯l:)(n),

(2)

Since/?(L) is dense in V, we mayextend L continuously from V to its dual space V’. Weshall use the same letter to denote both L or its extension depending on the context. It is customary to write L as -div (AV+ b) + cV + d, however we choose to write L as (,4, b, c, d, fl, V) to indicate the coefficients A, b, c, d, the domainit and the space of boundary condition V. It is convenient to have the coefficients defined on N’~ as we sometimes change domains. A pnrc second order operator is denoted by the triplet (A, it, V). The Dirichlet boundary condition corresponds to V = H~ (it) and the Neumannboundary condition to V = H~(ft). Strictly speaking, if the domain is smooth and V = Hi(it), the boundary condition is On.,u + b.nu = 0 on 0it, where 0~ is the conormal derivative and n the outward normal, so that this is a slight abuse of definition. Weconsider Lipschitz domains of the following types. A special Lipschitz domain (type I) is the open set located above the graph ofa Lipschitz map(I): II~ n-1 ~ ~ and the Lipschitz constant is, by definition, the quantity IIVOlloo. A bounded Lipschitz domain (type II) is an open, connected, bounded set in ’~ whose bo undary is covered by a finite numberof sets Ba: ClOf~kwhere Bk are open balls in ll~ n and ita is a rotated special Lipschitz domain. By definition, the Lipschitz constant of it is the infimum of max{Lipschitz constant of f~k} taken over all possible decompositions of ft in this way. The third type consist of those exterior domains(connected, open set, with compact complement) whose boundary is covered exactly as above. Unless explicit mention, we work with elliptic operators subject to Dirichlet or Neumannboundary condition on a Lipschitz domain of the above types in N’~, n>2.

2

THE GAUSSIAN PROPERTY: DEFINITION AND MAIN RESULTS

Weare given an operator L = (A, b, c, d, it, V), with for the momentarbitrary and V. Recall that A - L is the generator of an analytic semigroup on L~(it) in an open sector largzl < 7r/2 -co for some co ¯ [0,~r/2) and A ¯ II~. Let Kt(x,y) denotes the kernel of e-t~ for t > 0. Such a kernel mayenjoy decay and regularity as follows. Here, I is an interval (0, T) with T finite or infinite. (i)

[Gaussian upper estimate] For each t ¯ I, t(t(x,y) is a ~neasurable function on it x f~ and there exist constants Co and a > 0 such that II(t(x,y)l

~- e-- --v’--, t ¯

I, a. e. on [~x 0.

(3)

18

Auscherand Tchamitchiam

(ii)

[HSlder regularity in the first variable] For y E f~ and t ~ I, x ~ K~!.(x,y) is a HSlder continuous function in ~ and there exist constants cl > 0 and p > 0 such that c~ ~x-x’~ t’ t6I, x.x’~, y~. (4)

(iii)

[HSlder regularity in the second variable] For x ~ l~ and t ~ I, y ~ Kt(x, y) is a HSlder continuous function in ~ and there exist constants ce > 0 and p > 0 such that C2

DEFINITION1 We say that L has the Gaussian property (G) (resp. (G~oc)) the Kt(x, y) satisfies (i,ii,iii) for I = (0, ~) (resp. I = (0, T) fbr some REMARK 2 Recall that a HSlder continuous function on a set E extends to its closure. Thus, if L has the property (G) or (G~oc) then Kt(x,y) C~’(gt x f~) fo t ~ I, which shows that this definition includes interior and boundaryregularity. REMARK 3 By the semigroup property, if (G~oc) holds then the estimates (i,ii..iii) hold for all t > 0 provided we change the constaut ci to cie ~ for some a > 0. Sometimes, the methods of proof show that the exponential behavior for large times can be improved to polynomial growth (see [10]). Whenfl = IR’*, we refer to [17] for positive results on the Gaussian property.. When~ ~ ll~ ’~, the first result is that the Gaussian property depends on the principal part. A sketched argument can be found in Section 3.1. THEOREM 4 Let L = (A, gt, V) be a pure second order operator. Assume that it has the property (G~oc). Then, given any set of boundedcoelyicients, b, c and d,. the operator ( A, b, c, d, ~, V) has the property (G~oc). Wenext restrict our attention to pure second order operators. For real coefficients, things are well understood. See the references mentioned in the Introduction. PROPOSITION 5 Assume that ,4 has real entries main. 1. (A,f~,H~(f~)) 2.

and that [1 is a Lipschitz

do-

has the Gaussian property

(A, i2, H~(f~)) has the Gaussianproperty (G) if~ is o] type I or III, if ~ is o] type II.

The Gaussian property is not to be expected for many complex operators Lipschitz dmnains, even with constant coefficients.

on

PROPOSITION 6 If n >_ 5, there exists a conical domain ~ (with large Lipschitz constant) and an elliptic matrix A with constant complex entries such that (A, f~, H~ ([2)) does not have the Gaussian property.

GaussianEstimatesfor Elliptic i)ivergence Operators

19

This is based on [16]. However,we shall prove the following result. THEOREM 7 (Validity of the Gaussian property) Let L = (A, fl, V) be subject Dirichlet or Neumann boundary condition i.e. V = Hi or V = Hl, where A is complex and A E A(6). If f~ is a special Lipschitz domain (type (i)

L has the Gaussian property (G) when A is real.

(ii)

L has the Gaussian property (G) when n =

(iii)

If L has the Gaussianproperty (G) (resp. (G~oc)) then so does L’ = (A’, fl, V) whenever I[A - A’[Ioz < ~ for some ¢ = ¢(n,6) >

(ix,) If f~ has a small enough Lwschitz constant (depending on 6) and if A constant then L has the Gaussian property (G).

(v)

If f~ has a small enough Lipschitz constant (depending on 6) or is uniformly C~ (that is, the defining function of Of~ has uniformly continuous gradient) and if A is a small L~-perturbation of a BUC(f~)matrix then L has the local Gaussian property (Gloc).

(vi) If f~ has a small enough Lipschitz constant (depending on 6) and I[AIIBMO(a) is small enough then L has the Gaussian property (G). (vii)

If ~ has a small enough Lipschitz constant (depending on ~) or is uniformly C~ and if A is the BMO(f~)-perturbation of a wno(f~) matrix then L has local Gaussianproperty (Gloc).

If f~ is of type H then, except for (iii), replace in the conclusions above (G) (G~oc), and uniformly ~ by C~ i n ( v) a nd ( vii). For type III domains, the same modifications apply except for (i). REMARK 8 The proofs will show that the HSlder exponent is arbitrary in (0,1) for (iv) to (vii). Note that (vi) and (vii) include (iv) and (v) respectively worth distinguishing them and the proofs are quite different. Here is a corollary of (iii). For c~ > 0 and tt E (0, 1), let K:~,~, be the Banachspace of those complex-valuedfunctions pt (x, y) defined on (0, ec) × f~ × f~ which satisfy (3), (4) and (5), and equip this space with the norm co +c~ +c~ where the c~ are smallest constants in the respective inequalities. For A ~ A, denote by KtA(x,y) the heat kernel of (A,f~,V) and set KA: (t,x,y) --> Kt~(x,y). Here, f~ and V are fixed with the assumptions in (iii). Thus, (A, Ft, V) satisfies (G) Km belongs to some/(:a,,. Define a subclass in L~(Ft; M~(~2)) ~ = {A ~ A; (A, gt, V) has the Gaussian property (G) Part (iii) of Theorem7 tells us that ~ is an open subset of L°°(f~;M~(C)). particular, it is a neighborhoodof the class of real elliptic matrices on ~ in A. With the notations above we have

In

20

Auscher and Tchamitchiam

THEOREM9 For each Ao ¯ ~, there are constants a > 0 and p ¯ (0,1) such that that A -+ KA is complex analytic from a neighborhood of Ao in L~(l~; into ]~,~. In particular, there are constants ~ > 0 depending on ellipticity o~( Ao, and c >_ 0 and ~ > 0 depending on the constants in (G) for Ao such that fi~r all A ~ L°C(gt; Mn(C)) with [IA - A0[[~ < ~ then A ¯ 6 and [KtA(x,y) - KtA°(x,y)l - < cll A -.Aoll~ot -’~/~ exp

{ ~lx-yl} ’

(6)

for all (t, x, y) ¯ (0, c)c) × l’~ The proof of this result is an adaptation of that of Theorem26 in Chapter 1 of [17]. Of course, a si~nilar result holds replacing (G) by (G~oe), or by considering elliptic operators with lower order coefficients. Weleave details to the reader. Note that somerelated estimates for the difference betweentwo heat kernels o~ real symmetric operators with Dirichlet boundary condition on bounded C~-domains or on ~’~ in terms of local LP-averagesof A-,4o, for somelarge finite p, can be found in [18]. Here we impose more: na~nely L~-perturbations about .40, and we get more: the Gaussian property and the analyticity about Ao. This is one of the reasons tbr working with complex coefficients. The strategy to prove Theorem 7 is to replace (G) by an equivalent property, called (D). Compare [17], Chapter 1, when fl = ~’~. On the way we will obtain Theorem 4 and Proposition 6.

3

EQUIVALENCE BETWEEN THE GAUSSIAN AND THE DIRICHLET PROPERTY

PROPERTY

DEFINITION 10 Let E be an open set in ll~ n with E ~ gt ~ q). Let L = (A, [~, be an elliptic operator. A weak solution u of Lu = 0 in E ~ ~ is a function ~ U ¯ V ~ H (E ~ Ft) such that

/~

AVu- V~2 = 0, ~’~ ¯ Ht(II~’)),

Supp~ C ~, ~ala ¯

This definition of a weak solution takes into account boundary conditious on 0~. If E C ~, then we recover the usual definition. Notice that if V = H may not vanish at the boundary of ~. However, if V = H~ (~t) then we may a.ssume that Supp~ c E~t. By abuse of notation, we shall say that ¢p ~ H~(E) ~ in either case, which is consistent with the classical extension and restriction theorem for HI spaces on Lipschitz domains. Let B~ be the set of Euclidean open balls which are centered in ~ and have radii not exceeding someconstant p0 chosen as follows. If gt is a special Lipschitz domain, set Po = ¢0. If ~ is of types II or III, a compactnessargument(since c9~ is compact) shows the existence of P0 > 0 so that for any B centered in gt with radius not exceeding p0 then B ~ 0fl is either empty or contained in a Lipschitz graph up to a rotation in I~’*. As a consequenceand in any case, there exists ~" > 0 depeuding only on the Lipschitz constant of gt such that if B ¯ B~ then ]B

GaussianEstimatesfor Elliptic DivergenceOperators

21

DEFINITION11 Let L = (A, f~,V) be an elliptic operator. We say that L has the Dirichlet property (D) if there are constants # E (0, 1] and c > 0 such that any ball Bn E Ha, ifu ~ VNHI(Bn) is a weak solution ofLu = 0 in BnNfl, then for all0
Wesay that L has the local Dirichlet property (D~oc) if the same estimates hold with the restriction that balls have radii less than some number R0 > 0 (that may be related to ~2 and/or to the coefficients). Here and subsequently, we do not refer to the centre of balls in the notation as we implicitly assume that B~ has same centre as Ba. This is an extension of Definition 8 in Chapter 1 of [17] that was given only for interior regularity. This definition also takes into account boundary regularity. We have the following result. THEOREM 12 Let L = ( A, ~, V) be an elliptic operator. Then L has the Gaussian property (G) if and only if L and L* have the Dirichlet property (D). Similarly, has the local Gaussianproperty (G~oc)if and only if L and L* have the local Dirichlet property (D, oc). This result can be proved adapting the proof of Theorem 10 in Chapter 1 of [17]. So we do not write complete details in order to keep the length of the article reasonable. Someexplanations are nevertheless necessary. Before we do so, let us mention that what the counterexample of [16] disproves is exactly (D~oc) for an ad hoc constant coefficient operator subject to Dirichlet boundary condition (it has a weak solution that blows up at the boundary) hence (G~oc) is disproved by our result. This proves Proposition

3.1

(D) Implies

(G)

\Ve need to clarify three points to extend the proof given in Section 1.4.3 of [17]. 1) Morrey-Campanatospaces on 12: they are defined using the collection of balls in f~ with radii not exceeding 1 (or P0 if ~ is bounded). Such spaces are described in [19]. Then, the interior and boundaryMorreytype elliptic regularity theory goes through (step I and step 2) thanks to our careful definition of weaksolutions. The estimates depend on the number ~f introduced above. 2) Next, we used in II~ ~ a scaling argument to obtain correct rcsolvent kernel estimates (step 3). This is not possible on one domain, but it is possible if one works on the family of domains obtained one another by dilations as the Lipschitz constant (hence the constant ~,), the ellipticity constant and the various constants in the properties (D) and (G) remain unchanged by scaling (the number p0 dilates by scaling). 3) The last commentis to correct a mistake in [17] (step 4)..We asserted that the Ca~npanato spaces classically denoted by £~’ , 0 <_ A < n + 2 (denoted by

22

Auscher and Tchamitchiam

M~there), are complex interpolation spaces. Actually this is not known. But all we used was the following version of Stein’s complex interpolation theorem: if is an analytic family of operators on the open strip 0 < R.ez < 1, continuous on the closed strip, such that T~s : L2 -~ £~,),o and Tl+is : L~ f,2,~l --> ~ are bounded f~,~o with A0 (1 - 0)A0 + 0A~, ~ with uniform nor~ns for s ~ ]~, then To : L ~ ~ , 0 < 0 < 1. For a single operator, this is due to Stampacchia, and Campanatoand Murphy. An elegant proof by Spanne makes this result a simple consequence of the Riesz-Thorin theorem. See [20] and the references therein for the above. The :same proof for analytic families makes it a consequence of Stein’s theorem. Let us nowindicate a strategy of proof of Theorem4. If L = (,4, 9/, V) satisfies (G~oc), then L and L* satisfy (D~oc). Let ~ = (.4, b, c, d, Ft, V) and choose d so 0 ~ a(£). The Morreytype ellipticity theory of 1) applies to £ whenthe coefficients b, c, d are bounded. Then the dilation argument of 2) applies also to the resolvent (-£ + A)-~ for large IAl with A in the resolvent set, while for small IAI we use l) directly since 0 is in the resolvent set of £. A little care gives then the needed operator estimates on e -tc for all t > 0 and from there 3) applies without changes.

3.2

(G) Implies (D) : Gradient Estimates

The proof that (G) implies (D) in [17] applies verbatim and relies on gradient estimates and (boundary) elliptic Caccioppoli inequalities (similar to the parabolic ones in Proposition 13 below). V~re give the proofs of the parabolic Caccioppoli inequality and of the gradient estimates and leave further considerations to the reader. For the rest of this section, L = (A, ~, V) is subject to either Dirichlet or Neumann boundary condition and f~ is Lipschitz domain. I still denotes an iuterval (0, T) finite or infinite. PROPOSITION 13 (Parabolic Caccioppoli inequality) Given f ~ L’~(i]), the solution ’u~ = e-iLl of the parabolic equation o,~__~, = -Lut in [~ satisfies for t > 0.

for any ~ ~ Ct(I~n), real-valued, ellipticity constant of A. Proof:

bounded with bounded 9radient, where g is the

Since ut 6 V, we have ’h¥~~ ~ V. Therefore,

--g-( ~ ~2, whence = _ fn Out

23

GaussianEstimatesfor Elliptic DivergenceOperators

Set,M : l[~Vu~l[~ andobserve thatM < [l~p[l~lIVu~l[~ < eo since’u~~ H~([~). Usingellipticity andCauchy-Schwarz inequality we obtain

[]

and (8) follows.

PROPOSITION 14 Assume that e-tr is bounded fi’om Ll(ft) to L2(~) with norm -a boundedby ct for some a >_0 for all t E I. Then for almost all y ~ ~, K,(., y) ~ with

supIIV~K,(’,v)II2 < -~, ct-½t

(9)

for t ~ I. Proof: Take ~ = 1 and f ff L2(Ft)~ Ll(fl) in (8). From the hypothesis have Ilutl[2 ~_ ct-~llfll~ and using the analyticity of the semigroup on L2(Ft) obtain at ’~ <- ct~t-~i[f[[~" Hence, [[Vut[[~ _< ct-~t-~[If[[~. Since Vut(x) J’~ V.~K~(x,y)f(y)dy, it suffices to let f converge to a Dirac mass in the sense of distributions. [] Recall a well-known result, e.g., [61, [171,[12].

a proof of which can be found in many places. See,

LEMMA 15 Assume that Kt(x,y) satisfies (3) for I = (O,T). For any (0,:,r/2 -w), the holomorphic extension Kz(x,y) of the heat kernel satisfies for I arg z I <- ~ with Re z ~ I and one should replace t by Re z in the estimates. The same statement holds if (3) is replaced by (~) or (5). Consequently, for N*, tk(~ )kKt(x,y) satisfies the same estimates as Kt(x,y). Weare now ready to obtain the gradient estimates. PROPOSITION 16 (i) Assume that I(t(x,y) satisfies for all y ~ ~, t ~ I and r >0 one has <_l:,:_~t<_2~[V.~Kt.(x,y)[2 (ii)

(3) for I = (0, T).

<_ c Co t -~-~ r

e_~~..

Assumethat I(t(x,y) satisfies (3) and (5) for I = (O,T). Then, for all ~, tEI and r > O ’with 2ly - y’[ <_ ~ + r one has \ 1/2

[V,~Kt(x,y)

- V~Kt(x,y’)[ 2 dx~

The constants c, ~, ~ depend on the constant a in (3), ~ in (5), n and

(I0)

24

Auscherand Tchamitchiam

Proof of (10): Wehave to boundf,.<_l.~_yl<_2,. ]VzKt(x, y)[2 dx (all integrals occur one) for fixedyE ~,tEIandr >0. First, the inequality (8) applies to ut(x) = Kt(x,y). Indeed, by the semigroup property ut = e-2~-Lut/2 and ut/2 ~ L2(~) from the Gaussian upper bound. Pick ~(x) supported in 5r/6 <_ Ix-y[ ~_ 13r/6 with ~(x) = 1 if r _~ [x--y[ ~< 2r, [[¢P11~¢-< 1 and [IVcp[[~ <_ cir. On the support of ~p we have Jut(x) + t

~ ~

~

fir

<_ ct-~e--7-

for some fl > 0. Thus, we obtain

IV~Kt(x,y)l~dx<_~\t,~_l] 1+ e--F" _< ~ t-’-~- t,~) ~-~ for any ~’ < ~3. This is (10). Proof of (11): Wehave to bound J’~_0suchthat lY-Y’I<_r/2. Apply (8) to ut(x) = Kt(x, y) -K~(x, with~2 asin theproof o f (10) . O bserve that Ibr x in the support of ~2, we have r/3 ~_ Ix-Y’I ~- 8r/3. The HSlder regularity in the second variable gives us x

lu~(~)l+)tOut(

_< ct-~ I~! ~.--r( ~ ’ - ) n" ~

on the support of ~. Hence,

which is equivalent to (11).

4

VALIDITY

OF THE GAUSSIAN

PROPERTY

In this section, we prove Theorem7. To check (G) or (Gioc), Theorem12 tells us thatit suffices to check(D)or (Dio~).Depending on whetherwe regard~ or BJlOtypeconditions on thecoefficients, theproofs aredifferent.

GaussianEstimatesfor Elliptic DivergenceOperators 4.1

25

L°~ Type Conditions

In this section, we prove (i) to (v) in Theorem7. Wetake advantage of the local ture of the Dirichlet property in using bilipschitz transfor~nations and the reflection ~. principle to go back to the situation on I~ Let us first examinatewhat a bilipschitz transformation in I1~’* does to the Dirichlet property. Clearly, such a transformation preserves the class of Lipschitz domains, the class of elliptic operators, the class of weaksolutions as defined above. Since it mapsballs to sets which contain and are contained in balls of comparable measures independently of their radii, it also preserves the class of elliptic operators satisfying (D) or (D~oc). Here is a precise statement of these facts. LEMMA 17 (Invariance under bilipschitz transformations) Consider an elliptic operator L = (A,f~,V(f~)). Let ~: N’~ ~ N~ be a bilipschitz change of variables with jacobian matrix Set fl’ = ¢-~(Ft), A¢(y) = I det J4~(Y)I rj-~l(y) ¢(y)Jg~(y

(12)

and L¢ = (A¢, ~’, V(~’)) where V(fl’) is the subspace of H~(fl’) defined by foe-~ e V(~). Then

a. A~ A(6)impliesA¢~ A(c6)for somec > 0 that dependsonly and b.

If u is a weak solution of Lu = 0 on E ~ f~, then v = u o ¢ is a weak solution ofLov = 0 on ¢-I(E CI ft) = ¢-l(E)

c.

L satisfies (D) on f~ (resp. (D~oc)) if, and only if, L¢ does on

The proof is quite obvious so we skip it. Note that I’(fY) = Ht(9’) (resp. H~ (f~’))

whe,~V(~)=H~(~)(resp.H~(~)). The next result is concerned with the reflection principle, which is also a classic. LEMMA 18 (Reflection principle) There are constants c,d > 0 depending only on the Lipschitz constant of f~ with the following properties: Let L = (A, f~,V) with A ~ A(5). Let B ~ Bn with B ~Ofl ¢ ~ and let u ~ H~(B) ~ V be a weak solution of Lu = 0 on B C3 fL Then, one can find a. a matrix A~ ~ A(cS) with A~la = A, and A -~ A~ is continuous in °°, b. a set B~ C Nn containing B with B~ f3-~ = B ~ ~ and IBII c. a function u~ ~ H~ ( ~) such t hat div (A~Vu~) =0 in D’(B~),u If f~ is a special Lipschitz domain, the matrix A~ is the same for all balls and we shall denote by L~ = (A~,I~n,H~(I~’~)) the elliptic operator thus constructed.

26

Auscher and Tchamitehiam

Proof: First assume that fl = I~, that is f~ = {x E ~’~; x,~ > 0} using a coordinate system (Xl,... , x,~) in II~’~. Define the orthogoual sy~nmetryS of l~ ’* across 0fl by S(x~,... ,x,~-~,xn) = (x~,... ,x,~_~,-x,~). For f: I~ -~ C, let f~ (resp. fo) be its even (resp. odd) extension to ll~ ’~ by f~(x) f( Sx) (r esp. fo(X) = -f (Sx)) if’ x,, < 0. Given A E A, defiue A~ ~ A by A~(x) = A(x) if x,~ > 0 and A~(x) SA(Sz)S if x~ < 0. Take u~(x) to be the even (resp. odd) extension of u(x) under Neumann (resp. Dirichlet) boundary condition. Then one checks by using Green’s formula that if u is a weak solution of Lu = 0 on B ¢3 ~2 with the prescibed boundary condition, then L~u~ = 0 in ~’(B~) with B~ = B ~ fl U S(B t3 ~). Next, assmne that fl is a special Lipschitz domain. Choose ¢: ll~ ’~ -~ ~" to be a bilipschitz change of variables with ¢({x,~ > 0}) = fl and ¢({xn = 0}) = 0~, apply Lemma17 to obtain the desired extensions on pulling back the construction on to FI with ¢-~. Lastly, assumethat fl is of type II or III. Since balls in B~ have intersection with f~ empty or contained in a part of a (rotated) special Lipschitz domain, we are immediatly back to the preceding case. [] Wcnowcometo the proof of Theorem7, (i) to (v). First, as the Dirichlet property is a local property, it suffices to prove the statements whenF~ is a special Lipschitz domain, which we assume from now on. Secondly, if L = (.4, ~, V) is given, Lem,na18, it suffices to showthat L~ = (A~, ll~", H~ (I~)) satisfies (D) or (D~o~.). Proof of (i), (ii) and (iii): If L satisfies one of these items on f~, then so ~ on ll~ ~. Hence, we conclude using [17], Chapter 1, Section 4.6. Item (i) is, of course, consequence of De Giorgi’s theorem. Proof of (iv): Here ,4 (aij) is constant. Assume first that ~ = 1~. Take A~ = (A~j) as in the proof of Lemma18.. construction, a~j(x) = 0 i f 1 < i, j < n - 1 ori = j = n, and a~(z) = aijS (z otherwise, where S(u) = 1 if u > 0 and S(u) = -1 if u < 0. Thus A~ depends on the co-ordinate variable x,~. In such a case L~ satisfies the Dirichlet property (D) with p = 1 by [17], Appendix 2, p.153. Next, assume that ~ is a special Lipschitz do~nain with Lipschitz constant /~. Let ¢: I~" -~ I~" be a bilipschitz change of variables with ¢({x,~ > 0}) = ~ and ¢({z,, = 0}) = 0~. Moreover, ¢ can be chosen so that J~ (y) is an L~-perturbation of the identity matrix if Mis small. As a consequenceof (12), there exists a modulus of continuity w(M)such that

IIA¢- AiIL~(~.} < w(M) ~-~.

(13)

Wejust proved that (.4, I~_, V(I$~_ )) satisfies (D) with exponent p = 1. By applying ’~ ’~ (iii), t~A I’ I~ does as well ifw(M)~-~ is small enough and the e×ponent ~,II~+, ’(+)) can be chosen arbitrary in (0,1). Weconclude by Lemma17 that L = (A, g~, V(~)) satisfies (D). Proof of (v): By (iii), it suffices to assume A ~ BUC(~). Supposefirst that ~2 = 11~ and let us look at A~. Note that the "even" coefficients of A~(x) are continuous across the boundary while the "odd" coefficients are not

GaussianEstimatesfor Elliptic DivergenceOperators

27

unless they vanish on the boundary. Let ao(x) denote one of the latter boundary point. Then, lao(X) - a(z)S(x.)l = I(a~(x) - a(z))S(z.)l <

and z be a

-

where I’VA,~_ is the modulus of continuity of A on R~, so that liAr(x) - A:(x)ll W,~.~.(Ix - z l) where A:(x) has entries aij(z) if 1 _< i,j < n- 1 or i = j = n, and a~j(z)S(x,~) otherwise. Observe that (Az,II~’~,HI(I~’)) is elliptic (uniformly in z). Since has coefficients that dependon one co-ordinate variable, this operator satisfies (D) with exponent # = 1 and the constant c in (7) is uniform in z. By the pertubation result of [17], Chapter 1, Section 4.6, there exists a number eo > 0 such that if I]A~ - Azlln~(s) < So on some ball B, then weaksolutions of (AI~, IR", H~(I~’~)) B satisfy (7) with any/~ G (0, With these observations, we argue as follows. Choose R0 > 0 such that WA,~_(Ro)< ~o. Let B~(xo) be a ball centered at Xo e II~ n and R <_ Ro/2. If d(xo,Of~) < Ro/2 then we let z be the vertical projection of x0 on 0~t. Thus ]lA~(x) ,4 z(x)ll <_ W~,~(Ro) < ~0 sin ce ]z - x I <_ Ro on B~(xo), andthe above applies. If d(xo, Of~) >_ Ro/2 then A~ is continuous on Bu(xo) and [17], Chapter 1, Section 4.6 applies. Secondly, assume that ~t is a special Lipschitz domain with small constant M. Let ¢: ll~ ’~ ~ I~’~ be a bilipschitz change of variables with ¢({x, > 0}) = f~, ¢({x,, = 0}) = cq~t, and ¢ chosen so that J~(y) is an L~’-perturbation of identity matrix. Wededuce from (12) that I[A~ - d o ¢[Ioo _< w(M)~-~

(14)

where w(M) is a modulus of continuity. Since A o ¢ ~ BUC(t~_), we just proved that (Ao¢, ~_, V(I~ )) satisfies (D~o¢). Then (iii) implies that for Msmall (A¢, II~, V(~_)) satisfies (D~oc) as well. Weconclude by change of variable satisfies (D~o~). Lastly, assume that the domainis uniformly C~. The change of variable ¢, as well as its inverse, can be chosen to be in C~ (ll~") with bounded uniformly continuous gradient. Hence ,4¢ is directly seen to be in BUC(II~_)and we conclude readily. [] REMARK 19 Another proof of (v) consists tion.

4.2

in using the method in the next sec-

BMO Type Conditions

To prove (vi) and (vii) in Theorem7, we proceed differently reflection principle is not continuous for BMOtopologies. Weuse the following (semi-) norm on BMO(f~):

IIIII Mo/) = sup{wLr~(s),0

< s Po},

as A -~ ~ in the

28

Auscher and Tchamitchiam

where 1 wl,o(s ) ---- sup IB n Ft I n~ If(z)

- f~al ~ dx ,

(15)

the supremumbeing taken over all B ~ ~ with radii not exceeding s, and fE is the mean of ] over E. A function f ~ L~(~) belongs to vmo(~) ff wL~(s} has limit 0 whenevers tends to 0. This BMOspace is the one used by Coifman-Weissas ~ with the induced d~stance and Lebesgue measure i8 a space of homogeneoustype. Another natural definition of BMOuses hails (or cubes) that are contained in ~. This gives a priori a bigger space. But since ~ has Lipschitz boundary, Jones’ extension theorem [21], which asserts that the latter space agrees with the space of restrictions to ~ of n )-functions, implies that they are the same and that both (semi-) norms BMO(~ are equivalent. Weshall need three preliminary results, the last two being valid for arbitrary elliptic operators. LEMMA 20 Let BR E B~. Then, (f~3sn~

I~’fUl

2)1/2

is

a norm in HOt(Bt~) n

Proof: If V = H0~ (f~) then t (BR)n V = Hd(Bnnft) so th at Poincar0, inequ ality applies. If V = H~(Ft) then, as in Lemma18, extends u ~ H~(Bn) n to a function fie H~(B2R)xvith (fB~n IV’51~)~/~ <-- c(fBno~ I Vnl2)~/2 (by rescaling is independent of R and depends only on the Lipschitz constant of ~). Poincar6 inequality for ’5 showsthat

f~nn~lul~" < eR-~fm~nnlVul~" COROLARY 21 Let L = (A,f~,V) be any elliptic operator. For BR ~ Bn and F ~ L~(Bn ¢1 ft), there is a unique function w ~ H~(Bn) fl V solution of equation

f; AVw.v = fu

F.V~

V EHo(Bn)nV.

Moreover,

where c depends only on ellipticity. Proof: By the preceding lemma and the ellipticity of A, the sesquilinear form ~ fu,~ A%Tw" V--~ is coercive on Ho (Bn) ~ V. It suffices to apply Lax--Milgram le~nma to the continuous anti-linear functional ~ ~ f~.n~ F ¯ ~-~. (Remark that Hol(Bn) n v is stable under taking complex conjugation.) [] The following lemmais a consequence of Meyers’ interior Lp estimates and Lernma 18.

Gaussian Estimates for Elliptic Divergence Operators

29

LEMMA 22 (Meyers inequality at the boundary) For any L = (,4, f~,V), there exist constants e > 0 and c such that for all 2 <_p < 2 + ~ the following holds: if BR E l~n and u ~ HI(BR) N V is a weak solution of L in B~ n f~ then Vu ~ L’(Bn/2 N f~) and ( fBR/~nn WU[;)~I" < cR-"12+"l~( £Rna lVul~) I/~.

(16)

Let us comeback to the proof of Theorem7. Proof of (vi): Assumethat fi has small Lipschitz constant. Wehave to showthat L = (A, f~, V) has the Dirichlet property (D) provided[IAll.Mo(9)is small enough. Fix a ball Bn~ Ba. Let u ~ HI(B~) N V be a weaksolution of Lu = 0 in Bn ~ fl in the sense of Definition 10. Let 0 < r ~ R/2 and set AB~n~ to be the meanof Aon B,. ~ ~. This is an elliptic constant matrix. Thanksto Corollary 21, define w ~ H~(B~) R V by solving the elliptic problem fB

AB,.n~VU,.V~=

fu

Au,.n~VU.V~

V~ H~(B,.)~L

(17)

Let v = u - w. Thenv is a weaksolution of the operator (AB,.na,~, V) in B,. N aud we have

with c independentof u and r. Let 0 < p < r. Since the domainhas small Lipschitz constant, the operator (Ax,.nn, ~, V) satisfies (D). Hence,for someF e (0, 1)

_

(Pl

To estimate the last integral, weuse the definitions of u and wto write

= / ~B

(Amnn - A)Vu. V~

for all ~ E H~(B,.) A V. Using ~ = ~ aud the ellipticity obtain J B~n~

condition for A~,.nn, we

Auscher and Tchamitchiam

30

Next, choose p > 2 for which Meyers inequality at the boundary (16) applies and use HSlder inequality with exponents 2,p and q where 1/2 + lip + 1/q = 1. Then

Hence

Now,the John-Nirenberg inequality gives us < C(q, n) WA,~(r).

As WA,n(r)IIAN~Mo(a~, weobt ain £,.~ IVml~~ c~-~llAIl~Mo(~)£~,.~ IVul2" Hence, we find that, for some COllSta~t a > 0

S~ IW
+ IIAII~Mo(<~)

~ IW’I

provided 0 < p ~ r and r ~ RI2. If0 < r < p_< 2r and r < RI2, the above inequality is triviMly satisfied. Hence, changing 2r to r we obtain (18) whenever 0 < p < r < R. Thus we obtain from Campanato’s lemma (see Lemma 13 of Chapter 1 in [17]) that for any #’ ~ (0, #) there exists eo = so(a, #, #’) and c = c(a, p, p’) such that if il,4[iBMO(~)< e0 then

for all 0 < p ~ R. This proves (D). Proof of (vii): XVehave to show that L = (A, fl, V) has the property (D~o¢.) vided the L2-modulusof continuity WA.~(r) is small for s~nall r. If the domain has a small Lipschitz constant, we repeat verbatim the preceding argument. The only change is in the refinement of (18) where I~All~MO(n)is replaced by WA,n(2r). Thus we obtain again from Campanato’s le~nma that for any

GaussianEstimatesfor Elliptic DivergenceOperators

31

(0,/z) there exist ~0 = ~0(a, #, #’) > 0 c = c(a,It, # ’) such that if R0> 0satis fi es WA,~(2R0) < ~0 and if R <_ R0 then f P ,~ n-2+2p’

for all 0 < p <_ R. This proves that L has the property (D~oc). If the domain is uniformly Ca, it has locally a small Lipschitz constant. So it suffices to apply the above argument on balls with R _< R0 for some appropriately chosen R0 > O. [] Wethank P. Acquistapace to call our attention to an article by Q. Huang [22] where such arguments are presented to obtain elliptic regularity for vmocoefficients.

REFERENCES 1. J. Nash. Continuity of solutions of parabolic and elliptic equations. Amer. J. of Math., 80:931-954, 1958. 2. D. Aronson. Bounds for fundamental solutions of a parabolic equation. Bull. Amer. Math. Soc., 73:890-896, 1967. 3. J. Moser. On Harnack’s theorem for elliptic partial differential equations. Comm.Pure & Appli. Math, 14:577-691, 1961. 4. E. Fabes and D. Stroock. A new proof of Moser’s parabolic inequality via the old ideas of Nash. Arch. Rat. Mech.Anal., 96:327-338, 1986. 5. F.O. Porper and S.D. Eidel’man. Twosided estimates of fundamental solutions of second order parabolic equations and some applications. Russian Math. Surveys, 39(3):119-178, 1984. 6. E.B. Davies. Heat kernels and spectral theory. Cambridge Univ. Press, 1992. 7. D.W. Robinson. Elliptic operators and Lie groups. Oxford Mathematical Monographs, 1991. 8. N.T. Varopoulos, L. Saloff-Coste, and T. Coulhon. Analysis and geometry on groups, volume 100 of Cambridge Tracts in Mathematics. Cambridge University Press, 1993. 9. P. Auscher, A. McIntosh, and Ph. Tchamitchian. Heat kernel of complex elliptic operators and applications. J. Funct. Anal., 152:22-73, 1998. 10. P. Auscher. Regularity theorems and heat kernels for elliptic operators. J. London. Math. Soc., 54:284-296, 1996. 11. P. Auscher, T. Coulhon, and Ph. Tchamitchian. Absence de principe du maximumpour certaines 6quations paraboliques complexes. Coll. Math., 171:87-95, 1996. 12. W. Arendt and A.F.M. ter Elst. Gaussian estimates for second order operators with boundary condition. J. Operator Theory, 38:87--130, 1997. 13. M. Griiter and K.-O. Widman. The Green function for uniformly elliptic equations. Manuscripta Math., 37:303-342, 1982. 14. P. Kenig and J. Pipher. The Neumannproblem for elliptic equations with non-smooth coefficients. Inventiones Matematicae, 113:447-509, 1993.

32

Auscher and Tchamitchiam

15. X.T. Duong.H°°-functional calculus of elliptic partial differential operators in LO-spaces. PhDthesis, Mathematics, Physics, Computing and Electronics at Macquarie university, 1990. 16. V.G. Maz’ya, S.A. Nazarov, and B.A. Plamenevskii. Absence of De Giorgitype theorelns for strongly elliptic equations with complex coefficients. J. Math. Soy., 28:726-739, 1985. 17. P. Auscher and Ph. Tchamitchian. Square root problem for divergence operators and related topics, volume 249 of Ast~risque. Soc. Math. France, 1998. 18. Z.-Q. Chen, Z. Qian, Y. Hu, and W. Zheng. Stability and approximation of symmetric diffusion semigroups and kernels. J. Funct. Anal., 152:255--282, 1998. 19. M. Giaquinta. Introduction to regularity theory for nonlinear elliptic systems. Lectures in Mathematics. Birkhauscr, ETHZ/irich, 1993. 20. J. Peetre. On the theory of £p,~ spaces. J. Funct. Anal., 4:71-87, 1969. 21. P. Jones. Extension theorems for BMO.Indiana Univ. Math. J., 29:41-66, 1980. 22. Q. Huang. Estimates on genrealized Morrey spaces r.2’~ and BMO~for linear =¢ elliptic systems. Indiana Univ. Math. J., 45:397-439, 1996.

Approximate Solutions to the Abstract Cauchy Problem BORIS BJ~UMERDepartment of Mathematics/MS 084, University Reno, NV 89557

of Nevada,

ABSTRACT We introduce the notion of approximate, continuous solutions to deal with Cauchy problems which have, in general, only distributional or other generalized solutions. Weshow the equivalence of this notion of solvability with the regularized solutions introduced by I. Cioranescu and G. Lumer. As an example, we show that the backwards heat equation in f~ C ]Rd is wellposed in the approximate (or regularized) sense if and only if d =

1

INTRODUCTION

The aim of this article is to give an alternative point of view on regularized solutions (1) to the abstract Cauchy problem (ACP)

u’(t)

= Au(t);

u(O) = x; 0 <_ t <

as introduced by I. Cioranescu and G. Lumer[6],[7],[8]. continuous function v that satisfies

A regularized solution is a

v = A(l*v) + (l*k)x on [0, T), where here and in the following k will denote a regularizing, scalar valued L~oc-function with 0 E supp(k) and f,g

:= t ~ f(t-

s)g(s)

(l)~,,Ve investigate only scalar valued kernels in this article. Regularized solutions were defined with even more general operator valued kernels. Conditions on these kernels that still allow approximate solutions can be found in [2].

34

B.~iumer

denotes the convolution of f and g on ]R+. Any regularized solution v is the kconvoluted image of the "true" solution; i.e., if (ACP) admits a classical solution u, then v = k*u. Our approach uses the fact that we can always invert the process of convolution (see [2]): For any function e, e C0([0, T);X) and any regularizing function can construct a sequence ’u,~ ~ C([O,T);X) such that k * u,, -+ v nnitbrmly on compactintervals. This suggests the following definition of an approximate solution, converging to the "true" solution: DEFINITIONA k-approximate solution to (ACP) for some closed (~) linear operator A on a Banach space X with initial value x ~ X is a sequence of continuous functions u~ such that (i)

k*u~ -+ v for some v ~ Co([O,T);X),

(ii)

l,u,,

(iii)

k * (u,, - A(1 * un) - x) -~

e C([O,T);[7?(A)]),

Here [Z)(A)] is the domain of .4 equipped with the graph norm. Clearly, any k-approximate solution [u,] gives rise to a k-regularized solution v := lim k * u,,. One can think of the sequences [un] as elements in the generalized (a} function space C[t‘l ([0, T); X) := C([O,T); 11’11’ , the completion of the space of continuous flmctions with respect to the seminorms Ilfllt,,,T. := II k ,~ flirt. := sup0<~
2

PRELIMINARIES

Webriefly recall some theorems about the asymptotic Laplace trans~bnn and its inversion [1]. The asymptotic Laplace transform is an operator extension of the usual Laplace transform to all of L~oc, including functions that are not exponentially (2)Techniquesfor investigating the abstract Cauchyproblemwith nonclosedoperators are found in [31,[4], and [2]. (a)Fora moredetailed introductionto these generalizedfunctionspacessee [5] or [2]~

Approximate Solutions to Abstract CauchyProblem

35

bounded. It was first introduced by J.C. Vignaux in 1939 [13] and further developed by G. Lumer and F. Neubrander [12]. The asymptotic Laplace transform of a function f ¯ L~oc([0, T); X) is defined to be an equivalence class of analytic functions A(f~, X) defined in a post sectorial region(4) f~ of the complexplane; i.e, £A(f)={rEA(a,X):r~t~-~

e-~Sf(s)dsforallO
where r ~t q if r - q is of exponential decay(s) t. This set is always nonempty, meaning that CA(f) exists and is well defined for all f, and tim properties of the Laplace transtbrm extend flfily (see [12] or [2]). The asymptotic Laplace transform can be extended to transform generalized functions (6) f ¯ C[k] ([0, T); X) for k ~ n~oc[OT) with 0 e supp(k). If f E C[k] ([0, T); then g := k*f := [k*fn] ¯ Co([0, T); X), where f,~ ¯ C0([0, T); X)such that in C[k] ([0, T); X). The asymptotic Laplace transform of a generalized function f defined via CA(f) :=

CA(k) Thus, asymptotic Laplace transforms of generalized functions are, in general, coclasses of meromorphic functions. They have the property that for each element ] ¯ EA(f) we can construct a Miintz sequence(7) (B~),~er~ such that 1 where S := min{r; r ¯ supp(k*f)}. For a proof see [1]. For a proof of this fact and the following theorem (which is the backboneof this article), see [1]. The constants a,,,i,,~,,, N.,~ defined in the following theorem will be used throughout the paper. THEOREM 1 (The Phragm~n-Mikusifiski Inversion of the Laplace Transform) Let f ¯ Co([O, T); X), let f be a generalized funct ion in on e of th e Frechd.t space s C[~1 ([0, T); X) for some regularizer k ¯ L]oc[O, T) with 0 ¯ supp(k). Let ] ¯ CA(f) and (/3,,)ner~ be a M~ntz sequence such ~hat lim~ ~ In ~l](~n)l[ ~ O. Let e~ = max{~,’ ~ In I1~.,~.[(~,~)11)and n ~ ~. Define ~n,i

:=

~nie

choosea sequence -B’i

~ ~

~ ~nj

1 N,~ such that ~ ~., ~ T, as

--

~ni"

j=l

Then Nn

i=l

wherethe limit is taken in C0([0, T); X) or in C[~]([0, T); X), respectively. (’~)An open set f~ C ~ is said to be a post sectorial region if for all a < 7r/2 there such that {re i’° :r > b, 101 < a} C fL (5)A function r ~ #l(f~, X) is of exponential decay t if limsup.x_~c ~ .~1 In IIr(.k)ll _< (6)We use C{~"]([O,T); X) for simplicity of notation, of course the same construction for f ~ L~d~k]([O,T);X). (7)(~n)n~iN + is a M/i nt z seque nce if ~n

+l - fin >_I an d ~ n=~ 5’ ,’-~" "= ~

exists

b >

also works

36 3

B~iumer APPROXIMATE

SOLUTIONS

In this section we apply the Laplace transform results mentioned above to the abstract Cauchy problem. Weshow the equivalence of the notions of approximate solutions and the notion of regularized solutions for scalar valued kernels. Furthermore, for any problem that is wellposed in the regularized sense, we construct a sequence of exponential functions forming an approximate solution that converges in the best possible topology. Best possible in the sense that we construct a sequence u,~, independent of k, that will converge to u in the C[k] ([0, T); X)-topology for all regularizing functions k with u 6 C[k]([0, T); X). THEOREM 2 Let A be a closed linear operator on a Banach space X. Let k ~ LIo~[O,T) with 0 6 supp(k) and u,~ 6 C([O,T);X) for 6 ~. Thefoll owing m’e equivalent. (i)

Un is a k-approximate solution of (ACP).

(ii ) :=lir a k *u,is a k-regularized solution of ( A C P). Proof: (i) => (ii): Obvious by the closedness of (ii) => (i): Since 1 * v ~ C0([0, t); [/)(A)]) there exists EA(V) with T)(A) for all ), > 0. gn

wherefin is such that lim,~_~ ~A~In []~(fl,~)[ = 0 and the variables N,~ and a,~,~i are chosen as in Theore~n 1. Then u,~ ~ C~([0, t); [T~(A)]). By Theorem1, k * u~ uniformly in X on compact sets, and k * 1 * u,~ -~ 1 * v unifbrmly in [T)(A)] compactsets. Hence k * (u,, - A(1 * u,~) - x) -+ v - A(1 * v) - k * x = 0. Thus, the existence and uniqueness results for regularized solutions carry over to approximate solutions. COROLLARY 3 Let ~ # p(A). Let k 6 L~oc[O,T) wi~h 0 6 supp(k). f 6 C[~q([O,T);X). Suppose A ~-~ ]c(A)R(A,A)A/(A) 6 £A(g) for C0([0,T); X). Then

Let

i= 1

is a k-approximate solution to u = A(1 * u) + Proof: By assumption there exists .~ 6 £.A(g) with

-

=

Thus, by Laplace inversion, g is a regularized solution. With the same argmnent as ~ above, the assertion follows.

Approximate Solutions to Abstract CauchyProblem

37

Sufficient conditions on when an analytic function in some region in the complex plane is in the asymptotic Laplace transform of a continuous function can be found in [12] or [21.

4

WELLPOSEDNESS

In this section we explore someof the consequences and conditions for wellposedness in the following non-traditional sense. Wesay that (ACP) is wellposed if there exists a regularizing function k such that (ACP) admits a unique(s) k-approximate solution for all x E X. Wewill show that existence and uniqueness imply continuous dependence on the initial data and a nonempty resolvent set. Furthermore, we obtain a sharper version of Lyubich’s uniqueness theorem on the growth of the resolvent. These theormns are extensions of results obtained by G. Lumer and F. Neubrander in [12]. THEOREM 4 Suppose the abstract Cauchy problem has a unique approximate solution u(.,x) ~ C[~’I([O,T);X) for a11 x ~ X. (ACP)is ~vellp osed; i.e., ibr all 0 0 such that Ilu(.,a:)llt, < M,II:,,’II on [o,t]. Proof: Let S(.)x := k * u(., x). Thus, by Theorem2, Sx = A(1 * Sx) + (1 * k)x. By the existence and uniqueness property, we knowthat S : X -~ C0([0, T); X) a linear operator. Furthermore, Xn -+ x and Sxn ~ v imply that 1 * Sxn -+ 1 * v and, by the closedness of A, that A(l*v) = v - (l*k)x. By the uniqueness property, Sx = v, and hence S is closed. By the closed graph theorem, S is bounded and hence for all 0 < t < T there exists M~> 0 such that

Ilu(-,x)lh.,,IlSxl[,= sup IIS(s)xll<Md lxll. It was shown shown by, G. Lumer and F. Neubrander [12] that any function that is analytic in a postsectorial region and satisfies a certain growth condition is in the asymptotic Laplace transform of a. continuous function. Suppose a meromorphic function on such a region multiplied by the asymptotic Laplace transfor~n of a regularizer yields an analytic function satisfying the growth conditions. Then, this meromorphicfunction is an element of the asymptotic Laplace transform of a generalized function. Thus, the existence of a meromorphicresolvent(9) on a postsectorial region with some growth condition implies wellposedness. (8)Uniqueness,of course, is taken in the sense that the approximating sequencesrepresent the samegeneralizedflmctionin CDI([O,T);X). (9)Remark.Just havinga discrete point spectrumdoes not imply that the resolvent is ~neromorphic.Vector-valuedmeromorphic functions are quotients of vector-valuedand scalar-valued analytic fuuctions. Thus, the poles cannotforma Miintz sequence.

38

B~iumer

The next theorem shows that wellposedness has implications on the resolvent set and on the growth behavior of the resolvent of the operator ,4. (See [2] or [12]) THEOREM 5 If (ACP) is wellposed, then every Mfintz seq~ence has a subsequence (flnj)j~ sucl~ that fl,~ is in the resoh,ent set of A and lim sup ~ In I[R(~n~, A)II <

Proof: If (ACP) is wellposed, then there exists family (S(t))~6[0,T) such

=

a strongly continuous operator

+ k)x.

Furthermore, let 0 < t < T, and t

j~0

~(A)x := e-~S(s)x

ds.

By the previous Theorem, (1 * S)(t) ~ £(X, [/)(A)]) ~(),) ~ £(X,[D(A)]) and (~(,~))~d:

for all 0 _< t < T, and

is a strongly analytic

operator

[~ e £A(k). Then (~ - A)~(,~)x = ]¢(A)x + r.~(,~), where r.~ ,,~t 0 for all x ~ X. Furthermore, r(.)(,~) r.~ ~ 0, there exists M.~such that

~ £(X). Let 0 t, /2. Since

for all ,~ > 0. By the uniform boundedness principle there exists a constant Msuch that r(A) : x ~ r~(A) satisfies [Ir(A)[I _< -n(’-~). Let (~n),~eN be a Miintz sequence. Since 0 ~ supp(k), there exists a subsequence (.~.n~)je ~ such that limj_~ ~ In l~(~n~)l = 0 and thus there exists a constant .] such that ]~(~,~s)~ ~ e-~"~ > 2~r(~.n~)~ for all j > J. Then ~(.o,~) [] < 1/2. Thus Id + ~ is an invertible ,, ~(.~. ~ ) k(~.~ maps onto X and

Since/~(~) ¢ 0 is scalar-valued,

the operator

(~3,,~ - A)#(/3,,~) =~(/3,,~)(td

operator that

ApproximateSolutions to Abstract CauchyProblem

39

is a continuously invertible operator. Let

0(~,~):= (~’(~)(~d k(/G)" ] and define Then,

[IQ(fl,~j)ll ~ and (f

lnj -A )R(flni)=

Weshow next that (fl~ - A) is one-to-one. D( A ). Then

Id . Suppose fl.n~x Axfor some

(ez"~ t - 1)x = A (eZ"~ t - 1)z = A e’Z~’xds. J0

Since the solutions to ACPare unique, we deduce that S(t)x = (k * e~"~(’I)(t)x. Thus ~(,k)x is a scalar multiple of x for all ,~ ~ ¢. Hence,

o = (/G-A)g(Z,,~)z= ~(Zn~)(~ + ~(Z~)z). Therefore, II- xll = II [(fl"~)’~ I < Ilxll/2, and thus x = 0. Hence (fln~ - A) is ~,(,&~ ) one-to-one and thus R(fln~) = R(fln~, A) is the resolvent of A at fln~. Since IIQ(~)II _<2/l~(fln~)l,weknow that 1 limsup In IIQ(flns)ll < j-~oc ~n~ and hence the same holds for R(flni) = ~(flni)Q(fln~). As corollary to Theorem1 we obtain that existence and growth of the resolvent yield the tbllowing extension of Lyubich’s uniqueness condition, see [11] or [3]. COROLLARY 6 (Lyubich’s Uniqueness Theorem) If there exists (fl~),~e~ such that fin ~ p(A) for all n and

a Miintz sequence

1 lim sup _--In IIR(~n, A)/I < then every approximate solution is unique. Proof: Suppose u is a k-approximate solution of u’ = Au; u(0) = 0. Let fi ~ £A(U). Then there exists r ~ L:A(0) such that ~(~n) = R(fl~, A)r(fln) and thus

II~(fln)ll _
~

40 5

Biiumer EXAMPLE:

THE

BACKWARDS

HEAT

EQUATION

Consider the Laplace operator -A on a bounded open set fl C JR3t with smooth boundary and Dirichlet or Neumannboundary conditions. Then it is well known (see for example[9]) that the eigenvalues ,~a grow like n2/d. Thus, if d >_ 2, the point spectrum is forming a Miintz sequence. This implies that any regularizer ~ that regularizes the resolvent would have to be zero on a Miintz sequence; otherwise A ~-~ ~(A)R(A, A) can not be analytic on some region containing (w, c~). But in turn implies that k = 0. In one dimension, consider 12 = (0, ~). Weuse the eigenfunctions of the Laplacian; i.e., let en(x) := sin(nx) for the Dirichlet problem, and e,~(x) := cos(nx) for Neumannproblem. Every function f ¢ L=[0, 7~] can be expressed in terms of its "~ e,~. Let eigenfunction expansion f = Y’]-,~--0 a~en and thus R(A, A)f = ~,~=o~_--2~a 1

cx~

2

n -i

~(~1:= iz ,~0 ~-~ ~ Clearly, A ~ A’~c(A) is bounded and analytic in a right halfplane arid therefore ]¢ is the Laplace transform of a continuous function (m) (see, ibr example, [3]). order to show that k satisfies the conditions for a regularizer we have to show that 0 ~ supp(k). Weuse the basic fact of Laplace transform theory that min{x : x fi supp(k)} = - lira supA_~¢~ In [~(A)[. See, for example, [10] Satz 14.3.1 or [1]. Since 1 O-<-li~n2~up~ln

fi n2+A .:o n2 - A

=

° lim inf 1,~ In

<_

liminf 1 ~ ~-~ ~ n=O ln(1

<

lim inf ~-’.

2A ) A[ l/t2+ -- --

2

we obtain that 0 ~ supp(k). Since 1 oo [ 1

j2-A

c+~

tbr A in the open right halfplane, ~(X)R(A,A) is the Laplace trausibrm of a strongly continuous operator family. Thus we can apply Corollary 3. Let (~,~),~ be the.sequence of positive integers excluding the square numbers. Then

i=l

~=1 O°)The

function

~ exp(-~)sin(~)

k :

~ ~ ~sin

~ is

another

has zeros at ~ ~.

function

j:O whose

Laplace

transform

~ :

A

Approximate Solutions to Abstract CauchyProblem

41

is a sequence of exponential functions that converges in the best possible toPology towards the solution of the backwards heat equation u’(t,x)

- -Au(t,x);

u(O,x) f( x).

Notice that the sequence is independent of our choice of k, which might not have been the optimal choice. ACI(NOWLEDGMENT I would like valuable suggestions.

to thank Professor

F. Neubrander for many

REFERENCES 1. 2.

3.

4. 5.

6.

7.

8.

9. 10. 11. 12.

13.

B. B~iumer: On the inversion of the Laplace and convolution transfortns. Submitted. See http:/math.lsu.edu/~tiger/evolv/evolv.html. B. Bgumer: A Vector-Valued Operational Calculus and Abstract Cauchy Problems. Dissertation, Louisiana State University, 1997. See http:/math.lsu.edu/-~tiger/evolv/evolv.html. B. B~umerand F. Neubrander: Laplace transform methods for evolution equations, Conference del Seminario di Matematica dell’ Universit£ di Bari 259 (1994), 27-60. B. B~iumer and F. Neubrander: Existence and uniqueness of solutions of solutions of ordinary differential equations in Banachspaces. Preprint, 1995. B.B~iumer, G. Lumer, and F. Neubrander: Convolution kernels and generalized functions, in Generalized Functions, Operator Theory, and Dynamical Systems. C.R.C. Research Notes in Math., Chapman& Hall, 1998. I. Cioranescu: Local convoluted semigroups, in Evolution Equations (Baton Rouge, LA, 1992), 107-122. Lecture Notes in Pure and Appl, Math., 168, Dekker, NewYork, 1995. I. Cioranescu and G. Lumer: Probl~mes d’~volution r~gulal’isSs par un noyau g~n~ral K(t). Formule de Duhamel, prolongements, th~or~nes de g~n~.ration. C.R. Acad. Sci. Paris 319 (1994), sfiries I, 1273-1278. I. Cioranescu and G. Lumer: On K(t)-convoluted semigroups, in Recent Developments in Evolution Equations, Pitman Research Notes in Math. 324, 1995, 86-93. R. Courant and D. Hilbert: Methods of Mathematical Physics, Volume I, Inter-science publishers (a division of John Wiley and Sons, NewYork, 1953. G. Doetsch: Handbuchder Laplace Transformation. Birkhguser Verlag, BaselStuttgart, 1950. Yu. L. Lyubich: The classical and local Laplace transformation in an abstract Cauchy problem, Usepi Math. Nauk 21 (1966), 3-51. G. Lumer and F. Neubrander: Asymptotic Laplace transforms and evolution equations, in Evolution Equations, Feshbach Resonances, Singular Hodge Theory, Advances in Partial Differential Equations, 16, Wiley-VCH,1999. J.C. Vignaux: Sugli integrali di Laplace asintotici, A~tti Accad. naz. Lincei, Rend. C1. Sci. fis. mat. (6) 29 (1939), 345-400.

Smart Structures A. V. BALAKRISHNAN Flight

and Super Stability Systems Research

Center,

UCLA

ABSTRACT Let S(t), t >_ 0, denote a C0-semigroup over a Hilbert space 7/. We say that the semigroup is superstable if the indicator wo =t-~lim

(L°g"S(t)’[)

In this paper we give an example of structures. Specifically, we consider with embedded sensor-actuator strips there are no "modes" (eigenvalues) for superstability being a consequence.

1

= -cc’t

such a semigroup in the theory of "smart" Timoshenko models of smart beams -- beams -- with pure rate feedback, and show that a critical value of the feedback gain, the

INTRODUCTION

Let S(t), t _> 0, denote a Co-semigroup over a Hilbert space ~/. Wesay that the semigroupis superstable (see [1]) if the indicator wo = lim (Logl[S(t)[[)/t

=

The spectrum of the generator must then necessarily be empty. Super stability is a truly infinite-dimensional phenomenon-- 7/has to be infinite-di~nensional. Amongthe first examples of such semigroups is one given in Hille-Phillips [2, p. 664, et seq.]. A special case where the semigroupis actually nilpotent:

S(T) = O, T>0 arising as the solution ("disappearing solution") of a boundary value problem for hyperbolic partial differential equation was given by A. Majda in 1975 [3]. In this paper we shall show how they arise in stability enhancementof structures with self-straining material -- "smart" structures. Specifically, we consider Timoshenko models of smart beams -- beams with embedded sensor-actuator strips

44

Balakrishnan

-- with pure rate feedback, and show how super stability may be achieved at a critical value of the feedback gain. An example of a Timoshenkomodel limited to torsional motion only (alias a "smart string") was given in [4] where the semigroup is nilpotent. As noted in [4], what makes super stability possible is that these are examples of vibrating systems in which the mass-inertia operator is singular -- does not have a bounded inverse.

2

TIMOSHENKO

SMART

BEAM MODEL

Weconsider a beam of length ~, with a sensor-actuator strip along its entire length. In the Timoshenkomodel, limiting ourselves to one bending displacement ](t, s), 0 < t, 0 < s < e, and one rotation angle ¢(t, s), we have (superdot indicating time derivative, and the prime indicating space derivative): rn2i)(t,s)

- c4(v"(t,s)-¢’(t,s))

m6~)(t,s) - c2¢"(t,s)

- c4(v’(t,s)

0, 0<s <e, 0
¢(t, o) = v(t,o) c4(v’(t, g) ¢(t, g) ) + a~)(t,g) = c2¢’(t,e)

(2.1)

¢(t,s)) =

+ a~(t,e)

/ (2.2) (2.3)

Here the sensor strip generates a charge proportional to ¢(t, e), ~(t, and the actuator strip generates a momentproportional to the current, a being the feedback gain, 0 < a. What is characteristic of these smart structures models is that they may be considered a "singular" version of a beam with collocated point controllers in which the boundary mass/inertia terms are zero: Thus consider (2.1) with

.~,,~(t,e) + ~(.’(t,~)- ¢(t,~)) + ~(t,~) = 0 ~ ,no~;(t,~)+ c~¢’(t,e)+ ~(t,e) where m., me are the point mass and inertia respectively. This is "collocated rate feedback" which is knownto be stable -- see [5]. The total energy decreases to zero; the corresponding semigroup is strongly stable but not exponentiMly stable. The indicator

wo= 0.

(~,,.5)

Weshall show that (2.1), (2.2), (2.3) leads to a strongly stable dissipative tsemigroupfor all (~ > 0 and for a critical value of a the semigroupis superstabl.e, Webegin with an abstract formulation of the problem. Thus let

~. 7/ = L~[0,~]~× E ?Thisis of coursea "theoretical" result. Whatthe limitations are in practice that make~his phenomenon impossibleare beyondthe scopeof this paper -- suffice it to say that it is hungup with the questionof whetherwecan implementa perfect differentiator.

Smart StructureandSuperStabiiity

45

Define the linear operator A with domainand range in 7{ defined by

v(.) v’(.),v"(.)L2 [0,~] ¢(.) ¢’(.),¢"(.)e L2[0,/~] v(e) v(0)= ¢(0)

/)(A)

and

c4(¢’- v")

v(.) A ¢(’)

-c~¢" + c4(¢- v’)

~(e) = ¢(e)

c4(v’(t)¢(e)) c~¢’(e)

Thusdefined, ,’4 is closed, with densedomain,self-adjoint and nonnegative-definite. In fact [Ax, x] = c2 I¢’(s)l 2 ds + c4 Iv’(s)- ¢(s)l ~ ds, wherethe right side is the elastic energy. Wenote that zero is not in the spectrum of A. Further

implies that x is of the form

v’(.),¢’(.)

v(-) ¢(.) v(e) ¢(e)

e L2[0, e]; v(0) = ¢(0)

Withthese definitions, (2.1)-(2.3) goes over to the abstract waveequation M!i(t) + Ax(t) + aBB*2(t) whereB is the :’control" operator mappingE2 into 7{ defined by

and resemblesthe collocated point controller formulation[5], except that now 0

0

0

m6 0

0

0

0 0

0

0

0 0

0

m2

0

and is singular. In particular wecannotfollowthe usual procedure[5] for constructing the semigroupwhichrequires the inverse of M. Henceweproceeddifferently. Let 7{E denote the "energy" space: 27{E = D(V~)× n2[0,~]

46

Balakrishnan

where the elements are of the form: Vl ¢1

Vl

vl(e) , ¢~(e)

vx(e) ¢1(e)

V2

e T) vr~

2¢2 e L2[0, g]

with the "energy" inner product:

where Y=

V2 , Z =

V4

Note the absence of the kinetic energy in the boundary values. Thus defined, is of course a Hilbert Space. Next we define the operator A with domain and range in ~HEby:

V(A) = [ ¢ and V2

V!

A

vl( e)

:~ (v[(e) :_a

¢~(e) 732

-M O-~,40 where ’/~2V

V

-c~¢"- c4v’ +c4¢

rn6¢

Note that in particular we are implicitly requiring

v2(e) + CA(v~(e)-¢~(e))= 0, ¢~(e) + ~Ael(e)

SmartStructureandSuperStability

47

because of the condition that V2

:~ ¢~(e) It is easy to verify that A is closed with dense domain, and more importantly that .4 is dissipative: 2

RelAY, Y] = -c_~ i¢~(e)12 _ ~(v~(~) -

2 ~

el(e))

0

where Vl

y = vl(e) e z~(A). ¢1(e) V2 Hence A generates a Co contraction semigroup for each c~ > 0. Wecan now state our main result. THEOREM 1

Let the stiffness,

mass parameters be such that c2 c4

rn2 m6

Then for a

=

~

:

x/~m2

(2.6)

the spectrum of A is empty -- no "modes"/eigenvalues. PROOF Wenote first that the resolvent of .4 is compact for all parameter values. Weshall show that for the confluence of parameters assumed in the Theorem, the spectrum of J[ is empty. LEMMA 1 The eigenvalues of ,4 are the zeros of the entire function of A: D(A) = detH(A;e)

where H(A;g) =~ C1Hx2(A;e) + C2 !g12(A;g)

(2.7)

where eA(A) s =

, 0<s<e

(2.8)

48

Balakrishnan 0

0

1

0

0

0

0

1

A~u~

0

0

1

-c

0

0

c+

A~

(Here A(A) is to be distinguished from

C1

=

where

m2 m6 _2 = C4 -- ; t~ = --C2 ;

/)2 PROOFBy straightforward

calculation

C =

c4 C2

of the eigenvalues and eigenfunctions.

The particular confluence of parameters can be motivated by considering the eigenvalues for large IAI.

Asymptotic

Analysis

For large [A[ we can use the approximation: 0

0

1

0

0

0

0

1

0

0

0

>,2u~

0

0

A(A) 0 leading to the eigenfunction solution AY = AY v

Y = Av

v(s) = A.v sinhAu2s ¢(s)

= AcsinhAu6s,

SmartStructureandSuperStability

49

(the system (2.1) is decoupled and we go back to the model in [4]), and A is determined by the end conditions at s = g: O~/~V(e)

-I-

C4(Vt(e)-

=* Av(aAsinhAu2e

¢(e))

+ c4,~u2coshAu2~)

- A¢c4sinhA/1~

= 0

c2¢’(e) + ~a¢(e) A¢c2A/1~ coshAu~f + AaAcsinhA/2~f = 0. Hence (a sinh

+ c2/1~ cosh A/1~)(a,\ sinh A/12t! + c4/22 cosh A/22~)

Hence either asinhAu6£

+ c2v6coshAu6*

= 0

or~

asinh A/1~ + c4/12 coshA/1~ = 0. Hence for O~ :

C2126

= C4/22

we have sinhA/16g + coshv~ = 0 = sinhA/12~ + coshA/12~ or we have no eigenvalues, for [A[ sufficiently large. Let us thus show that D(A) is an entire function with no zeros for O~ = C2/26

= C4//2

and in particular

To simplify the notation we shall take c=l;

~ = 1.

(2.9)

The main calculation is: LEMMA 2

D(a)

=0

as lal ~ ~o.

PROOF Weshall need to be brief to comply with the page limitation, problem is in evaluating e A(’~)s, 0<s and in particular H~2(A, s).

The main

50

Balakrishnan

The specialization (2.6), (2.9) yields C1 = C2 0 Au6 As may be expected the eigenvalues of A(,~) play an important role. Now

and using (2.9) the roots can be expressed:

and we simplify further by taking v2:l.

Defining

= -A V/~ +i $,

ReA < 0

ReA >0 ReA < 0 we see that Pl, P2 are analytic in the respective half-planes, and we can calculate

H(A,s)

=

1

~ sinh#ls + ~ sinhp2s A- coshlZl8 -I- coshp2s

-i cosh p~ s + i cosh #2 s Ni~sinh#ls+

~i~sinhlt=s

~ sinh Pl s + cosh tt~ s + ~ cosh#2s + coshp2s

~ sinh #1 s - ~ sinh + i cosh #1 s - i cosh

where either determination of #1, #~ can be used. Hence D(A) (P -~ +

P- ~)sinh,~lsinhp~ + + ~-

+

2c oshp~cosh#2

sinh #~ cosh#l +

we can readily see that e2)~

~ -y-

as 0¢.

+

sinh Pl cosh

SmartStructureandSuperStability

51

Next let r(£)

D’(A)- 2D (£) D(£)

To show that

we take ReA > 0 and note that A(All(A) + A22(A))e(~1+"2) + A(All(A)

~,.(~)

where A~(£) = a~(£)’

+ a~_(A)(#~ +t t2)’

A22(A) = a:2(£)’

+ a~l(£)(p~

A~2(A) = a12(~)’

+ a21(~)(#~-#2)’

-

A2~(A) = a2~(£)’

a~(A)(#~-#2)’ -

2a:~(A)

where all(A)

=

1 +

, al2(A)

=

+$+

1+

~

a2~(£) =

+~--

1+

~

But (Al1(~) + A22(~)) (a11(~) + a22(~)) and

Hence

= 2

(an(A) + a~2(X))’

+

52

Balakrishnan

A similar argmnent goes for Re A < 0. Finally, to prove the Theoremwe simply note that the number of zeros D(,k) a circle of radius R: n(R)

= 1 J~c’

2~--~ D’(A) . D(~) dA, CR-- [A [IAI
_ 1 /e r(A) 2~ri n

dA

and the estimate of Lemma2 shows that n(R) --~ as R - -~ Hence for every R>0, or,

D(~) has no zeros. Q. E. D. COROLLARY ,4 generates

a superstable

semigroup:

PROOFThe proof, which will appear elsewhere, makes essential use of the fact that we have a true semigroup -- zero is in the spectrum of the semigroup; as well as the finite beamlength. Thus we have shown that the boundary value problem:

~(~,s) - (v"(t,s) - @’(t,.))= 0, ,~(t,~) - ¢"(t,~)(v’(t,s)- ¢(t,s))) = o,o <.~ ¢(t,o) = ~(t,o) v’(t, e) ¢(t, e)+ ~(t, e) ¢’(t,e) $(t,e) = has no eigenvalues -- rio "modes." Further, the solution decays faster than any exponential. REMARK It would be interesting

to see what an FEMapproximation yields.

ACKNOWLEDGEMENT Research supported in part by NASAGrant NCC 2-374. I am indebted to J. Lin for checking the manycalculations. Special thanks also to I. Lasiecka for her helpful comments.

SmartStructureandSuperStability

53

REFERENCES 1.

2. 3. 4.

5.

Balakrishnan, A.V. (1996) On Superstability of Semigroups.In: Proceedings 18th IFIP TC-7 Conference on System Modelling and Optimization, July 1997. Pp. 1-8. Hille, E. and Phillips, R.S. (1957) Functional Analysis and Semigroups. Colloquium Publications, American Mathematical Society. Majda, A. (1975) Disappearing Solutions for the Dissipative WaveEquation. Indiana University Mathematics Journal, 24: 1119-1133. Balakrishnan, A.V. (1996) Vibrating Systems with Singular Mass Matrices. Proceedings of the First International Conference on Nonlinear Problems in Aviation & Aerospace, Daytona Beach, Florida, 9-11 May1996. Balakrishnan, A.V. (1997) Dynamics and Control of Articulated Anisotropic Timoshenko Beams. In Dynamics and Control of Distributed Systems, ed. H.S. Tzou and L.A. Bergman. Cambridge University Press.

On the Structure of the Critical Spectrum of Strongly Continuous Semigroups MarkBlake St. John’s College, OxfordUniversity, Oxford, England SimonBrendle GraduateSchool of Science and Technology, KobeUniversity, Kobe, Japan Rainer Nagel MathematicsInstitute, University of Ttibingen, Tiibingen, Germany The failure of the spectral mapping theorem cr(T(t))

{0} = et ~(A), t >_O,

(SMT)

for a strongly continuous semigroup (T(t))t>o with generator A on a Banach space X is the main reason for manydifficulties arising in the investigation of the asymptotic behavior of these semigroups (see [5], Chapter IV and V). To overcomesome these difficulties, the essential spectrum aess (T(t)) has been used by manyauthors and for manyconcrete equations. Morerecently, the critical spectr"o,m acrit(T(t)), smaller than ~ress(T(t)), has been introduced in [9] and then applied in [3] to perturbed semigroups. In this paper, we prove symmetry properties of the critical spectrum thereby explaining certain phenomenaarising in the counterexamples to (SMT)discussed Wrobel [14]. In the final section, we study a nonautonomousdifferential equation with infinite delay and show, by replacing the essential spectrum by the critical spectrum, howone can obtain new criteria for uniform exponential stability.

1

THE CRITICAL

SPECTRUM

Webriefly recall the main definitions we associate the sequence space

and results from [9]. To the Banachspace X

S.":=e~(x) ={(~,,),~e,~ c X:sup fl:c,JI endowed with the sup-norm. The strongly continuous semigroup T = (T(t))t>_o on X will be extended to a (not necessarily strongly continuous) semigroup ~" (~(t))t>_o on :~," by

56

Blakeet al.

This semigroup has the space of strong continuity flit := {(x,~),~er~ e ~’: limsupHT(t)x.~ - x,~ll = 0}, ~$0 n6N

which is closed and (~(t))~0-invariant.

Therefore, on the quotient space

the quotient operators ~(t)(:~

+ .¢r) := ¢(t)~

~ + .t~w ~ ~,

are well defined and yield a seinigroup ~ = (~(t))t~o of bounded operators. DEFINITION 1 The critical

spectrum of the semigroup (T(t))t>o acra(T(t))

while the critical

:= a(:~(t)),

is defined as

> O,

growth bound is

there exists M,o such that wcrit(T) := wo(J-) := inf {wE IR: 1{7~(t)ll <_ M~,e~,t for all t > 0 }’ The critical spectrmn is smaller than the essential spectrum and, e.g., trivial for eventually norm continuous semigroups. However, it is large enough to ensure the following modification of (SMT)(see [9], Theorem3.2). THEOREM 2 For a strongly one has a(T(t))

continuous senqigroup (T(t))t>_o

{0} = ew(a) U a~rit(T(t)) \

and therefore the spectral bound s(A) and the critical termine the growth bound wo(T) by

{0}, t

with generator

A

> O,

growth bound Wcrit("]") de-

wo(T) max{s(n), w~rit(T)}. In the following proposition, we give a condition on the approximate point spectrum Aa(A) or on the resolvent R(A,A) of the generator .4 in order to produce points in the critical spectrum. To that purpose, we need the following topological property of the exponential function (see [14], Lemma4.3). For abbreviation, put F := {# E C: lit I = 1}. LEMMA 3 Let (A,,),~e~ be a sequence of co~nplex numbers satisfying limn-~ReA,~ = w and lirn.-~oo }ImAm]= o~. Then, for ahnost all t >_ 0, each point # ~ ewtF is ~n accumulation point of the sequence (eA-t),~e~. PROPOSITION 4 Let t _> 0, # ~ C and (An),,e~ be a sequence of complex numbers satisfying limn-~o~ Jim AnI = c~ and limn-~oo e ~’‘t = it. For the generator A of a strongly continuous semigroup (T(t))t>o we assume that either (A,~).~w Aa(A)

(*)

57

Critical Spectrumof Strongly ContinuousSemigroups or

(A,~).er~ p( A)

and

lim

(**)

IIR(An, A)ll :

Then we obtain

~ e ~.(T(~)). PROOF fying

Under both assumptions, there exists a. sequence (x,~).ne~ C D(A) saris-

for all n ¯ N and lira [[A,~x,~ - Ax,d[ = O. Then we have

for all h >_ 0. Wenow put lim,,~c R,e A,~ =: w. Using Lemma3, we obtain

supIIT(h)x. - x.II _>’°~’+1 for almost all h > O. Therefore, the semigroup (T(t))t>o is not uniformly continuous on (xn)~e.~, i.e.,

On the other hand,

II~z. - T(t)xnll= lira Ilea"t=~- T(t)x.II = lim

/o’

e’X"(t-S)T(s)(Anxn

- Ax,J

=0. This implies (#xn -T(t)x,~),,eN

¯ co(X)

From this we conclude that # is an eigenvalue of ~(t), hence # acrit(T(t)).

I

58

Blakeet al.

EXAMPLE 5 Let X := C0(fl) or X := LV(~,p), where fl is locally compact and (~,#) is a a-finite measure space, respectively. Further, let (Tq(t))t>o be the strongly continuous semigroup generated by the multiplication operator IQqf := qf for some function q : ~ ~ C satisfying sup Re q(s) Then one has

# e ocr (Tq(t)){0} if and only if there exists a sequence

of complex numbers such that lim,,_~ Ihn Anl = oc and lim,~-~oo e ~-t = #. For the,, proof we refer to [9], Example2.8 (ii).

2

ROTATIONAL

SYMMETRY

OF

THE

CRITICAL

SPECTRUM

Herbst [6] and Wrobel[14] noted that if (SMT)fails in the knownexamples, it fails dramatically in the sense that the sets a(T(t)) ta(A) become quite lar ge. In thi s section, we show how this phenomenoncan be explained by the critical spectrum. THEOREM 6 Let (An)ne~ be a sequence of complex numbers satisfying lim~_~ Re An = w and lim,~_~ IIm A,~I = oc. For the generator A of a strongly continuous semigroup (T(t))t>_o we assume that either ()~,~),,~

C Aa(A)

(,)

or

(A..),~eu

p(A) and

lira [IR(A,,,.4)I[

: oc.

(**)

Then, for almost all t _> 0, we have e’~’tF C a~t(T(t)). PROOFBy Lemma3, each point tte e~tF is an accumulation point of the sequence (eA-t),~e~ for almost all t _> 0 . Proposition 4 implies that each point # ~ e~tF belongs to a~r~t(T(t)) for ahnost all t > 0 . [] Using the following abscissa of uniform boundedness of the resolvent

F(A)

e a:

there exist sw and C~, such that r + is ~ p(A) and HR(r + is, A)l I < C~, for r > w and Isl _> s~ ’

we obtain that the corresponding circles are contained in ac~u(T(t)) for almost all t >_ 0. (For other abscissas and their significance for the asymptotic behavior of semigroups see [13].)

59

Critical Spectrumof Strongly ContinuousSemigroups COROLLARY"7 We have

for almost all t > O. In particular,

we obtain the estimate s~(A)

In case X is a Hilbert space, it has been shownin [1], Theorem4.4, that s~(A) COROLLARY 8 Let X be a Hilbert

space.

Then

e~or"(TIr c ac~t(T(t)) for almost all t ~ 0. For multiplication semigroups, it follows from the description of the critical spectrum in Example 5 that the condition in Theorem6 always holds for points in the critical spectrum. COROLLARY 9 Let (T~(t))t~o be a multiplication semigroup as in Example 5. T~ena~.~(T~(t)) consists of full circles for ~lmost all t ~ 0, i.e.

~,(T~(t)) . r =~,(r~(t)) for almost all t > 0. Finally, we consider positive semigroups on Banachlattices certain dichotomic behavior.

(see [8]) and obtain

COROLLARY 10 Let (T(t))t~o be a positive semigroup with generator Banach lnttice X and assume that the peripheral spectrmn

A on

~+(A) := {~ e ~(A): a~ = ~(d)} is cyclic. If WorSt(T) < s(A), then we have a+(A) = {s(A)}, i.e., Re A < s(A)

~ora~~ e ~(d)~ {~(A)}. PROOFIf s(A) + ir a( A) fo r so me r ~ ~, the n the cycl icity of a+(A) implies s(A) + inr ~ a(A) for all n ~ N. By the ~sumption Wcr~t(T) < s(A) and by condition (.) in Theorem6, we conclude r = Semigroups satisfying the condition Wcr~t(T) < s(A) are called asymptotically norm continuous. This class of semigroups has been studied intensively in the papers by Martlnez and Maz6n[7], Blake [1] and Thieme [12]. The opposite situation is considered in the following result proved by W. Arendt and O. E1 Mennaoui (oral communication).

6O

Blakeet al.

Let (T(t))t>o be a positive semigroup with THEOREM 11 (Arendt-E1 Mennaoui) generator A on a Banach lattice X. If s(A) < Wcrit(T), then

[ds(A),d~o’’’(r)]c ~c,.~(T(t)) for all t > 0. PROOFWe first assume that s(A) < 0 ~ w~.(T). By positivity exponential stability, i.e., there exist constants M, g > 0 such that

this

implies

~]T(t)xH ~ Me-~tJ]Az]~ for all x e D(A) (see [5], Proposition VI.I.14 or [10], Theorem1.4.1). Assumenow that 1 ¢ acr~t(T(t)) for some t ~ 0, which, by Theorem2, implies that

1 ¢ ~(T(t)). Then R(1,T(t))x

= ~ T(nt)x >_

for all 0 _< x ~ D(A). The density of D(A) ~ X+in X+then implies R(1, T(t)) >_ Therefore, we use [11], Exercise V.5.(e) to conclude that r(T(t)) hence

~o(T)< By Theorem 2, we must have

Wcr~t(T) < which is a contradiction.

Thus, we conclude that

for all t _> 0. By rescaling, we obtain [e~(A), et~’’’(T)] C a¢,.it(T(t)) for allt >0.

[]

A comparison of these results with the known examples of semigroups having s(A) wo(T) le ads us to the foll owing openprobl em. PROBLEM 12 Do there exist strongly continuous semigroups (T(t))t>o such that the critical spectrum a~rit(T(t)) does not contain full circles for almost all t ~: 0?

61

Critical Spectrumof Strongly ContinuousSemigroups 3

SMOOTHING

OPERATORS

In this section we give another characterization of the critical growth boundWc.rit(7~) of a strongly continuous semigroup (T(t))t>_o with generator A on a Banach space X. For a systematic discussion of this and other growth bounds we refer to [2]. DEFINITION13 An operator S E £(X) is called a smoothing operator for (T(t))t>_o, if limh$0 II(T(h) - Id)SII = Moreover, we denote by $ t heset of a ll smoothing operators in £(X). PROPOSITION 14 The subspace 8 is a closed right ideal in t:(X) containing all resolvent operators of A. PROOF It is trivial to check that $ is a right ideal. To see that R(~, A) E $ notice that t ~ T(t)R(A, is right n___orm con tinuous for t _>0. To show that $ is closed, let C := lim h¢ollT(h) - Idll. If C = 0, then $ = £(X). Otherwise take Sn ~ S in £(X), Sne 8, and let e > 0. Then there exists N ~ such that [IS - Snll <_ ~ for n _> N, and II(r(h) - Id)SII <_liT(h) - Idll IIS - S,dl + ll(T(h) for h, _> 0. Letting h $ 0 we obtain lim hJ,0ll(T(h) Id)Sll <_ e.

[]

For B ~ £(X) we now define p~(B) := inf{llB- S[]: S ~ $}, p~(B) := limh+ol](T(h) - Id)Bll, p3(B) := lim ~_~IIAR(~, A)Bll. PROPOSITION 15 The functionals

RI,P’~,P3 define equivalent seminorms on I:(X).

PROOFIt is easy to check that each pi defines a seminorm on £(X). Let C lim h,ollT(h) -Idll. Then C > 0 unless (T(t))t>_o has a bounded generator in which case each p~ is identically zero on £(X). Wesuppose that this is not the case, and show that p~(B) <_ on(B) <_ p.~(B) <_ Cpt(B) for all B e £(X). Suppose w > Wo(T), so there exists M> 0 with IIT(t)ll A>w, 5>0, andxeX. Then we have IIAR(,~, A)Bxll = II(~R(~, A) - Id)Bxll

<_ w~ for al l t _>0. Let

62

Blakeet al.

Letting A -~ ~ and then 5 J. 0, we obtain that pa(B) _< p.~(B). By Proposition lIAR(A, A)B

14, AR(£,A)B E $, so p~(B) <_ pa(B) since IIAR(A,A)BII

Finally suppose pl(B) < ~. Then there exists S 6 8 such that [IB - S[[ :_< a. Then

<_ lim h,o[lT(h) -- Idll

liB-

< C~. This holds for all c~ > pl(B), so we conclude p2(B) < Cp~(B).

[]

COROLLARY 16 The following assertions hold. (1) S ~ ~ if and only if AR(A,A)S -~ S in L(X) as A -+ co. (2) S is the closure in/:(X) of the right ideal generated by the resolvents R(A,

e p(A). PROOF (1) follows from the equivalence of the smninorms P2 and Pa in Proposition 15. To prove (2) we let $’ be the norm closure of the right ideal generated by the resolvent operators R(A, A), A ~ p(A). Then $’ C S by Proposition 14, and 8 by (i). PROPOSITION 17 The critical growth bound w~it(T) boundof the function t ~-~ pi(T(t)) for i = 1, 2, 8.

4

coincides

with the growth

AN APPLICATION

Let X be a Banach space. On the Banach space C0(IR+, X), we consider the operator af := -f’(.)

A(.)/(.)

with domain D(G)

f is differentiable, the function -f’(.) + A(.)f(.) f~/e C0(IR+,X) : belongs to Co(IR+,X), and f’(0) -Bf(O) - Lf

/

Here, (A(s))s>o is a bounded strongly continuous fmnily of bounded operators on X, B is the generator of a strongly continuous semigroup on the same Banach space X, and L is a bounded operator from Co(IR+, X) into X. The operators (A(s)).~>o generate an evolution family H = (U(s, r)),>~>o on X solving the evolution equation

=

t _>o.

Weassume the asymptotic behavior of (U(s, r)),>_~>_o and in particular bound ~o(H) := inf {w ~ ~

there exists/k/~, snch that I[U(8,"F)[I~ l~,’~w gw(s-r) for s _> r }

the growth

Critical Spectrumof Strongly ContinuousSemigroups

63

to be known. Weare now interested in solutions satisfying the boundary condition given by B and L at s = 0. Note that, without this boundary condition, the operator G would be the generator of the evolution semigroup corresponding to (U(s,r))8>_~>o (see [4] or [5], Section VI.9). Westart by verifying that G is the generator of a strongly continuous semigroup. LEMMA 18 The operator G generates on Co(II¢+, X).

a strongly

continuous semigroup (T(t))t>o

PROOFIn case A(.) = 0, this can be found, e.g., in [5], Theorem VI.6.1. the general case, the assertion follows from the bounded perturbation theorem for semigroup generators (see [5], TheoremIII.1.3). Next, we prove that the semigroup (T(t))t>o satisfies translation property (see [8], B-IV).

a generalization

of the

LEMMA 19 Let (U(s,r))8>~>o be the evolution family generated by (A(s))s>o. Then we have ~V(s, O)(T(t - s)f)(O) [U(s,s- t)f(s-

(T(t)f)(s)

for s _
for each f E C0(ll~+, X). PROOF

It suffices to take f ~ D(G). By definition of (T(t))t>_o, we have

ff

--~(T(t-

s + r)f)(r)

= A(r)(T(t-

s

This implies [U(r,O)(T(ts)f)(O) fors_~t, [U(r,s-t)f(s-t) fort_<s,

(T(t-s+r)f)(r)= hence

[U(s,O)(T(t- s)f)(O) for s _< t, = [U(s,s - t)f(s for t _< s,

(T(t)f)(s)

LEMMA 20 Let (V(t))t>_o A(0) + B. Then we have

be the strongly

(T(t)f)(O)

= I/’(t)f(O)

continuous

semigroup generated

by

+ V(t -

for each f ~ Co(I~+,X) and t _> O. PROOF As above, it suffices to prove the assertiou for f ~ D(G). In this case, we obtain -~(r(t)l)(O)

= (A(0) B)(r(t)l)(O) +

64

Blakeet al.

Hence, u(t) := (T(t)f)(O) is the solution of an inhomogenousCauchy problem for the operator A(0) + B. Therefore, we obtain the solution (T(t)f)(0)

= V(Qf(0) + V(t - a)LT(a)f

for t_> 0. Putting these facts together, we obtain the following integral equation for the semigroup (T(t))tE0. LEMMA 21 The semigroup (T(t))t~o (T(t)f)(s)

satisfies

U(s,O)(V(t - s)f(O) = ( U(s,s- t)f(s-

the relation

+ fg V(t- s-a)LT(a)f

for s ~ t, for t 5~ s

for each f ~ Co(~,X). Weare nowwell prepared for the main result of this section. THEOREM 22 If the semigroup (V(t))t~o norm continuous, then

generated

by A(0) + B is immediately

u~,,~,(T) = ~0(U). PROOFFor each f ~ Co(~, X), we have (T(t)f)(s)

~U(s,O)(V(ts)f(0) + f~-~ V(~ - a)LT(a)fda) for

= ( U(s,s - t)f(s

for t ~ s.

~Ve now put

{

u(~,0)(~.’(~- ~)f(0)

(R(t)f)(s)

+

.f~-s V(t - s - a)LT(a)f := ~ (t - s)~(s,,O)(V(t -

for s

-

fort-l<.s w0(U) we find a constant Msuch that.

Therefore, we obtain

~cr~*(7") < ~0(U). On the other hand, since (T(t)f)(s)

= U(s, s - t)f(s

Critical Spectrumof Strongly ContinuousSemigroups

6~

for t ~_ 8, we must have

Wc,ri,(T) = ~o(U). Using Theorem2, we can draw the following conclusion. COROLLARY 23 If the semigroup (V(t))t>o norm continuous, then

generated by A(O)+B is immediately

wo(T) =max{s(G),wo(U)). In particular, if (U(s, r))8>~>o is uniformly exponentially stable s(G)< 0, th en (T(t))~>o is uniformly exponentially stable. Weremark that a(G) and therefore s(G) can be described by an operator-valued characteristic equation analogous to the scalar case in [5], ExampleV.3.9. This idea will be pursued in a subsequent paper.

REFERENCES 1. 2. 3. 4.. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

M. Blake, A spectral bound for asymptotically norm continuous semigroups, J. Op. Th., to appear. M. Blake, Asymptotically norm continuous Semigroups of Operators, Ph.D. thesis, 1999. S. Brendle, R. Nagel and J. Poland, On the spectral mapping theorem for perturbed strongly continuous semigroups, Arch. Math., to appear. C. Chicone and Y. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, Amer. Math. Soc., to appear. K. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Text in Math., Springer-Verlag, to appear. I. Herbst, The spectrum of Hilbert space semigroups, J. Op. Th. 10 (1983), 87-94. J. Martinez and J. M. Maz6n, Co-semigroups norm continuous at infinity, Semigroup Forum 52 (1996), 213-224. R. Nagel (ed.), One-parameter Semigroups of Positive Operators, Lect. Notes in Math., vol. 1184, Springer-Verlag, 1986. R. Nagel and J. Poland, The critical spectrum of a strongly continuous semigroup, Adv. Math., to appear. J. van Neerven, Asymptotic Behaviour of a Semigroup of Linear Operators, Birkh~iuser, 1996. H.H. Schaefer, Banach Lattices and Positive Operators, Springer-Verlag, 1974. H. Thieme, Balanced exponential growth o[ operator semigroups, J. Math. Anal. Applo 223 (1998), 30-49. L. Weis and V. Wrobel, Asymptotic behavior of Co-semigroups in Banach spaces, Proc. Amer. Math. Soc. 124 (1996), 3663-3671. V. Wrobel, Stability and spectra of Co-semigroups, Math. Ann. 285 (1989), 201-219.

An Operator-Valued Maximal Regularity

Transference Principle and on Vector-Valued Lp-spaces

PHILIPPE CLI~MENTTechnical

University

JAN PRESS Martin-Luther-Universit~t

Delft,

The Netherlands

Halle-Wittenberg,

Germany

ABSTRACT A recent vector-valued multiplier theorem for operator-valued Fourier multipliers in Lp(I~; X), 1 < p < ~, X a UMD-space,is combined with the vector~valued transference principle to prove existence of operator-valued R-boundedH functional calculi. These results are applied to maximalLp-regularity of abstract operator equations as well as to evolutionary integral equations.

1

INTRODUCTION

In recent years it has become apparent that the concept of families of R-bounded operators plays an important role in operator-valued multiplier theory. A family T C /~(X,Y)) of bounded linear operators on a Banach space X is called bounded if there is a constant C > 0 such that for all N E N, elements xj ~ X, selections Tj ~ 7", and N independent symmetric {-1, 1}-valued random variables sy on a probability space (g/, E, p) the inequality N

N

(I.I) j=l

is valid. This can be seen as a strengthening of the concept of uniform boundedness of the family T. The history of this property goes back to a paper of Bourgain [3] where this concept is used implicitly. It was first explicitly defined in the paper of Berkson 67

68

CI6mentandPriiss

and Gillespie [1] whereit was termed Riesz-property. Cl(~mentet al. [4] reinterpreted this notion as randomized boundedness. It is well-knownthat the classical Mikhlinmultiplier theoremin L~,(II~) is valid the vector-valued case, i.e. in Lp(l~; X) for scalar Fourier multipliers if and only if the Banach space X is a UMD-space,a space with the uniform ~nartingale difference property - equivalently the Hilbert transform is bounded on Lp(I~;X) for some p e (1, o o); see Bourgain [3], Zimmermann[22], and Priiss [15]. For the case of operator-valued Fourier multipliers the situation is disappointing: The Mikhlin theorem is valid in the operator-valued case if and only if X is a Hilbert space; this result is due to Pisier [14]. Very recently Weis [21] proved that, in case X is a UMD-space,the operator-valued version of the Mikhlin theorem is true, provided the bounds in the Mikhlin conditions are replaced by R-bounds. He also obtained so~ne partial converse of this result. Via this multiplier result Wcis has also given a new characterization of the property of maximalLp-regularity of parabolic evolution equations of the form i~+Au=f,

u(O)=O

The results of the paper of Weis are the motivation of this note. Our goal is to combine his multiplier theorem with extensions of the transference principle of Coifmanand Weiss due to the authors. Following Hieber and Priiss [12], we obtain operator-valued functional calculi for generators of bounded Co-groups on UMDspaces, as well as for their negative squares, and for tensor extensions of positive contraction semigroups on Lp(f~). These results are applied to derive maximal regularity results for sums ,4 + B in the spirit of Da Prato and Grisvard [8], avoiding imaginary powers of operators like in the Dore-Venni [10] approach. On the expense of more special structures of A we are able to relax the assumption on B. This way a number of new results emerge, in the abstract framework as well as for more concrete problems like evolutionary integral equations. The plan for this paper is as follows. Wefirst show in Section 2 that R-boundedness is necessary for operator-valued Mikhlin theorems. This is done in a fairly general setting and extends and simplifies the corresponding result of Weis [21] in several directions. A proof of the multiplier theorem is given, which differs in various respect from that of Weis [21]; it builds more upon recent work of Clement et al. [4] on Schauder decornpositions. Section 3 contains the main results of this paper, the operator-valued functional calculi which are based on the multiplier theorem and the transference principle. In Section 4 we introduce the notion of 7~-sectorial operators. Wepresent a direct proof that in a UMD-spaceX, operators with bounded i~naginary powers are T~-sectorial. Wealso show that operators in Lp(G; IRN) which admit Poisson estimates in the sense of Prfiss [16] are Tg-sectorial. Section 4 is devoted to the above mentioned applications to maximal regularity problems of sums A + B and of evolutionary integral equations. Weemphasize that some of the applications of our results in concrete Lp-spaces can also be deduced from another recent paper of Weis [20]. Acknowledgement: The authors thank L. Weis for a number of discussious led to improvements in the paper.

which

69

Maximal Regularity on Vector-ValuedL~-Spaces 2

OPERATOR-VALUED

FOURIER

MULTIPLIERS

Let X be a Banach space and consider the spaces Lp(~; X) for 1 < p < ~x). We denote by D(iI(; X) the space of X-valued C~-functions with compact support we let V’(~;X) := B(D(iI~),X) denote the space of X-valued distributions. X-valued Schwartz spaces ~(~; X) and S’(~; X) are defined similarly. Then given Fourier multilplier M~ L~doc(~; B(X, Y)), where Y denotes another Banach space, we may define an operator T : ~-~(~; X) ~ S~(~; Y) by ~neans TO := ~-~M~¢, for all

Y¢ e ~(~; X),

(2.1)

where ~ denotes the Fourier transform. Since ~-~(~; X) is dense in Lp(~; X), we see that T is well-defined on a dense subset of L~(~; X). We next want to show that R-Boundedness of the family ~ := {M(p) : £(M)} is necessary for Lp-boundedness of T, where £(M) denotes the set Lebesgue-points of M. For this purpose we have to show that there is a constant C such that for given N points pj ~ ~(~), N vectors xj ~ X, N independent {-1, 1}-valued randomvariables ej on a probability space (~, ~), the inequality

is valid. Choose a function ¢ e ~(~), nonnegative, symmetric, 0 ~ ~ ~ 1, such that f~ ¢(p)2dp = 1, and let Cn(P) = ¢(pn), and Cn(t) ¢( t/n); se t ¢ = Then ~¢.,(p) n¢(pn), and ~¢, ¯ ¢, e ~ (~), as well as

f

~n(p)~n(p)dp

Therefore we have lira

f M(p)iY@~](p Po)~.(P -

po)dp = M(

in B(X, Y), for each Lebesgue point P0 e £(M). This yields

J

~ lim L £ 11 ~ejei°itT[ei°~’¢’(’)xJ](t)ll:’dt(i

70

ClementandPriiss

here 1/p + 1/p’ = 1. Since T is bounded fl’om Lp(I~IX) into Lj~(I~;Y)., by Corollary 3.18 of C16mentet al. [4] the set {eia’Te -iT" : a,r e I~} is R-bounded in B(Lp(~; X), Lv(~; Y)), hence we may continue ~~lim (f~ I~@,~(t) ~ dt)~/P’~

~ ~ ~j¢~(t)xj]~dtd#

Thisshowsthat;M is necessarily ~-bounded, in particular M is me.bounded. PROPOSITION1 Suppose X, Y are Banach spaces, 1 < p < L~,toc(I~; 13(X, Y)). Denote the set of Lebesgue points of M by £M, and let defined by (2.1) be bounded from Lp(~;X) into Lp(II~;Y). Then {M(p) : is R-boundedin 13(X, Y), in particular M L~(I~; 13 (X, 1" )). Wenow turn to sufficient conditions for T to be Lp-bounded. The following is the operator-valued version of the vector-valued Mikhlin theorem of Bourgain, McConnell [13], and Zimmermann [22]: .it is due to Weis [21]. THEOREM 1 Suppose X, Y are UMD-spaces, 1 < p < oo, let {0}; B(X, Y)) be such that the following conditions are satisfied.

M ~ C~(I~

(i) 7¢({M(p) : p e ~ \ {0}} =: to0

(ii) rc({pM’(p):p e~, \ {0}}=: ~ < c~. Then the operator T defined by (2.1) is bounded from Lp(~; X) into L~(II~; Y). Proof. W.o.l.g. we may restrict attention to multipliers which vanish on (-oo, 0). In fact, if M(p) is given, set M+(p) = M(p)x~+(p), M_(p) = M(p)x~_(p), where XA denotes the characteristic flmction of the set ,4. Once we know that the operator corresponding to the multiplier M+is bounded, by reflect.ion we also get boundeduess of that one with symbol M_, hence also the operator with symbol M is bounded. Step 1: Consider the dyadic decomposition [2j, 22+1), j E ~, of (0, cx:)) and a fixed subdecompositionA~t := [2 j + (l-1)2 j-k, 2j +12J-k), l ---- 1... k. Let t he multiiplier be first constant on each subinterval A~t, i.e. k, M(p)= M~:t, p~ A~t, j ~ Z, l= l...2 where M~i ~ B(X, Y). Wedefine projections on Lp(R; X) and in Lp(l~; Y) by means of the Riesz projection R; note that R is bounded in these spaces since 1 < p < c~

71

MaximalRegularity on Vector-Valued/,,-Spaces

and X and Y are UMDby assumption. For p E ~ we sat Ra := eiP’Re -ia" . Then

(2.2)

Then by the boundedness of the Riesz projection, the family {D~ : j E Z,/ = 1... 2~} forms a Schauder decomposition of RLP(II~; X) resp. of RLp(II~; Y). By a theorem of Bourgain (cf. Zimmermann[22]) we know that the Schauder decomposition {Aj : j ~ Z} is unconditional; note that Pj~ = Aj. Moreover, boundedness of the Riesz projection R yields also by Corollary 3.18 of Clementet al. [4] that the set {e~’Re-~r’: a,r ~ R} is R-bounded in B(L~(R;X))) as well as B(Lp(R;Y))). Therefore the set

ez, is R-bounded in L~(R; X) as well as in Lp(R; Y). Next we rewrite I

which gives the following decomposition of T.

j

j

r:l

By Theorem3.4 of Clementet al. [4], the first sum yields a boundedoperator if the set {Mj~h : j ~ Z} is R-bounded. For the second sum, we have to show that the .~v ~2~ k(M k k :j Z}is R-bounded, . T o seethis we estimate family tz_,~=~, ~,. -- Mj(~_I))Pjt as follows. k2

~ 2

- Mj(,_~))Pj, : j e Z}) _< ~-~ 7~({(Mj~r- Mk(,._~ )P)}: j e Z}) r=l

r=l

2~

_

- Mk(r_~) : j ~ ~ 2

_

- M~(,_~) : j e Z})] sup~ ~({P)~ : j /~{1 ..... 2}

r:[ ~ 2

r=l

Therefore in this special case T will be Lfbounded if we have the following two properties. (i) 7~({Mj~ : j ~ Z}):= k < oo;

72 (ii)

ClementandPriiss E,=I ~({(My,

k

If this is the case, ~hen

where Cp(X, Y) denotes a constant only depending on X, Y and p. Step 2 Suppose we are given a sequence of operator-valued Fourier multipliers M,~ ~ L~,~o~(~; B(X, Y)) such that the corresponding operators T,~ defined by (2.1), i.e.

%~ := ~-~[M~], ~ ~ v(~;x), is uniformly Lp-bounded, i.e.

Suppose further Weclaim that then T is bounded as well, with

In fact, for ~¢ ~ D(~; X) we obtain from L~,toc-convergence of the multipliers T,¢~Td

Mn

asn~in

For such ~, the sequence T~¢ is bounded in Lp(~;Y). By reflexivity of Y and 1 < p < ~, L~(~; Y) is reflexive. Therefore we may extract a subsequence Tn~ ~ ~ g in Lp(~;Y). By means of test functions of the form ~ @ y* with y* ~ Y* and ’¢ ~ S(~) we see that g = T~, and so Tn¢ ~ T¢ in L~(~; Y), ~ well

Thus it remains to approximate the given multiplier Mby discrete multipliers M~ like in Step 1 in such a way that conditions (i) and (ii) of Step 1 hold uniformly k. Step 3 Let M(p) be given such that the assumptions of the Theorem are valid. Wedefine the discrete approximations by means of

M~:= M(~+ ~’~), j ~ g, ~ = 0,..., ~. Then by assumption (i), there follows g({M~0:j ~ Z}) 5 ao hence a~ 5 ~0 fi)r k ~ N. On the other hand we have

~ - M)i,_~ ) = M(:~+ t: ~-~) - i(:~ + (t =

M(2j + (l - 1)2 ~-k + s2J-k)ds

= 2-k 2JM’(2 j + (1 - 1)2 j-~ + s2~-k)ds.

MaximalRegularity on Vector-ValuedL~-Spaces

73

Employing Lemma3.2 of Clement et al. [4] on the absolute closed convex hull of a family of R-bounded operators we obtain

This showsthat a~k _< nl for all k E N. Therefore from Step 1 we obtain the uniform boundC (n0 + ~1) for the approximating operators Tk in B(Lp (~; X); Lp (ll~; Y)), from Step 2 we obtain T E B(Lp(~; X); Lp (ITS; Y)),

since the Multipliers

3

M~: corresponding to T~ converge to Ma.e. in/~(X, Y).

OPERATOR-VALUED

TRANSFERENCE

(i) Suppose T(t) = -At i s a bounded C0-group wi th ge nerator -A on a UMD space E, set ~4f T := supteR ItT(t)ll~(~). Wewant to prove that A admits an Rbounded functional calculus. This will be explained in the following. First we extend the Phillips calculus for bounded C0-groups to the operatorvalued case. Suppose K ~ LI,~(~; I3(E)), which means K(.)x ~ L~ (If(+ ; E) for each x ~ E. Then H(p)x = ]~ e-~tK(t)xdt,

t ~ I~, x ~ E,

(3.1)

is well-defined and H(.)x Co(I~;E) by the vect or-valued Riem ann-Lebesgue lemma. If K(t) commutes with T(t) we may define H(A)x = ~ e-AtK(t)xdt,

t e l~, x

(3.2)

as a strong vector-valued Bochner integral. This definition defines an algebra homomorhismfrom/,1,~(N; B(E)) to B(E)~ as described in more detail in Hieber and Priiss [12]; there the scalar case was considered. Now let (~,E,#) be a measure space, 1 < p < oc, and set E := L,(Ft;X), where X denotes a UMD-space. Define A0 := d/dt, the derivation operator on Lp(N;X), with domain Z)(A0) = Hp~(II~;X). A0 generates the translation on Lp(IR;X) which is strongly continuous and bounded. Then the transference principle of Coifmanand Weiss [7] extends to the operator-valued case. PROPOSITION2 Let (f~, E,#) be a measure space, 1 < p < oo, and X a UMDspace. Suppose T(t) := -At i s a bounded Co-group on Lp(~; X) wit h bound MT

74

ClementandPriiss

IIT(t)llB(L,(a;x)).Then

supte~

IIH(A)IIB(L,(n;x)) ~_U~llH(Ao)ll~(~,,(~;x)), where H(p) is given by (3.1), S(Ao) are defined by (3.2).

(3.3)

LI,~ (II~; B(X)) commuting with A, an d t t(A)

This result is proved along the lines of Clement and Pi’iiss [6], Appendix, in view of the assumed commutativity. It is well-known that L~(fl; X) is a UMD-spaceif X is UMD and 1 < p ,< c~. Therefore choosing (~,p) a one point space, we may replace L~)(~; X) by Proposition 2. Next we are going to apply the multiplier theorem, Theorem 1 of Section 1, to estimate the norm ]IH(Ao)~I~(L~(R;x)). In fact, from Theorem 1 we obtain ][H(Ao)[~(L~(~;x)) ~ C~(X)[~({H(p) 0}) + ~({pH’(p) : p ~

0}],

where ~(T) denotes the R-bound of a family T B(X). This es timate ma y be worse than the natural one from the Phillips calculus, i.e.

IIU(Ao)i[,(L,(~;x))~ MTVar[IKll~(x), but it will be better for the following class of functions H. Consider the space ~(~)

:= {M ~ C~(~ {0};B(X)):

M(p)(iA-A) -~ = (iA-A) -~M(p),A,p

~ ~}, (3.4)

and set n:M(~) := {M e ~: {M(p): p # 0} {pM’(p) : p

0}ar e R-bounded }.

(3.5) Then ~ (~) forms a Banach algebra

when equipped with norm

IIMII~:: ~({M(p): p # 0}) +

n({pM’(p):

p 0}).

Let X ~ ~(~) be such that 1 E X(P) ~ O, X(P) = Ipl ~ ~, andx(p) : o for p ~ 2. Given M e nM(~), we set M~(p) := M(p)x(p/n)(1 x( np)). Th en by Kahane’s principle there exists a constant C such that n({M,(p) :n e ~,p # 0}) C~({M(p): p

0}) =: a 0,

as well ~ by Lemma3.2 of Clement et al. [4] n({pM,,(p) : n ~ N,p 0}) g n({pM’(p) : p 0}) =: ~ (~; B(X)) and M~has compact support, it follows Since M,, ~ BUC 1p ~ eiptMn(p)d defined by

~(,,(t) belongs to Ll,s(~; B(X)). In fact, integrating by parts we tK,~(t)z

=

i £ eiO¢M~(p)xdp,

75

MaximalRegularity on Vector-ValuedL~-Spaces

hence K,~(t) and tK,~(t) are bounded uniformly in t E II~. Moreover, since X is a UMD-spaceby assumption, it has finite Fourier type, which means that there are p, q E (1, cx~) such that the Fourier transform is bounded from Lp(ll~; X) into Lq(~;X); cp. Bourgain [2]. Therefore we obtain tK,,(t)x ~ Lq(~;X) and so Kn(t)x LI(II~;X) fo r ea ch x ~ X.NowTheorem 1 im pl ies boun dedness of Mn(Ao) and the inequality

IIM,~(Ao)lb which is uniform in n ~ N. Consider on ~he o~her hand ~he multiplier arguments employing ~he Fourier ~ype of X, L(t)

(~p)~ N(p) := ~ tp). Then by similar

1 ~ e~p~ N(p)dp = i ~ e,~N,(p)dp, : ~

t E

belongs to L~,s(~; ~(X)). Defining in a similar ~vay N,~(p) := ~M,~(p) as L~(t), it is clear that L,~ ~ L in L~(~;B(X)), hence N~(A)

~ N(A) laB(X),

~ n ~ ~.

From the algebra property we have N~(A) = M~(A)(A2(1 A)-4), we obt ain for x ~ ~(A~) ~) ~ ~(A Mn(A)x = Nn(A)(1 A)4A-~x = ~ e- AtLn(t)(1 +

A)4A-~xdt.

As L~x ~ Lx in Lj (~; X) for n ~ ~, x ~ X, by domiuated convergence we obtain M~(A)x ~ M(A)x, where we define M(A)x = N(A)(1 A)4A-~x = ~ e- AtL(t)(1 +

A)4A-~xdt,

for x ~ ~(A~)~(A~). Thus M~(A) is uniformly bounded and converges on a subset, hence M(A) is uniquely defined and bounded, by the Banach-Steinhaus theorem. This shows that. the map M ~ M(A) is bounded. Wehave proved the following result. THEOREM 2 Let X be a UMD-space and let T be a bounded Co-group on X. Denote by -A its generator and assume Af(A) = Then the extended Phillips calculus given by (3.1) and (3.2) extends uniquely the Banach algebra TiM(R) which is defined by (3.4) and (3.5), as a bounded algebra homomorphism ¯ : nM(m ~ ~(X). There is a constant C > 0 such that

IIM(A)IIB(x)_
co : {~xe c \ {o}:I arg~ ± ~/21< o}.

76

ClementandPriiss

PROPOSITION3 Let X be a Banach space, and suppose H : Co -4 B(X) is holomorphic and R-boundedfor sozne t9 > O, i.e. n{H(A) : A 6 Co} < oc. Then for M(p) := H(ip) ’we have Proof. Cauchy’s theorem yields for k ¯ No p~M(~’)(O)= (ip)kI-I(~l(ip)

= (i/sinO)kk!

e-i~SH(¢(1 + e is sinO))ds/2,,r,

hence by convexity of R-bounds we obtain 7~{O~M(k)(p) : p ¢ 0} <_ k!(sin0)-~{H(~)

Co} < ~. [ ]

COROLLARY 1 Let X be a UMD-space and T be a bounded Co-group on X. Denote by -A its generator and assume Af(A) = O. Then A admits an R-bounded H°°-calculus on each double-cone Ca, ~ > O, in the following sense: The Phillips calculus given by (3.i) and (3.2) extends uniquely to the Banach algebra 7~7-/°~(Ce) := {g H°o(Ce;B(X)) : g()~)T(t) = T( t ¯ I~, ,~ ¯ Co, "R(H(Co)) as a bounded algebra homomorphism

There is a constant C > 0 such that

IIH(A)IIe(x) Cn(H(C~)) < ¢x) , H ¯ 7"i7 4~(Co). Next we consider a corresponding R-bounded H°%calculus for -A2 where as befbre -A denotes the generator of a bounded Co-group on the UMD-spaceX, with Af(A) = 0. In fact, with E0 = {A ¯ (2 \ {0} : I arg I <0}, fo r a given holomorphic function G : E0 -~ /3(X) we define H(A) = G(-A2); g ¯ H °°(Co/2; B(X)) This yields COROLLARY 2 Let X be a UMD-space and T be a bounded Co-group on X. Denote by -.4 its generator and assume ~’(A) = O. Then 2 admits an Rbounded H~’-calculus for each sector Eo, 0 > O. (ii) Next we deal with the case of tensor extensions T ® I of positive contraction semigroups T on Lp(~), 1 < p < cx), to Lp(~; X), where X is a UMD-space. starting point is now the operator-valued version of the Phillips calculus for Cosemigroups which is defined as follows. Let K ¯ L~,,(IR+;B(E)), where E Banach space, such that K(t) commutes with a given bounded Co-semigroup S(7) on E. Then with G(A)x :=

e-atK(t)xdt,

.~ ¯ C+, x ~ E,

(3.6)

77

MaximalRegularity on Vector-ValuedL~-Spaces we define a(B)x

:= S(t)K(t)xdt,

A E e+,

(a.7)

where -B denotes the generator of S. This definition creates an algebra homomorphism from the Ll,s-functions commuting with S to /3(E), as described in more detail in Hieber and Priiss [12] for the scalar case. Now, let X be a UMD-space, 1 < p < oo, and suppose T(t) is a positive Cosemigroup of contractions on L,(12), [br some measure space (12, E, ~)..Then tensor extension T of T is defined on Lp(ft; X) by means T(t) ~ Cj ® x~ := ~[~(T(t)¢i) J

J

where Cj ~ Lp(gt), xj 6 X, and j runs over a finite set. Positivity of T(t) and its contraction property ensure that the semigroup 7" defined this way on a dense subset of Lp(ft; X) can be extended to a C0-semigroupof contractions on Lp(~; X). Wecall T the tensor-extension of T, for details we refer to Cl6ment and Egberts [5]. For such semigroups a Coifman-Weiss inequality was proved in C16ment and Pr/iss [6]; it extends to the operator-valued case. PROPOSITION 4 Let (12, Y., #) be a measure space, T a positive Co-semigroup of contractions on Lp(f~), 1 < p < oo. Let X be a UMD-spaceand let 7" denote the tensor-extension of T to Lp(F/; X) with generator -A. Then there is a constant C > 0 such that [IG(A)[[B(Lp(a;.~)) C[IG(Ao)I[B(Lp(:~;x)), where G(A) is given by (3.6),

(3.8)

K ~ L~,~(N+;B(X)), and G(A), G(Ao) are

Observe that commutativity of T and G is already built in, since here we consider only functions G : C+ -~ B(X), which extend canonically to B(L,(Ft;X)). proof of Proposition 4 follows from C16mentand Priiss [6], Appendix, in a straight forward manner. Having this inequality to our disposal we may nowproceed as in Step (i) of this section to derive the following result. THEOREM 3 Let (12, E, p) be a measure space, T a positive Co-semigroup of contractions on Lp(12), 1 < p < oo. Let X be a UMD-spaceand let T denote the tensor-extension o] T to Lp(f~; X) with generator -.4. Assume Af(.4) Then the extended Phillips calculus given by (3.6) and (3. 7) extends uniquely the Banach algebra

There is a constant C > 0 such that

where

78 Proof. by

ClementandPriiss The proof is similar to that of Theorem2. This time we approximate G(A) G,~(,~) := (1 + A-)-:G(A n

1),

n

which converges to G(/~) in /3(X), locally uniformly on C+. As in the proof Theorem 2, Gn(~) = ~n(£), for some Kn e LI.~(~;B(X)), hence by Proposition 4 G~(A) is well-defined and uniformly bounded. Then we consider H~(A) ~ (~)~G,(~)

~2

and H(~) := ~G(~) The identities G,(A)x

are valid,

= H,(A)(1

A)4A-~x, x

E n( 2) rq

hence G,~(A)x -~ G(A)x := H(A)(1 A)4A-~x asn ~ ~o.

COROLLARY 3 Let (f~,#) be a measure space, T a positive Co-semigroup of contractions on Lp(Q), for some p (1 ,~). Let X bea UMD -space and T be a the tensor extension oft to Lp(f~; X) with generator -M and assume JV’(A) = O. .A admits an R-bounded HC~-calculus on each sector Eo, 0 > 7r/2, in the following sense: The Phillips calculus given by (3.6) and (3. 7) extends uniquely to the Banach algebra ~7-/m(Eo) := {a Hec(Eo;B(X)): 7~(G(E0)) < as a bounded algebra homomorphism ~I’: 7~7/~(P~e)--~ /~(Lp(fl; There is a constant C > 0 such that

This result is deduced from Theorem3 in the same way as Corollary 1 is obtained from Theorem 2.

4 " 7~-SECTORIAL

OPERATORS

The concept of T/-bounded families of operators leads immediately to the notion of ~-sectorial operators. DEFINITIONA sectorial

operator is called

~-sectorial

if

hA(0) := 7~{t(t + A)-~ : t > 0} < The g-angle ¢~ of A is defined by means of ¢~:=inf{8~ where

(0, r~):

nA(r~-8)


T~A(O):= 7~{~(A+ A)-1 : largAI <_ O}.

Maximal Regularity on Vector-Valued/,,-Spaces

79

This definition makes sense for the following reason. Suppose T~{t(t + A)-I : t > 0} < ~; then by means of the Taylor series (~ + A) -1 = Z(~ - t)’~(t

+ -n-~

which is convergent for I,k - t[ < 1/Mt, with M:= supt>o{tl(t + A)-~ : t > 0}, we obtain by the convexity of R-bounds 7~{~(/k + A)-I : larg~[ _< 0} _< CZ(sinO)"~[Ti{r(r+A)-~ ’~+~ : r > 0}] = CTC{r(r + A)-~ : r > 0}/(1 - (sin0)TC{r(r + A)-’ : r > Thus whenever (sin0)~{r(r + -~ : r > 0}< 1, the n T~A(0) < c ~, and ther e is such ~ > 0. This shows that the ~-angle of an TGsectorial operator is well-defined and the 7~-angle of A is always not smaller than the the spectral angle of A. It has been shownby Weis [20] that the concept of 7~-sectorial operators is wellbehaved under perturbations, like the class of sectorial operators. The class of operators with bounded imaginary powers does not have this property, and it is smaller, at least in case the underlying space is UMD. THEOREM 4 Suppose X is a UMD-space and let OA. Then A is Ti-sectorial and ¢t~ <_ OA.

A E BZT’(X) with power angle

Proof. The proof is based on representation formulas due to Priiss and Sohr [17], p. 436. Suppose A ~ BZT)(X) with power angle OA := limlsl~o~ Isl-~ log [[A/S[l~(x ). Then the first identity reads (1 + rA)-~x : ---1pV2~ri

- ~(rA)-"SXsinh~r_____~rcds+ ~.z, x ~ X,r > 0. (4.1)

The second one is 1 £ (l+re~¢A) -~ =(l+rA)-~+~--~ (rA)-i~( e¢~-l)sin-~ s,

~rds

I¢1 <~r--0A, r>O. (4.2) Suppose n ~ N, z3 ~ X, ;~3 = r3e~, r~ > O, I¢~1 <- ~r -0 are given, where 0 > Oa is fixed. Wehave to prove that there is a constant C > 0 such that

j=l

j=l

is valid, where the functions ej are independent symmetric random variables on some probability space (f~, ~, #) with values in {-1, 1}. For this purpose we decompose 1 (1 + r¢e~o~A)-~ = ~ + T~+ S¢+ R~, where Tj = jf~ (rA)-isCf(s)ds,

CT(s) = ¢~s - 1) 2isinh~rs1 ,

80

ClementandPriiss 1 f~ Sj = ~i (rA)-iS~b~

Rjx

~(s)ds

= PV-

7r ¢~(s)- sinh(rrs) (rA)-i~xdS,

X(s) s

x E X;

8

1

here X meansthe characteristic function of the the interval [-1, 1]. By the triangle inequality we estimate separately. Using Kahane’scontraction principle we get for the first term in virtuc of

because of 0 > OA, and for r~ < 0 - OA. The same type of estimate applies to the term involving the Sj. This time we have I¢~(s)l _< ce-’~.~., s ~ ~, and so

where again r/is small. The third term is more sophisticated, it is only here where we use the UMDproperty of the underlying space X. Webegin with Kahane’s contraction principle and also apply the boundedness of ,4 ~’ for Is I < 1. For t e [-1, 1] we have

it ~ ¢jrjtR~zjllL,(~;x), II ~ ¢fl~jxyllL.(~;x)<_CIIA j=l

j=l

81

MaximalRegularity on Vector-Valued L;Spaces hence integrating

over t E [-1, 1] 21’ ~ ejRjxj’[PL~(n;x) <- C I_I~

Cjrj Rjxj PLp([~.X) j=l

j=l

j:l

A gjrj

1

xj

where we used the boundednessof the Hilbert transform, and once more Kahane’s contraction

principle

in the last

inequality.

These estimate

prove the theorem. []

We next consider an important class of operators on spaces X = Lp(G), where G C II~ ’~ is open and 1 < p < ~, and which give a large variety of concrete examples of ~-sectorial operators. DEFINITION Let A be a sectorial operator" in X = Lo(G; N~). A is said belong to the Poisson class P(X) if the resolvent (z + A)-~ 4 A admits a kernel representation

[(z

+ A)-t

fl(x)

for z ~ E._¢, and the measurable kernel t~=(z,Y)I

(4.3)

= £ %(x,y)f(y)dy,

Clzl~-lP(Iz -

7~ satisfies

Yl lzl’/m), x,

the Poisson estimate y e a, z e ~-

(4.4)

wherep : (0, ~) ~ (0, ~) is continuous nonincreasingand such

~

p(r)r~_~ dr

The Poisson angle ¢~ of A is defined and (~.~) are valid.

as the infimum 4 all ¢ > 0 such that (~.3)

EXAMPLE As a typical example for an operator ~(L,(G; N)) c onsider G = N~andA th e L~-r ealization of a operator of order m with constant coefficients, i.e. (A~)(~)

= (~(n)~)(z)

= ~ ~%(~),

A belonging to the class sys te m of di ffe rential

=,

82

Clement ant] Priiss

where am e CN×N. If A is elliptic in the sense that a(A(i~)) ~¢,1\,{0} fo r al ~ E II~ ’~ \ {0}, then A E/)(L~(G; II~N)) p(r)

e -~r(l+s) (1

s"-2ds rn-l’ -~ 8)

(4.6) r > 0,

where ~ > 0 is a constant, and the Poisson angle CAP equals This result extends to manyother (systems of) differential operators with nonconstant coefficients on domains G C I~’~, for example to boundary value problems of the Agmon-Douglis-Nirenbergtype on sufficiently smooth domains, with sufficiently smoothcoefficients. There is a large literature on Poisson estimates, however.., here we refer only to the monographsof Davies [9] and Robinson[18], and to Hieber and Priiss [11] for further examplesand discussions. THEOREM 5 Suppose ,4 is a sectorial operator on Lp(G;I~N), G C ’~ open, belonging to the Poisson class 7)(L~(G; ~v )) Then A is 7~-sectorial in L~(G; ~N ), 1 < p < ~, with angle ¢~ ~ ¢~. Proof. Let k ~ N, fi ~ Lv(G;~N), $j ~ E._¢ begiven, Tj = $~(~j + A) -~, and let ~ be symmetric independent random variables on a probability space (~, E, with values in {-1, 1} be given. Wehave to prove the estimate

where Mis a universal constant, not depending on k, Aj, For this purpose we first use Khintchine’s inequality.

where K depends only on p. Next we employ the Poisson estimate. ITjfj](:r)

~ Cp~* [fj](x),

x e e,j = l ...N;

here pj(x) = r~p(~xIrj), rj = ~/’. Extending fj by 0 to al l o f ~" we may assume G = ~" in the sequel. Therefore we obtain

Estimating further we get

Next we apply the maximalinequality (see e.g. (16), p.51 of the new book of Stein, [~91). supypj * g(x) ~ co . Mg(x), x ~

MaximalRegularity on Vector-ValuedL.Spaces

83

where Mdenotes the maximal functiofi My(x) : sup a,cr-"’

: Ig(x - y)ldy,

¯r>0 JlY

and Co = f~ p(r)r’~-tdr.

Inserting this estimate we obtain

Since the maximalfunction Mis bounded in L,(ll(")

for p > 1, we get further

Finally applying Khintchine’s inequality once more we arrive at

which proves the assertion.

5

MAXIMAL

[]

REGULARITY

Wewant to apply the transference results from Section 3 to maximal regularity of problems of the form #u + Au + Bu = f. As a result we are able to relax the assumptions of the Dore-Venni theorem on B; we merely need to require that B is ~-sectorial. For the Dore-Venni theorem we refer to Dore and Venni [10], Prfiss [15], and Priiss and Sohr [17]. For this purpose we distinguish three cases, as before. (i) Suppose X is a UMD-space,-A the generator of a bounded Co-group in X with Af(A) = 0, and let B be a closed linear operator in X which commuteswith A in the sense of resolvents, such that q-iB are 7~-sectorial. Assumealso that B is invertible. Weclaim that A + B with natural domain 7)(A + B) = 79(A) C~ ~(B) is invertible. For this purpose we use Corollary 1 for the functions M(A):= -1 B(A+B) and N(A) = A(A + -1. Fr om the co ndition th at B is 7~-sectorial we see that Corollary 1 is applicable. This yields the first part of THEOREM 6 Suppose X is a UMD-space, let -A be the generator of a bounded Co-group in X, X’(A) = O, and assume B is a closed linear invertible operator X commutingwith A in the resolvent sense, such that =l=iB is T~-sectorial. Then A + B with domain I)(A + B) = D(A) fq D(B) is closed and invertible. is also sectorial with spectral angle CB< rc/2, then A + B is sectorial as well, and CAq-B <__ 7f/2.

Prbof. Let us consider first the case where B is 7¢-sectorial with TO-angle smaller than 7r/2. Then -B generates an analytic C0-semigroup in X, which is even exponentially stable since B is invertible by assumption. Nowwe easily see (z + A + B)-~x

e-Ate-Bte-Ztxdt,

z ~ C+, x ~ 79(A) ~ 7:)(B).

84

Cl~mentandPriiss calculus (z + A + B)-1 = H:(A), where

Thus by the functional Hz(A)

e-~te-~te-Btdt

-1. = (A + z + B)

By the assumption on B we obtain from Corollary 1 IIB(z + A + B)-llI + (l + IzI)II(z

+ A -1 <_ C,

and so the assertion follows from standard arguments. In the general case we employ a variant of the Grisvard formula. (.4+8)_

1 1 fr ~-- 271"----~

(’~ if" A)-I(’~ --

B)-ld)~’

where F denotes the contour F = F_(-F+) with F+ = i(~c, rle -i¢ + r[ie -i¢, -ie i¢1 + (-i)[r,

i¢. ~x))e

Replacing A by A and evaluating the integral we obtain by the functional calculus (A + B) -~ = H(A), with H(A) = (A + -1. The assumptions on B by Corollary 1 then imply boundedness of BH(A), hence the assertion follows by standard arguments. [] (ii)

Weturn to the problem -A~u + Bu = f,

in a UMD-spaceX. Using Corollary 2 this time we obtain in the sa~ne way’ THEOREM 7 Suppose X is a UMD-space, let -A be the generator of a bounded Co-group in X, Af(A) = 0, and assume B is a closed linear invertible operator in X commutingwith A in the resolvent sense, such that B is Ti-sectorial. Then -A~ + B with domain I:)(-A ~ + B) = D(A’2) ~ I)(B) is closed, invertible, and also sectorial with spectral angle ~h-A~+B<_ (ill) The next result a e (0, 2).

is based on Corollary 3 where we have built in a power

THEOREM 8 Suppose (fl, E, p) is a measure space, T a positive Co-semigrvup in LB(~), for some p (1 , ~c ), and le t X bea UMD -space. Let T denote the tens orextension of T to Lp(~; X) with generator -.A and assume Af(A) = O. Let an invertible Tg-sectorial operator in X with spectral angle ¢~3 + (~r/2 < 7r, where ~ e (0, 2), and let 13 denote the canonical extension of B. to L~(f~; X). Then JI~ + 13 is closed, invertible and sectorial on Lp(f~; X), with spectral angle CA¢,+I~< max{aTr/2, CB}. Proof. Here we use Corollary 3 for the function H(~) = B(/~" + -~ which is R-bounded on a sector E0 for some ~ > ~r/2 in case ¢/3 + aTr/2 < rr. []

85

MaximalRegularity on Vector-ValuedL~-Spaces

For applications, let B be a sectorial and invertible in a UMD-space X, and consider the evolutionary integral equation. u(t)

+ a(t-

s)Bu(s)ds

= a(t-

s)f(s)ds,

(5.7)

in Lp(IR+;X), where a E Ll,toc(lI~-) is a fixed scalar kernel, and f e Lp(N+;X). This problem is studied in detail in the monographof Priiss [15], also concerning maximalregularity. Here we can add the following results. (i) Choosing A d/ dt in Lp(IR~_) with domain 7)( A) = 0W~ (II~_) = { u e W]( u(0) = 0}, the abstract equation A~u+Bu= f in Lp(IR+;X) beco~nes the fractional evolution equation (5.7) with a(t) ga(t); he re th e ke rnel g~ is given by

g~(t) : t~-~/r(a), t > a ~ (0,2). Theorem 8 then implies that this problem adnfits a unique solution u ~ Lp(IR+;:D(B)) for each ] ~ Lp(ll~_; X), provided B is 7¢-sectorial a~r/2 + CB~ < 7r. In addition we have u ~ H~(II~+ ;X), and there is a constant C > 0 such that

(ii) Assumea(t) is a completely positive function; cp. Pr/iss [15], Section 4. It is well-known that then problem (5.7) can be written in abstract form as Au +/3u = in Lp(II~+;Lq(G; If{N)) where A is the tensor-extension of an operator A such that -A generates a positive C0-semigroup of contractions in Lq(ll~+; X); see Clement and Priiss [6]. Therefore by Theorem 8 with a = 1 problem (5.7) admits a unique solution u ~ Lp(ll~+ ; X) for each f ~ Lp(ll~+ ; X), and there is a constant C > 0 such that [113UllLp
in X and ¢~ < 7r/2.

(iii) Assumethat a(t) is 0,-sectorial, i.e.

~(A)¢ and l arg3(A)l< 0., fo r ReA >O, and 1-regular, i.e. sup{l~’(A)/3(),)l : ReA > Then setting H(A) = (1 + ~(A)B)-~ we see by the help of Lemma8.1 of Priiss [15] that M(p) = H(ip) satisfies the conditions of Theorem1, provided B is ~-sectorial and 0, + CB~ < ~r. Then by Theorem 1 we may deduce that problem (5.7) enjoys maximalregularity of type Lp as in (ii). Observe that the assumption on the kernel a(t) is precisely the same as in Theorem8.6 of Priiss [15] where it is shownthat the corresponding operator A admits bounded imaginary powers on Lp(N+;X). These results were known before only in case B ~ BIP(X) and angle CB~ replaced by the (larger) power angle 0s, which by Theorem3 are stronger conditions.

86

Clementand Priiss

(iv) In the special case X = Lq(G;~N) with 1 < q < ~c and B an operator of class P(X) of Poisson class, by Theorem5 B is T/-sectorial and ¢~ _< ¢~. Therefore for each of the above classes of kernels we obtain maxinmlregularity in the class Lp(I~; Lq(G; ~N)). This considerably improves Theorem 3 in Priiss [116]. These results follow independently also from Weis [20]. (v) We can even go a step further. Suppose a E L~.to~ is 1-regular. Then by means of Proposition 1 and Theorem 1 we may deduce that (5.7) has the property of maximalregularity of type Lp if and only if {(1 + ~(A)B) -~ : ReA > 0} is ?~-bounded.

REFERENCES 1. 2. 3.

4. 5. 6. 7.

8. 9. 10. 11. 12. 13. 14. 15.

E. Berkson and T.A. Gillespie. Spectral decompositions and harmonic analysis on UMD-spaces. Studia Math., 112:13-49, 1994. J. Bourgain. Someremarks on Banach spaces in which martingale difference sequences are unconditional. Ark. Mat., 22:163-168, 1983. J. Bourgain. Vector-valued singular integrals and the H~ - BMOduality. In D. Burkholder, editor, Probability Theory and HarmonicAnalysis, pages 1-19, NewYork, 1986. Marcel Dekkcr. Ph. Clement, B. de Pagter, F.A. Sukochev, and H. Witvliet. Schauder decompositions and multiplier theorems. Studia Math. (to appear). Ph. Clement and P. Egberts. On the sum of maximal monotone operators. Diff. Integral Eqns., 3:1127-1138, 1990. Ph. Clement and J. Priiss. Completely positive measures and Feller semigroups. Math. Ann., 287:73--105, 1990. R.R. Coifmann and G. Weiss. Transference Methods in Analysis, volume 31 of Conf. Board Math. Sci., Reg. Conf. Series Math. Amer. Math. Soc., Providence, RhodeIsland, 1977. G. Da Prato and P. Grisvard. Sommesd’op5rateurs lin~aires et ~quations diff~rentielles op~rationelles. J. Math. Pures Appl., 54:305-387, 1975. E.B. Davies. Heat Kernels and Spectral Theory. Cambridge University Press, Cambridge, 1989. G. Dore and A. Venni. On the closedness of the sum of two closed operators. Math. Z., 196:189-201, 1987. M. Hieber and J. Priiss. Heat kernels and maximal Lp - Lq-estimates for parabolic evolution equations, to appear, 1996. M. Hieber and J. Priiss. Functional calculi for linear operators in vector-valued Lp-spaces via the transference principle. Adv. Diff. Eqns., 3:847-872, 19!)8. T.R. McConnell. On Fourier multiplier transformations of Banach-valued hmctions. Trans. Amer. Math. Soc., 285:739-757, 1984. G. Pisier. Someresults on Banach spaces with local unconditional structure. Comp. Math., 37:3-19, 1978. J. Priiss. Evolutionary Integral Equations and Applications. Birkh~user Verlag, Basel, 1993.

Maximal Regularity on Vector-ValuedL~-Spaces

87

16. J. Priifl. Poisson estimates and in£ximal regularity for evolutionary integral equations in Lp-spaces. Rend. Istit. Mat. Univ. Trieste, XXVIII:237-321,1997. Suppl. Vol. 17. J. Priiss and H. Sohr. On operators with bounded imaginary powers in Banach spaces. Math. Z., 203:429-452, 1990. 18. D.W. Robinson. Elliptic Operators and Lie Groups. Oxford University Press, Oxford, 1991. 19. E. Stein. HarmonicAnalysis. Princeton Univ. Press, Princetoni 1993. 20. L. Weis. A new approach to maximal Lp-regularity. Preprint, 1999. 21. L. Weis. Operator-valued Fourier multiplier theorems and maximal Lpregularity. Preprint, 1999. 22. F. Zimmermann.On vector-valued Fourier multiplier theorems. Studia Math., 93:201-222, 1989.

On Anomalous Asymptotics

of Heat Kernels

A.F.M. TER ELST Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600 MBEindhoven, The Netherlands DEREKW. ROBINSONCentre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, Canberra, ACT0200, Australia

ABSTRACT Let A1, A~, A3 denote a vector space basis, formed by right invariant vector fields, of the Lie algebra fl of the three-dimensional Lie group G of Euclidean motions of the plane. Wedemonstrate that for m :> 4 the semigroup kernel Kt associated with the strongly elliptic operator =H (-1) "~/~ ~i=l "~ ANsatisfies m-th order Gaussian boundsfor all t >_ 1 if, and only if, two of the A~ span the nilradical of ft. If this condition is not satisfied the kernel has an anomalousasymptotics. It behaves like an m-th order kernel in one direction and like a second-order kernel in the other two directions. No such anomaly occurs for the kernels associated with the operators H-- (- ~i=1~ A~)~ m/~. AMSSubject Classification:

1

22E25, 35B40, 35B27.

INTRODUCTION

The rate of dissipation of heat on a manifold is determined by the available volume. This geometric principle is illustrated by the behaviour of the heat kernel K associated with a right invariant sublaplacian on a Lie group G of polynomial growth. Then one has bounds (1) for all t > 0 where V(t) denotes the volume of the ball of radius t measured with respect to the subelliptic distance (see [10], TheoremsIV.4.16 and IV.4.21, or [11],

90

ter Elst andRobinson

TheoremVIII.2.9). The asymptotic properties of the kernels associated with higherorder operators is less clear. Let ~ denote the Lie algebra of G and A~,... ,Ad a vector space basis of formed by right-invariant vector fields. First, let K denote the semigroup kernel corresponding to a power of the Laplacian H=(_ ~-~d=1 Ai) 2 .m/2 . Then one has the m-th order analogue

v(t)


(2)

of the bounds (1) for all t > 0 The upper bounds are a consequence of the Gaussian bounds established in Theorem 3.1 below and then the lower bounds follow from [6], Corollary 2.4. Secondly, consider the kernel corresponding to the operator H =(-1) "~/2 ~=~ a A~ . Then the situation is more complex. If G is nilpotent the bounds (2) are again valid for all t > 0 because the Gaussian upper bounds K follow from [7], Theorem3.5. More generally, if G is the local direct product C ×t N of a compact Lie group C and a nilpotent Lie group N then the bounds (2) are valid. The key upper bounds (2) are a consequence of Theorem4.3 of with rn = m. The special form of H allows one to verify that Conditiou I~ of [2], Theorem4.1, holds and hence all the equivalent conditions of Theorems4.1 and 4.3 are valid. In particular H satisfies the strong G~rdinginequality (~o, HT) >_ # sup IIA~11.22

(3)

for some # > 0 and all ~o ~ D(H) where we have used the standard nmlti-index notation. Conversely, Dungey[1], Theorem1.1, has shownthat if H satisfies (3) and K satisfies Gaussian bounds for all t > 0 then G is the local direct prodnct C xt N of a compact Lie group C and a nilpotent Lie group N. Therefore it. is of interest to examine possible upper bounds and the possible asymptotic behaviour of K for groups which are not of the special form C xt N. The purpose of this note is to investigate this problem for the simplest such group, the three-dimensional group of Euclidean motions in the plane. Our analysis establishes that for large t the bounds (2) are the exception rather than the rule. Indeed manym-th order operators have the large time characteristics of second-order operators in somedirections. Let G denote the three-dimensional, connected, simply-connected Lie group of Euclidean motions, [~ its (solvable) Lie algebra and n the (two-dimensional) nilradical of ~. Further let I" I be the modulusassociated to a fixed basis of g and note that different bases give equivalent moduli. Fix a basis a~, a2, a3 of 9" Next let A1, A2, Aa denote the infinitesimal generators of the one-parameter groups t ~ L(exp(-ta,:)) where L is the left regular representation of G in L2(G). All the operato~t’s we consider arc m-th order polynomials in the Ai with the commonfeature that the corresponding semigroup kernels K are smooth functions satisfying Gaussian bounds [Kt(g)] <_ct-al"e~te-b(l~l )~1( .. .. ~), with b,c > 0 and w >_ 0, uniformly for all g e G and t > 0 (for details see [10], Chapters I and III, or for a short proof, [5]). Our interest is to derive bounds of this nature with the optimal behaviour as t ~ co. THEOREM 1.1 Let H (_l).m/_~ = ~i--~" ~ conditions are equivalent.

with m>

_ 4 even. Thefollowing

91

Anomalous Asymptoticsof Heat Kernels There exist b, c > 0 such that ’/( .... ~) IKt(g)[ <_ct-3/me-~(l~l"~t-’)

(4)

for all g E G and t > O. II. III. IV.

There exists a c > 0 such that ct -3/m < ][Ktll~ for all t > 1. limt-~ t 3/m Kt(e) exists and is not The nilradical rt is spannedby two of the basis elements al, a2, a3.

The theorem implies that the geometric bounds (2) and the good Gaussian bounds (4) are only valid for large t for very special bases. This contrasts starkIy with the situation for powers of the Laplacian, Theorem3.1 below, for which the Gaussian bounds are satisfied independently of the choice of basis al, a~, a3. Note that Theorem 1.1 gives examples for which K satisfies the Gaussian bounds (4) but H does not satisfy the strong G£rding inequality (3). Indeed (4) together with (3) imply that G is of the form C xt N, by [1], Theorem1.1, which is a contradiction. The next result gives detailed bounds on the kernels of Theorem1.1 for general bases. To state it we need an explicit description of G. First, there is a basis b~,b~,b3 of l~ satisfying [b~,b2] = b3, [bl,b3] = -b2 and Ibm, ba] = 0. Then n = span{b~, b3}. Secondly, define the homeomorphism G by O(Xl, x2, x3) = exp(xlbl) exp(x2b2) exp(x3b3) Then there is a c > 0 such that c-~ ]x] ~ ]O(x)] ~ c ]xl for all x ~ 3. ( Actually, if the modulus] ¯ ~ is defined with respect to the basis b~, b2, b3 then ]O(x)] for all x ~ R3.) Thirdly, for b,t > 0 and n ~ 2 even introduce the Gaussians ~ G(~) ~,~ R: ~Rby

= z. on the commutative group R THEOREM1.2 Let H = (_1)~/~ ~i=t A~ with m ~ 4 even and assume the nilradical ~ is not spanned by any pair of the basis elements a~,a~,a3. Then the following is valid. I.

There exist b,b’,c > 0 such that

for all t > 0 and (x~,x~,x~) ~ z where * denotes co nvolution on R~. In particular there are b, b’, c > 0 such that

¯

e ~ ct-(m+l)/me-b(Iz~l’"t-~)t/(

....

for Ml t > 0 and (z~,z~,za)

ift51

~) (~-b((~+x~)"/~t-’)~/(~-~l

~a

-b’(~+~)t-’) Ve

ter Elst andRobinson

92 II.

limt-~ t (m+l)/m Kt(e) exists and is not zero.

The asymptotic behaviour of the kernels K associated with the homogcueous operators H = (-1) "q2 ~i3=~ A/~ can be described in muchgreater detail. Wewill demonstrate that the kernel is accurately approximated for large t by the kernel of 3. an m-th order, weighted strongly elliptic operator with constant coefficients on R The above results extend to subelliptic operators H = (-1) m/-~~i=~ ~ A~ ’~ with at, a~ an algebraic basis of 1~. The subelliptic geometry changes the detail of the small t estimates but not the large t estimates. One has normal Gaussian behaviour if r~ contains one of the ai and anomalousbehaviour if this is not the case.

2

PROOF OF THEOREMS 1.1

AND 1.2

We begin by establishing crude upper bounds on the kernel K by standard arguments based on Sobolev inequalities and perturbation theory. The bounds are ~). established on L_~(R Let B~, B~, Ba denote the representatives of b~, b.~, ba in the ~left2"egu~lar representation of G on L~_(G). Then B~, B~, Ba transfer to operators B~, B~, Ba on a) L:(’,R by use of the homeomorphism~. Explicitly, if ~ ~ C~(Ra) then (~I(~)(Z)

= (Bl(~O

o O-1))(O(x))

(/~2qo)(x) = (B~(~oo (I)-t))(O(x))

) ,

= - cos xt (02~o)(x) +

(/~a~o)(x) = (Ya(~oo gh-t)) ((I’(x)) x, (O~o)(x) cos x~ (0a~o)(x for all x = (x~,x~,xa) ~ Ra where 0~ = O/Oxk. Since b~,b:,b3 is a basis of 1~ there is a non-singular, real-valued, matrix (uk~) such that ak = ~a=t u~b~. Then A~,A2,A3 transfer to operators ~k= ~=~ uktB~- on L~(Ra) for all k ~ {1,2,3}. a) Hence the operator H is represented on L2(R by 3

=

=

A~..qo

3 for all ~o ~ oo C~ (R). Next for all p ~ R3 introduce the multiplication operator Uo by (U~)(x) e~"~o(x) and set ~ = UoftU~~. Further let ~ and/~ denote the forms associated

with ~ and/~ on L~(R3). LEMMA2.1 There is a ~/ >_ O and for each e > O ace >0 such that

-

_<

+

for all ~ ~ D([~) where ~o~(p) = p~ + p~ "~p3 +~/(p~~ + p~). Moreover, if a~ = ~.b~ with ~ ~ R and a~,a3 span r~ then one may choose 7 = O.

93

Anomalous Asymptoticsof Heat Kernels PROOFOne has U,~It:U~ ~ = ,’~ + Lt(p) with Lk(p) : vtlp~ + vk2(p2c~ P3Sl) + Uk3(p2sl + p3CI)

where cl (x) = cos xl and 81 (Z) : Sill Xl. Weconsider Lk (p) both as a multiplication operator on L2(R3) as as a function on R3. Moreover,

~((P)

_~((P)

3

= Z (((~k -- Lk(P))n~, (~k + (, k~,

~n &~))

-~

(5)

k:l

with n "- m/2. But n-1

=

c,~;~ (p) A~o /=0

where the coefficients

have the form n;l

~ "~-

~X-’c I,jl .....

P j~ i:1

The sum is over allp ~ {1,... ,n} and j~,... ,jp >_ 0 such that jl+...+jp+p+l = n (~) andthe Q,j~ ..... i, are numerical constants. Nowone immediately obtains bounds 2 2 1/’2ifj >_ 1. Therefore, 2 ’2 2 ~/’2 and [[Aa -JL ~(P)[I~ < cy(pe+pa) ][nt~(p)[[oo < co(p~+P’~+P3) fixing 1’ > 0 one has bounds (k) 2 2~n--I ~ 2(’-~)/ 11%~ (p)llo~ <-~Co(p~+p~+p3) +cl "/(p,2 + P~)<- cc~’~(P)

(6)

Next it follows from (5) that 3

n--1

k=l

l=O 3

(7)

n-! ~ 2

k----1

)

I=0

Then the leading term in (7) is bounded

-~ II&dl~ II%~(p)ll~ll&~’ll~ < e~,(~)+e

2 k=l

k=l

I=0

-~ --

for all s > 0. But

’/~+,~-~/("-~)c (~},,,( p)11oo"/°’-° IIc~l0)11oo 114~o11~ <,~h(,~) II ~’I1", _~ (~ ~(~)l/2

(~- I/(n-l)cn/C2Cn-l))033,(/9)1/2

Ilfl12

for all 5 > 0, I ~ {0,... ,n - 1} and k ~ {1,2,3}, by (6). Then the first statement of the lemmafollows by adding the contributions and choosing 5 appropriately.

94

ter Elst andRobinson

Finally~,.if al = u~lb~ then u12 = 0 = u~3 and ifa2,aa span n then u~ = 0 =: u31. Hence (A~kLk(p)) for all j _> 1 andk E {1,2 ,3 } and th e bo unds(6) are vali with 3’ = 0. Then the subsequent arguments are also valid with 3‘ = 0. [] Since m >_ 4 and G is three-dimensional one has a Sobolev inequality

I1 o11 _ 0 such that

for all ~o ~ D(h) and all t _> 1. But standard estimates give bounds IIS$ll _< Metb’~(P)ll~ll~ and Ihp(S~o)l <_ Mt-~etb’~(P}ll~oll2~ for all ~ e L2(R3) ~nd all t > 0 where Sp is the semigroup generated by ~p. Hence one obtains bounds 1~S~]~2~ ~ cetb~(p) for all t ~ 1. Since the corresponding kernel Kp satisfies 118t/~11~118,/~11:~ one immediately obtains crude bounds on ~he kernel K. LEMMA2.2 There are b, c > O and T >_ O such that I/Q((I)(x))I

<_ At) -a/" inf

for all x E R3 and t > O. Moreover, if az = u~zb~ with uli Ett and a2, a3 .span then one may choose 3’ = O. PROOFThe bounds for t E (0, 1] follow frmn the standard small time Gaussian bounds. The bounds for t _> 1 are a consequence of the previous reasoning. [] If 7 = 0 then the bounds of the lemmacan be reexpressed as

IKt(O(x))l < c(1

A t)-Slme

uniformly for all t > 0, i.e., one has Gaussian bounds with an additional polynomial growth factor (1 + t) ~/’~. If, however, 3’ > 0 one has bounds (’+~)/"’ -bCl’*’l’t-~)~/("-~ IKt(O(x)) t-ll"~e I_
\(G(’Ob,t *’-’b’,t)(x2,x3)fJ(2) ~ (8)

uniformly for all t > 0 and x = (x~,x~,x3) E p~3 and these are of the type given in Theorem 1.2 but again with the additional growth factor (see [2] Proposition 2.10.I). Next we use arguments based on periodicity to remove this factor. The operator ~ on L~ (R3) is strongly elliptic with periodic coeftlcients. Therefore one can analyze it with the Bloch decomposition as in [9]. Let Ho be the operator on L~([-zr, 7r] 3) obtained by replacing each 0t, in ~ by Ok -- iOt,, where 0 E C:~ and the 0k on L2([-~r, zr] 3) have periodic boundary conditions. Then H0 has a. compact resolvent with a simple eigenvalue 0 and eigenfunction ~o0 = l, the constant function with value 1. Therefore by holomorphic perturbation theory there exist 6, ~ > 13 and holomorphic functions A0 : f~ ~ C and 0 ~ ~o0 from f~ into L2([-~r, zr] 3) such that

Anomalous Asymptoticsof Heat Kernels

95

H099o= Ao(8) 990 and Ao(O)is the unique eigenvalue Ho with [Ao(0){ < e for all E fl, where Ft = {0 E C3:10[ < 5}. (Cf. [9], Proposition 2.7.) The behaviour of the Ao depends on whether Condition IV of Theorem1.1 is valid or not. If it is we again make a technical restriction which we later remove. LEMMA2.3 If rt is not spanned by any pair of at,a2,a3 then there exist c > 1 and ct, c2 > 0 such that

IAo(0) - o(0)1c(O ? + og for all 0 ~ ~, where ~o(0) = c10~~ + c~(O~+ 0~). II.

If at = ~ub~ with bql ~ V~ and a2,a 3 span n then there exist c >_ 1, # > 0 and a homogeneous m-th order polynomial ~o : R~ ~ R such that

-

<_.lel

’n for all 0 ~ ft. Moreover,the coefficients oleo(O) are real and ~o(t?) >_ttlOl a. for all ~ ~ R PROOF

For all k ~ {1, 2, 3} and OE R3 set ~) L~

= i(PklO \

1 q-

Pk2(Cl

02 --

81 03)

t)

Pk 3(81 e2

"-[ - 121 03) /

and L(o~) = -t~k~c91 - ~2(cl 02 - sl 03) - ~’ta(st 02 c~Oa) In particular L~~) = Weconsider L~a) ~) as a multiplication operator and L(~ as a partial differential operator on L2([-u, u]a). Then He = (-1) m/2 ~,=~ (L~k) + L(o~))m. Note that if .~b is a linear combination of ct and st then Ho~/a = ~¢, where 3 ~ = ~=~ ~,~. Since ~o and tO ~+ 99e are holomorphic and ~ ~ He is a polynomial one can write

Ao(0)=EA(~)(0 ~l----0

) , 99e=E99(’~)(o)

and

rt:O

He= H(~)(O) ~.-~0

for all 0 ~ ~, where each A(’*)(O), 99(n)(0) and H(")(8) is homogeneous in ~, if ~ is sufficiently small. Then ~(o)(~) = O, 99(0)(8) = 1 and H(°)(t?) H(")(O) ~(n)(8) = He990= Ao(0

(9)

~,~0

for all 0 ~ R3 with [0[ < 5. Comparingthe linear terms gives Ho~:~(1)

(0)

H(1)(O) 1 =1(1)(0) 1 an

(2u) 3 ~(~)(0) = (1, Ho 99(t)(0)) H(U(O)= 0

96

ter Elst andRobinson

ThenH0~o(~)(0) = -H(L)(O) 1 is a linear combination of c~ and si. Since ~o(~)(0) linear this implies that there is a linear function Tt : R3 -~ C such that @~)(0) rl(0) 1 v- IHO)(O)1 for al l 0 ~ R3with [01< 6. Comparingthe second order terms in (9) gives (2~)a A(~) (0) = (1, (~) (0) 1) - -~(1, 0) (0) H(t) (0) 1 = (1, H(~(0)1)

~-~llUU)(0)111 ~

Wecalculate both terms. One has 3

H(t)(O) l =

3

~o

~

~o

~ = ~ ~

k=l

~o

1

k=l

(lO)

3

k=l

Then 3

IIH(’)(0)1[1~= 32-’(2~r) Z

-

//m-l//m-l((//k203 k,l=l

02)(//1203 3

Ill

%" (/~k2 02 %" //k3 03)(///2

2-’(2~r)3(0~ +032)

//m-

1//

~

02 %" //13 03))

m-1 ut~ , u~,3

k=l

for all 0 E R3 with 101 < 6. Similarly, 3

(I,H(~)(0)

= Z( (L(ok))m/~-’L~k)I,’~’O , k=l

. (

3

= 2-1(27r)3(0~ + 0~)

k=l

Soa(~)(o)= 2-’~,,(oI+ol)

But it follows from the Cauchy-Schwarzinequality that

.m12 ,nil m/2, for all l E {2, a}. So c~ R0. Moreover, since ,t//~ ,u:~ ,//al ) ¢ (0,0,0) Cauchy-Schwarzinequality implies that c~ = 0 if, and only if, there are p~, p~ ~ R

97

Anomalous Asymptoticsof Heat Kernels

,n/e-1 ,,/2 such that Ukl ukt = plu~ for all k E {1, 2, 3} and IE {2, 3}. If, however,n is not spanned by any pair of al, ae, a3 then there are k~, k~ ~ {1, 2, 3} with k~ # k~ such that uk,~ # 0 ~ Uk~l. So if, in addition, c2 = 0 then there are pa,p3 @R such that ukd = p~uk,1 for all l E {2,3} and i E {1,2}. Then a~, = uk,a(b~ +p2b2+ p3 b3) for all i E {1,2} and ak~ and a~ are linearly dependent. Therefore c2 > 0. Conversely, if 11 is spannedby a pair of a~, o.2, (23 then it is easy to showthat c2 = 0. Note that the coefficieuts of 0~~ in ~(")(~) and (’0 (~) equal ~(")(0o) a (’0 (0o), where for simplicity we assume that ~o = (1,0,0) G ft. Then ~oO)(0o) = vl(0o) u-~HO)(~o) 1 = T~(~0)1 by (10). Let n E {2,... ,m} and suppose there constants p~,... ,Pn-~ G C such that ,k(J)(~o) = 0 and ~(J)(~o) = Pj 1 j e {1,... , n - 1}. Comparingthe n-th order terms in (9) at ~o gives n-1

,~(’0 (~0) 1 = H(’0(0o) pjH(’~-J)(O°) 1 + Ho~(’~) ( j=l

Since L(~) is an operator of multiplication with a constant it follows that ~o H(J}(~o) l = 0 for all j ~ {1,...,m- 1}. So ifn < m then (2~r) 3,~(")({?o) = (1,Ho~(")(Oo)) = 0 and Ho ~(")(~o) = 0, which implies that T(’0(~o) some p, e C. Alternatively, if n = m then (2n)3A(m)(~o) = (1, H(m)(~o) (- 1)’m/2~=1(1 (r(~)~m 1) = a v. S o i f n is not s panned by an y p air of a~,a~,a3 then c1 = ~ and one sets £o(~) = c~ e7+ 2-~c:(eJ+ 0~). This establishes Statement I of the lemma. Finally, if a~ : ~llb~ and a2,a3 span n then for each k ~ {1,2,3} the operators L~~) and L~k) commute. Hence it follows as in the last part of the proof of Statement I that ~(~)(0) = 0 for all a with ~0] < 5. O bviou sly A(m)(0 ) ex(_l)mi2 ~=i(1,(L~k))ml) for all tends to a real homogeneous polynomial of degree m. It remains to show that A(")(8) # 0 ibr all ~ a wit h 0 < [~ ] < 5. Let ~ ~ R a with ]0] < 5 a nd suppose that A(m)(0) = 0. ~ ll(L~))~/’21il~ = (2~)uA(~)(0) (L~k))m/21 = 0, for each k, almost everywhere. Since (L~k))m/21 is continuous one deduces that (L~k)l)(x) = 0 pointwise. Setting x = 0 gives Z~I vmet = 0 for each k. But the matrix (vkt) is not singular and hence ~ = 0. This completes the proof of the lemma. Proof of Theorem 1.2 Suppose that 11 is not spanned by any pair of a~, a2, a3. Let ~o, c~,c2 and c be as in Lemma2.3.I. Then there is a c3 ~ (0, c~ A c~/ such that ReAo(0) _> c3~ ~ +~+0~) for all ~ ~ fl, if5 is small enough. Set (-1)m/2c~ n - c~(O~ + 0~). Then ~ is a weighted str ongly ell iptic ope rator on R3 (see [4]). Using the crude Gaussian estimates (8) one can argue as in the proof Theorem3.5 in [9] to deduce that there are d, ~ > 0 such that

98

ter Elst andRobinson

for all t _> 1, where/~ is the kernel of the semigroup generated by 3. Hence there is a c" > 0 such that (11) for all t _> 1. Since (-1)"/20{ ’’ and -(0~ +0.~) commutethe kernel ~ has Gaussian bounds

for all t _> 1 (see [2], Proposition 2.10.III). Then one can combine(11) and (12) interpolate with the bounds (8) as in the proof of Corollary 3.6 of [9]. It follows that for all ~ > 0 there are b, b’,c > 0 such that

o ¢(z)

< ct

’/(’-’) {G(")

’ (13)

for all t >_ 1 and x = (x~, x2, x3) ~ 3. The b ounds ( 12) a nd ( 13) i mply t he f bounds of Theorem1.2.I in case t > 1. The bounds for t < 1 follow from the local Gaussian bounds and Propositions 2.10.I and 2.10.II of [2]. Finally it follows from (11) and Fourier theory that lim t (m+U/mKt(e) = lim t (m+l)/m ~t(O) = Irt3 dpe-(~’PF+~(P]+V])) = 2rr F(m-t) -1 c-~Um(c2m) and the proof of Theorem1.2 is complete. - /’m REMARK 2.4 Note that actually lim gm+~)/,~ K~(9) = 2~r F(m-~) c~ ~ (c~m)- l for all 9 ~ G, by the same argument. Proof of Theorem 1.1 "I=~II". Since H is self-adjoint it follows from Corollary 2.4 of [6] that Kt(e) >_ -a/m for so me c > 0 and al l t > 0. This imp lies Condition II. "II~IV". If Condition IV is not valid then Theorem1.2.I implies that there is a c > 0 such that []Kt[[~ _< ct -(m+Wm for all t >_ 1. This contradicts Condition II. Obviously Condition III implies Condition II, so it suffices to show the implications IV=~I and IV~III. Wefirst prove these implications if a~ = ~u b~ and a2,a3 ~ Let # > 0 and ~0(0) be as in Lemma2.3.II. There exist ca ~ R such that = (-1) m/2 ~lc, I--m 0 mnce ~o(0) = ~’~l~l=mCa c~ f or a ll 0 fi R3. Set ~ ~o(0) _> p [0[ mfor all ~ ~ R3 it follows that/) is a strongly elliptic operator on 3. If ~ is the kernel of the semigroup generated by ~ then it follows as in [9] that there isac> ()such that IIK~o~-~tll~ <_ ct-a/mt-~/ .... for allt_> 1. Arguing

Anomalous Asymptoticsof Heat Kernels

99

as in the proof of Theorem1.2 the Gadssian upper bounds of Condition I follow. Moreover, lira

t3/mgt(e)

= lim t3/’n/~(0)

= [ doe-~°(°) ¢ 0

which is Condition III. Finally suppose that Condition IV is valid. Wemay assume that a2, a3 E ft. Then u~ ¢ 0. Set a = u~l(u13 b2 - Ul~ b3) E ft. Let t~ be the inner automorphism of G induced by expa, so ~(g) = (expa)gexp(-a). Then ff2(expak) = expgk k fi {1, 2, 3}, where gl = u~ ba, (z2 = a2 and g3 = a3. Let U be the unitary mapfrom -1 L~(G) ontoL2(G) definedbyUT= ~ o ~-1.Then UHU =(-1)m/2 ~k=13 Amk, where A~ is the generator in the direction gk. So the kernel /’( of the semigroup -~ has Gaussian bounds and lim,-~oo a/m generated by the operator UHU t exists and is not zero by the foregoing arguments. But Kt =/~ o ~. Since ¯ is an automorphismof G the Conditions I and III for K follow from those for/;/. Also here Condition III can be strengthened (or weakened) to lim t 3/m Kt(g) = for all g ~ G with c ¢ 0 a constant independent of g. REMARK 2.5 The foregoing ar~mnents apply with very little alteration to subelliptic operators H = (-1) m/2 ~i=~ Ai with a~,a.2 an algebraic basis of 1~. The subelliptic geometrychanges the local singularity of Kt from t -3/rn to t -4/m but the behaviour for large t remains unchanged. Examination of the proof of Lemma2.3 shows that the previous condition for normal Gaussian behaviour for large t is replaced by the requirement that rt contains one of the ai. If this condition is not fulfilled one has the anomalous second-order asymptotics.

3

POWERS OF THE LAPLACIAN

The next theorem shows that the anomalous behaviour exhibited by Theoreln 1.1 cannot occur if H is replaced by a power of the Laplacian. Note that the following argument is independent of the group structure and applies equally well to powers of operators on a space of polynomial growth. THEOREM 3.1 Let G be a Lie group with polynomial growth, a~,... , aa, an algebraic basis of the correspondingLie algebra fJ and A~,... , Ad, the represcntatives in the left regular representation on L2(G). Further let I’1 ~ and V denote the associated ~2~m/2 subelliptic distance and volume. Finally let H = (- v’d’ z_,i=~ ,~, with rn >_ 2 even, and K the corresponding semigroup kernel. Then there exist b, c > 0 such that IKt(g)l <_cV(t)-~l"~e-b((t~l’I’~t-’)~/~ .... ~ and sup ~/°~-’~ I(AiKt)(g)l <_ ct-I/’mV(t)-l/me-b((M’)"~t-’) l<_i<_d’ ]or all g ~ G and t > O.

100

ter Elst andRobinson

PROOFThe bounds are well known for m = 2 and the general bounds follows from this special case in three steps. The first step reduces the proof to the case D’ = D _> 4, where D’ and D are the local and global dimension of G corresponding to the algebraic basis. If D’ > D define G2 = HD’-D x R3 where H is the three-dimensional Heisenberg group, if D’ < D define G2 = TD-D’ x Ra a. and if D’ = D define G2 = R Then consider the group 0 = G x G2. Choose a full basis of the Lie algebra 92 of the group G2 and consider the algebraic basis of 0 obtained by the union of the algebraic basis of G and the full basis of G2. Let/~’ denote the local dimension of 0 with r2spec~ to the corresponding basis and/~ the dimension at infinity. It follows that D’ = D _> 4. Next let A~ denote the Laplacian corresponding to the full basis of~. and ~2 = H~®I+I®A~, whereH,~ = -y~.d’ A? Note that there is a i=1 ~ ’ c > 0 such that P(t) >_ cV(t)V~(t) for all t > 0 where ~ and V2 are the volumes on ~,, and G~. corresponding to the appropr~te bases. Nowsuppose the estimates of the proposition are valid for the kernel K associated with (H2)"~/~. But with ~ = (g, g2) ~ 0 one has

K,@)= [ d~ ~,((,q,~)) 2

for all g ~ G. Hence -g((l~l’)"~-’)’i"~-’) I i’~(t) -lIra- ~,f~ dg2e-f’(la=l’"t-’ ),/( . .... ]Kt(g)] <_5 V(t)-ll" <_cV(t)-llmg-g((l~l’)’~t-1)ll( .... Thus the estimates on G follow from those on the larger group ~. Therefore we now assmne D’ = D ~ 4. Then V(t) behaves like ~ f or a ll t > 0. The second step consists of deriving the Gaussian bounds on the ball {g : Ig]’ ~ til’}. First, let S(2} denote the semigroup generated by the (sub)Laplacian H.2. Then there is a c > 0 such that 11S~2)N2~~ ct -~/4 for all t > 0. Secondly, if n > D/4 then

~ F(n - D/4) ~-~+~/~ = c~ A-,~+~/,t I1(~I + H2)-=112~~ c(nW for all k > 0 by Laplace transformation. Hence if S is the semigroup generated by H then

II&ll>+~<__c~ A-~+DI411(,~I + H~)~&II~-,~ < ~,,’

/~-n+D/4(An

+ t-2n/m)

for all ,~, t > 0 with a suitable cn > 0, where the last boundfollows from spe.ctral theory. Thirdly, setting ,~ = t -~/’~ gives bounds I[Sdl~m <_ 2c~ t -z)/(~’~). Hence

IIS¢,llo~
for all

~t-~/’% -°‘c’~’’ " ’ IS~’,(~)i <_c’~. b > O, g ~ 0 andf > 0 with I.ql’ <_

Anomalous Asymptoticsof Heat Kernels

101

The final step of the proof is to derive the bounds on {g : ]gl’ >- tl/m} ¯ The starting point is the Cauchyintegral representation 1 j(~ + H,~)_ St = ~i d~e~t()~I 1 where n = rn/2 and F is a curve running from infinity with arg A = -~r+e around the origin in the sector A(r- e) to infinity with arg ~ = ~- e for somefixed with A(O) = {z ~ C~{O}: largz~ ~ 0}. If A ~ C~{-~,O] and a ~ {0,1} define A~ = lAl~e ~gx. Set w = e ~i/~ and

_

= /=1 ,lCk

for all k ~ {1,... , n}. Then, by partial fractions, (M + g;) -~ = ~ ck (~’/~)~-~

(~k

k=l

for all ~ ~ C~(-~, 0], where ~k = --e -~/’* A~/’~w~. Therefore [K~(g)[ ~ ~(2~)-~ [c~.[ ~ d[A[ [e~t[ ~A[-~+~/~[R~.(.q)~

(14)

k=l

where R, denotes the kernel of the operator (#I + H~)-~. Note that n-~/n for all A ~ A(n-¢) and k ~ {1,... ,n}. Moreover, there exist b,c > 0 such that

In.()l uniformly for all p ~ A(r - c/n) and g ~ G~{e} (see, for example, the appendix of

[3]).

Let t > 0 and g ~ G with ~g~ ~ t ~/m. Choose F to be the contour in the complex plane formed by connecting the two line segments Ln,~ = {A ~ C : arg ~ = ¯ (~-¢), ]A] ~ R} and the arc AR= {A ~ C: argA ~ [-~+¢,~-e], ]~] = R}, where R = (bin -~ ]g]~ t-~) m/(m-~) is chosen such that the function x ~ x t- bx~/m~gV on [0, ~) attains its mininmmat R. One then estimates for each k ~ { 1,... , n} that,

~ c~ t-2/(m-1) (]g]~)-D+2m/(m-1)e-w((~g]’)’"t-~)~/~ ( ~ t -D/me-w((]g]’)~t-~)~/(~-~ since ]g~’ ~ t 1/’~ and D ~ 4 ~ 2m/(m - 1), with w = (bm-~)m/(m-~)(m 1) Moreover,

<_ 2ce-2-’l~tc°se(Igl’)-D+2t-1/n

~

d~ 12--1÷l/ne-2-

l ~, cos

~

102

ter Elst andRobinson

since D > 2, with w’ = 2-1(bm-1) ’n/(’n-t) cos¢. A combination of the last two estimates and equation (14) gives the required Gaussian bounds for the kernel. The bounds on the derivatives follow by a similar argument. First, one estimates [JAiStll2-*~ as before but using the bounds IIAiS}2)H2~~ ~ ct-D/4t -l/2. Then one has the resolvent bound [1,4~(~ I + H~)-n[]~o~. ~ c ~-,~+~/4+t/z. Consequently one obtains l[AiKt~]~ ~ at -(D+~)/’n. Secondly, on the set {9: ]g]’ ~ t~/’n} one has

I(A~ZC,)(g)I ~ ~(2~)-~Ic~l ~ dial I~I-~+I/’~I(A~R~)(g)I k=l

in place of (14) and

’I(A~.)(g)l~ a (Igl’)-(~-~)~-~’""~’~’ in place of (15). The latter estimate again follows from the appendix of [3]. The only difference is the introduction of an extra factor ([g[’)-~, which is bounded m. t-l~ REMARK 3.2 The situation is quite different for higher derivatives. If n _> 2 one has bounds [[A~Kt[[~ <_ ct -I~1/’~ V(t) -1/m for all t > 0 and a with [a I = n.if, and only if, G = C ×~ N is the local direct product of a connected compact Lie group C and a connected nilpotent Lie group N. If m = 2 this result is contained in [8], Theorem1.1. The general case can be deduced from the special case. REMARK 3.3 It also follows that d’ -~=~ A~, with m _> 4, satisfy the G = C ×t N. The inequality (3) for a with lal = m/2. But then by [8],

the powers H = Am/2 of the sublaplacian A = strong G~rding inequality (3) if, and only if, H is equivaleut to IIA~A-’"/4112~2 <_ c~ for all Theorem4.4, this is equivalent to G = C ×~ N.

REMARK 3.4 The foregoing arguments also apply to powers of n-th order operators. If H generates a holomorphic semigroup with a kernel satisfying n-order Gaussian bounds and if HTM also generates a holomorphic semigroup then the latter semigroup has a kernel which satisfies Gaussian bounds of order nm. Weconclude with a specific illustration

of the power of these results.

EXAMPLE 3.5 Let Bt,B2,B3 denote the representatives of the standard basis b~,b~,b3 of the Lie algebra of the Euclidean motion group. Then the kernel of H = (B1 + B2)~ + B24 + B~ satisfies fourth-order Gaussian bounds by Theorem1.1. Hence the kernel of the operator ((B~ + B:) 4 + B24 + B])~ satisfies eighth-order Gaussian bounds by Theorem 3.1.

ACKNOWLEDGEMENTS This work started whilst the second named author was visiting the Eindhoven University of Technology in 1998 and Was completed during a second visit in 1999. These visits were made possible by financial support of the EUTand travel funds from the CMA.

Anomalous Asymptoticsof Heat Kernels

103

REFERENCES 1.

DUNGEY, N.,

Higher order operators and Gaussian bounds on Lie groups of polynomial growth. Research Report MRR053-98, The Australian National University, Canberra, Australia, 1998. 2. DUNGEY,N., ELST, A.F.M. TEa and ROBINSON, D.W., Asymptotics of sums of subcoercive operators. Colloq. Math. (2000). To appear. D.W., Functional analysis of subelliptic 3. ELST, A.F.M. TEa and ROBINSON, operators on Lie groups. J. Operator Theory 31 (1994), 277-301. 4. ELST, A.F.M. TErt and ROBINSON, D.W., Weighted strongly elliptic operators on Lie groups. J. Funct. Anal. 125 (1994), 548-603. D.W., Elliptic operators on Lie groups. 5. ELST, A.F.M. TEFt and ROBINSON, Acta Appl. Math. 44 (1996), 133-150. 6. ELST, A.F.M. TEF~ and ROBINSON,D.W., Local lower bounds on heat kernels. Positivity 2 (1998), 123-151. 7. ELST, A.F.M. TEa, ROBINSON,D.W. and SIKOrtA, A., Heat kernels and Riesz transforms on nilpotent Lie groups. Coll. Math. 74 (1997), 191-218. 8. ELST, A.F.M. TaFt, ROBINSON,D.W. and SIKOFtA, A., Riesz transforms and Lie groups of polynomial growth. J. Funct. Anal. 162 (1999), 14-51. 9. ELST, A.F.M. TEFt, ROBINSON,D.W. and SIKORA, A., On second-order periodic elliptic operators in divergence form. Research Report MRR99-016, The Australian National University, Canberra, Australia, 1999. 10. ROBINSON,D.W., Elliptic operators and Lie groups. Oxford Mathematical Monographs. Oxford University Press, Oxford etc., 1991. 11. VAFtOPOULOS, N.T., SALOFF-COSTE, L. and COULHON, T., Analysis and geometry on groups. Cambridge Tracts in Mathematics 100. Cambridge University Press, Cambridge, 1992.

On Some Classes of Differential Operators Generating Analytic Semigroups* ANGELO FAVINI Dipartimento Bologna (Italy)

di Matematica,

Universita’di

Bologna,

GISI~LE RUIZ GOLDSTEINCERI and Department of Mathematical University of Memphis, Memphis TN 38152 (USA) JEROMEA. GOLDSTEIN Department of Mathematical Memphis, Memphis TN 38152 (USA) SILVIA ROMANELLI Dipartimento Interuniversitario di Bari, 70125 Bari (Italy)

0

Sciences,

40127

Sciences,

University

of

di Matematica, Universita’

INTRODUCTION

Inspired by the fundamental papers by Feller [10], C16ment and Timmermans[3] and Timmermans[17], we began a systematic study of the analyticity (in different function spaces) of the Co-semigroups generated by second order differential operators of the type ~ Au := au" + ~3u where a and ~3 are real-valued continuous functions in an interval:I of R, a(x) > in the interior of I and a is allowed to vanish at the endpoints (see e.g. [5], [6], [7], [8]). Valuable contributions to this study were also given by Campiti, Metafune and Pallara [2] and Metafune [14]. An updated treatment of the analytic semigroups generated by second order differential operators in spaces of continuous functions is included in a section of the forthcoming monograph by Engel and Nagel [4], while a detailed study of the degenerate case (i.e. a vanishing at the boundary) in various spaces can be found the book by Favini and Yagi [9]. *Worksupported by M.U.R.S.T.60%and 40%and by G.N.A.F.A.(C.N.R.) 105

106

Faviniet al.

Here we consider the space C[0, 1] of all real-valued continuous functions on [0, 1] equipped with the supremumnorm and we focus on the problem of analyticity for the semigroup generated by A, whenever its domain contains the so-called Timmerroans ( Wentzell, resp.) boundary conditions lim Au(x) = tj E C

x3 --~

(li m Au(x) 0, resp.) j .~:--4

- O, 1.

3

Our main theorems prove that whenever a and ~ are assumed to satisfy suitable additional assumptions easy to handle then the semigroup generated by A with domain D(Y) = {u e C[0, 1] ~ C2(0, 1)[ lim Au(z) = 0, for j = 0, 1} x--~ 3

coincides with the semigroup generated by A with domain D(T) = {u e el0, 1] V~ C2(0, 1)1 m Au(x) e R,for j = 0, 1} x--4 3

and is analytic on C[0, 1]. Moreover, in some cases we state that D(V) = D(T) = D(N), provided that D(N) := {u ~ C[O, 1] V~C-~(O,1)lau’~ ~[0, 1], l ira a(x)u’(x) =0}. x--~0,1

Notice that the semigroup generated by (A, D(V)) is positive and contractive (see e.g. [4, Chapter VI Lem~na4.16 and Theorem 4.17]).

1

GENERATION WITH WENTZELL BOUNDARY CONDITIONS

AND

TIMMERMANS

In this first section we recall some basic results due to Clement and Timmermans [3] and Tim~nermans[17] concerning generation in C[0, 1] and, in particular cases, we express their main assumptions in a more convenient way. Let I denote an open interval (r~,r~), where -co _< r~ < r~ _< +co and )Y the two point co~npactification of [r~,r~]. C(~) is the Banach space of all real-valued continuous functions on ~ equipped with the supremumnorm, while Co(I) := {u e C(]) I lim u(x) = 0}. In particular,

for I = R we define

C[-co,+co]

:={u G C(-co,+co)[

lira u(x) e R, lira .T-+-- O~

Co(-co, +co) e c[-co, +coll lira u(x)

=

0}.

u(x) e

Differential OperatorsGeneratingAnalytic Semigroups

107

Then, according to [3, Theorem2], [i7, Theorem3 and Remarkp.215], the following result holds. THEOREM 1.1 Let ~, ~ be real-valued continuous functions on I and define the functions W, Q and R as follows

on I, with a(x) >

W(x) : = exp(- ~ ~(s) Q(x)

:= a(z)W(x) ’~

W(s) °

~

R(x): = w(x)

where Xo denotes an arbitrary but fixed point in I and x E (r~,r.~). (i) The operator (At.’, D(V)) given D(V) : = {u ~ C(]) N C;(I)l (~u" + Co(i)} A~.u : = ~u" + ~u’,

u ~ D(V)

generates a Co-semigroup on C(7) if and only W e Ll(rl,xo)

or Q ~ L~(rl,xo)

or both

(H-l)

and W e L~(xo,r2) orQ ~ L~(xo,r~) or both.

(H-2)

(ii) The operator (AT, D(T)) defined D(T): = (u e C(~) C~(I)I c~u" + ~u’ e C(I)} Avu : = au" + ~u’,

u ~ D(T)

generates a Co-semigroupon C(-[) if and only ’if R ~ L~(r~,xo) and R ~ Ll(xo,r2). Moreover, i] all assumptions (H-i) (i=1,2,3) are satisfied, A~,, D. -- AT generates a Co-semigroup on C(-

(H-3) then D(V) = D(T)

In other words, the operator A~, whose domain contains Wentzell boundary conditions generates a Co-semigroup on C(~) if and only if r~ and r2 are not entrance points, following the terminology by Feller [10], while the maximal operator AT generates a Co-semigroup if and only if (H-3) holds. Whenever conditions (H-i) (i=1,2,3) hold, the operators A~ and Av coincide and generate a Co-semigroup Let us examine some particular cases. (1) Let I := R, ~ _-__ 1 and ~ ~ 0 and let us define as above the corresponding operators (Av,D(V)) and (AT,D(T)) denoted by (~,D(~)) and (~T,D(:~)), respectively.

108

Faviniet al.

Since W(z) -- 1 and Q(x) = x = R(x), then Q ~ L~ (-co, 0), Q ~ ~ ( 0, + oc) a Theorem1.1 (i) implies that the operator (J~v,D(I~)) generates a Co--semigroup on C[-~,+~]. Moreover, R ¢ L~(-~,0), R ~ L~(0,+~) and Theorem 1.1 implies that (A~, D(~)) generates a Co-semigroup, too. Hence D(P) = D(~),

{,~ ~ c[-~, +~] ~ c~(-~, +~)1~"e c[-~, +~l} = {~ e c[-~, +~] a c~(-~, +~)1u"(~)Co(-~, +~)}. (2) Let us consider I := (0, 1). Let us suppose that c~ 1 (0, 1) with (~(x) > 0 in (0,1)

(1.1)

and that the operators (Avl,D(V~)) and (ATI, D(TI)) are defined as (Av,D(V)) and (AT, D(T)) ~. respectively, provided that ~ = a If a satisfies ~ x(1 - x) (1.2) ~i dx = +co, then the conclusion of Theorem1.1 holds. Thus, we deduce that Av~ and AT~ generate the same semigroup on C[0, 1] and ~ntzell boundary conditions for (c~u~) ’ = ~u" + ~u~ are equivalent to (~u~) ’ ~ C[0, 1]. Indeed the functions W, Q and R becomerespectively

fo

w(~)- ~(~) ~(~)’ Q(x) =

~(s) X

2

1

R(x)- ~(~) for any x E (0, 1) and since Fubini’s Theorem ½Q(x) dx=

-

satisfies

½(

(1.2), then W~ L~(0, 1). Moreover,

½ ds

~-~)a~ = - ~-~d~ = -~

and

Q(x) 9~½1 Therefore (Av,, D(V~)) satisfies

dx j~_l = -g~ 1 - sd~ = +~.

(~.4)

(H-1)-(H-2) and hence it generates a Co-semigroup

onC[0,1]. On the other hand,

fo

R(x) dx = +o~

R(x) dz = -c~,

(1.5)

2

and consequently (ATe, D(T~)) is a generator in C[0, 1], too. Hence Av~ = A75 and the proof is complete.

109

Differential OperatorsGeneratingAnalytic Semigroups More generally,

let us consider the ophrators Av and AT where

1 z(1 - z)

f0

aeCo(O,1)NC’(O,

(1.6) ~(~i

dx= +(x)

and

-- e L1(0,1)

(1.7)

and denote by W1, Q1 and Rt the following functions

W,(x)

5(})

Q~(~):=

~(~), 1

R~(x) : = ~, x (0 ,11. The assumption (1.6) allows to obtain as in (1.3)-(1.5) I I) and

(1.8)

1 R1 ¢ LI(0, ~), R1 L’(~,1)

(1.9)

Then -~ezp(-

-~

.

ds) = W1(x)¢(x),

where ¢(x) := exp(- ~ ~(S~s~’(S) d ) and assumption (1.7) yields also that

0 < Co< ¢(z) < ci. Now, taking into account that Q(x)

-~

o~(x)W1

(z)¢(x)

(s)¢ (s) ds

we obtain the following estimates

- Q~ (x) dx < Q(x) dx <

co/,

C1

--

"~’ 1 S} a(x)W, (x)

’Ji’

Co

= 1C1- o~ Q,(x)d.~. Co

W, (s) ds)

110

Faviniet al.

In a similar way it is possible deduce the corresponding estimates for the integral of Q in (0, ~-)2 ¯ Therefore, (1.8) implies that Q ~ LI(0, ½), Q 1, ~ 1). Moreover, from R(X)

-~"

W(x)

~V l(x)~(x) l(s)~(8)

og(8)W(8)

t~ (8)W

it follows that -- R~ (x) dx <_ Cofl C1 -

R(x) dx < -- R~ (x) dx ~1

~1

-f_ll ~o -

and analogous estimates hold for the integral on (0, ½). Hence, applying (1.9) obtain that, R ~ LI(0, 1),2 R ~ LI(~, Thus, Theorem 1.1 guarantees that (Av,D(V)) and (AT, D(T)) coincide and generate a Co-semigroup on C[0, 1]. Weremark that the assumption 3-- - Ea’LI(0,1)

can be compared to the con-

dition in [18, p.431]. 2

ANALYTICITY WITH WENTZELL BOUNDARY CONDITIONS

AND TIMMERMANS

The main aim of this Section is to check that suitable conditions on the coefficients ~, 3 yield at the same time analyticity for the semigroup generated by the operators (Av, D(V)) and (AT, D(T)) above and the equality D(V)= D(T To begin with, we give a direct proof that under the assumptions as in [7, Theorem 2.2], we have (Av, D(V)) = (AT, D(T)) and it generates an analytic semigroup on

c[0, THEOREM 2.1 Assume that 6~ verifies

(1.1), (1.2), ~ EC~[0, 1] , fl ~ C[0, 1] and

~--~e C[0,II.

(2.1)

Then (Av, D(V)) = (AT, D(T)) and it generates an analytic semigroup Cli 0,1]. Pwof The translation

group T(t)u(x) := u(x + t), x,

is a Co-group on C[-~, +~], whose generator K is given by du

Ku:= ~, D(K): = {u e C[-oo, +oo]l u’ e C[-cc, +oo}}. Notice that for u ~ D(K) necessarily lim.~-~+oo u’(x) = O. If, for instance, Uoo := lima,+co u’(x) > 0, then for sufficiently large Xo we have that

u’(x) >-~-> 0, Z>Zo.

Differential OperatorsGeneratingAnalytic Semigroups

111

Thus, we obtain

,~(x) - ~,(Zo)= u’(~)(x>_~-~- ~o) where 5 is a suitable element of the interval (xo, x). This implies lim,~+~ u(x) = +ee, in contradiction with u E D(K). Hence lim.~_~+~u~(x) = 0 and lim~_~_~u’(x) = 0 is similarly proven. Now,since the square of K generates an analytic semigroup (see [16, A-II Theorem 1.15]), it follows that the operator 2= d2 ~ with domain D(K2) ={u E C[-c~, +cx~]l there exist u’, u" ~ C[-oc, +c~]} ={u ~ C[-cx~, +c~][ there exist u’ G C(-c~, +co), u" ~ C[-c~, +o~]} generates an analytic semigroup on C[-cx~, +oe] and the arguments as above assure that any u E D(K2) verifies lim u’(x)=

lim u"(x)=

0.

(2.2)

On the other hand, as it was pointed out in Section 1, for a -- 1 and ~ = 0, the domains D(f’) and D(~) coincide. Consequently, e =A~’~ = A--~. an d th ey generate an analytic semigroup on C[-oo, +c~]. Let us assume that r/ and introduce the operator (L, D(L)) by Lu :=u" + ~ flu D(L) :={u e C[-~, +oo] fl C~(-oe, +o~)l u" + r~u’ ~ C[-oo, +oo]}. Taking into account [13, Corollary 3.1.9], it is readily seen that (L, D(L)) generates an analytic semigroup on C[-oo, +~]..At the same time, the operator (S, D(S)) where D(S) := D(K~) and Su := u" + ~?u’, u 6 D(S) generates an analytic semigroup on C[-oo, +oe], because K2 2does and r~K is K bounded with K~-boundequal to zero (see e.g. [12, Corollary 6.9 p.42]). Notice that ~(t)Ku(t) = ~(t)u’(t) ~ 0 as since u’ -+ 0 and r/E L~(-~, Let us show that D(S) = D(L). Indeed, D(S) C_ D(L) and since for ,~ sufficiently large, both equations ,ku - Lu = I ~ C[-oo, +oo]

~v - Sv = y e C[-~, +o~] have a unique solution, we conclude that D(S) = D(L). Hence, S can be characterized by D(S) = D(K2) = {u e C[-~, +~]lu" + r/u’

e C[-~,

112

Faviniet al.

But every u ¯ D(S) satisfies lim u’(t) =

lira u"(t) =

so that Z~(S) C_ {~,¯ C[-~o, +oo] r~ C~(-oo, +~)1 lim Su(t) = O} C_ D(K2) = D(S). Therefore

{~¯ c[-oo,+o0]n c2(-oo,+~o)1u"+ ~u’¯ c[-oo,+oo]} = {~¯ c[-oo,+oo]n c~(-oo,+oo)l lim ~"(t) +,~(t)u’(t) Now, let us come back to our operators AT and Av. Introduce the mapping

¢(x) :=jf~xVh-~’ d s x ¯ (0,1). This is invertible and its inverse ¢ induces an isomety J¢ from C[0, 1] onto C[-oo, +oo] by means of Jeu(t)

= u(¢(t)),

as in [7, Lemma3.1]. Moreover,if we define

~(t):=2/~(e(t))-~’(¢(~)) 2~

’

t¯

R,

then we deduce that (AT, D(T)) is similar to (S, D(S)) = (L, D(L)) via the isometry Je, following the same argument as in [7, Theorem 2.1]. Hence, Av = AT and it generates an analytic semigroup on C[0, 1]. [] Wealso have the following consequence THEOREM 2.2 Assume that c~ verifies

(1.1),

(1.2),

~ ¯Ct[0, 1] ,/~ ¯

C[0, 1] and

-- ¯ C[O, 1].

(2.3)

Then (Av, D(V)) (AT, D(T) ) and it generates an analytic sem igroup on Cli1]. Proof. Previous arguments allow us to affirm that Av = AT generates a Cosemigroup on C[0, 1]. Weonly need to prove that this semigroup is analytic. Indeed, for any x in (0, 1)

/~(.~)_/~(x) d(x) L) ~ + d(x____

Differential OperatorsGeneratingAnalytic Semigroups and hence ~ E C[0, 1] in view of (1.2),

113

~ ECt[0, 11 and(2.3 ). []

Now, in order to present an alternative proof of the analyticity in the above Theorem 2.2, we show a Lemmawhich may be of independent interest. LEMMA 2.a Assume that ~ verifies (1.1), (1.2) and v~ e CI[0, 1]. Then, for each real-valued Junction m defined in [0, 1] such that re(x)

> O, forall

x e [0,11,

m e Cr[O, 1l,

(2.4)

the operator (Ms, D(M~)) defined

\m

D(M~):={ueC[O, 11C~C’~(O,1)llim. m(z) ~’ (x):0, j=0,1} generates an analytic semigroup on C[0, 1]. Proof. Similar arguments as in the proof of Theorem2.1 (see also [7]), lead to the conclusion that the operator Bu := V~u’ with domain D(B) := {u e Co(O, 1) a C~(0, 1)1 lim ~u’(x) = 0, j = generates a Co-group on C[O, 1]. Hence, according to [16, A-II Theorem 1.5], the square (B~, D (B~)) generates an analytic semigroup on C[O, 1]. Moreover, following [7] we can affirm that D(B~) = {u e Co(O, 1) cl C~(0, 1)1 lim. ~-(~u’(a) lim.c~(x)u"(x) x--* 3

and

B~ = v~

u"+~---~u’ .

Consequently, for each u ~ D(B’~), we have m ~

=m

u"

~’m

=B~u + kBu, where k .-

2v ~ m ~ C[0,

11.

= 0, j = 0, 1}

114

Faviniet al.

Since mB is B2-bounded with B2-bound equal to zero, the operator (M~,D(B~)) generates an analytic semigroup on C[0, 1] according to [12, Corollary 6.9 p.42]: In order to conclude the proof we observe that, if u E D(B2), then

and u E D(M~). By [16, Corollary 1.34 p.47] we deduce that D(B~) = D(M~) and the assertion follows. [] ALTERNATIVE PROOF OF ANALYTICITY IN THEOREM 2.2 observe that the operator

Firstly, we can

Tu = au" + Bu’, u ~ D(T) can be written in the form Tu = aexp(- ~~(S) d s)[exp(~ ~

~, ~fl(S) ds)u~]

u e D(T).

Let re(x) := a(x) exp(-

ds).

~3(s) ds) = a(-~)

Observe that 0 <mo<_ re(x) <_ ml,

¯ e [0,1]

for suitable constants too, ml, and m~ C~ [0, 1]. Since

and D(V) coincides with the domain D(M~) considered in Lemma2.3, we deduce that (Av, D(V)) generates an analytic semigroup on C[0, 1]. The assumptions on ~ together with the results proved in Section 1 allow to conclude that (Av, D(V)) (AT, D(T)). COROLLARY 2.4 Under the assumptions of Theorem 2.1 (Theorem 2.2, resp.), m is a real-valued function defined in [0, 1] which satisfies (2.4), then (mAy,D(I/’)) = (mAT, D(T)) generates an analytic semigroup C[0, 1]. Pro@It suffices to check condition (2.1) ((2.2), resp.) for the perturbed coefficients a := ms,

b :=

Differential OperatorsGeneratingAnalyticSemigroups

115

Indeed we have

~ e c[0,11, while in the respective case we have !rn

e c[0,11

and the assertion holds. [] One more analyticity result can also obtained as follows. THEOREM 2.5 Let us assume that a,b E C[0, 1] with a(x)

>0,

b(x)>O

~ e C1 [0,

and

xE(0,1)

11, v~ ~

1 x(1 - x)

fo

C1 [0,

1]

:

Then, the operator Mu := a(bu’)’ with domain D(M):= {u e C[0, 1] N C2(0, 1)] lim a(x)(bu’)’(x)

0}

x-~0~I

generates an analytic semigroup on C[0, 1] and D(M)= {u e C[0, 1] n C~(0, 1)1Mue C[0, 1]}. Proofi We begin observing that for any u ~ D(M) we have Mu= a b u" + a b’ u’. Let a := ab and fl := ab’. Then ’a b’

v~b

~ - ~ - ~ ~ c[o, if. Hence all the assumptions of Theorem 2.I hold for ~ and fl and the proof is cornpleted. [] REMARK2.6

Let us observe that if v~ e C~[0, 1], 0 < aWeC[0, 1],

where Wis still

(2.5)

defined as in Theorem1.1 and

o < do< a(x)w(~)< d~ i,,, [o, 1]

(2.6)

116

Faviniet al.

we have the following estimates do f~ ds d, J½ ~

~(s) ~(

<- -Q(x) - a(x)~V(x) a(s)

1

d~ ~ ds

-~ <x <~,

< Q(x) < ~ a(s)’

dx

= (

W(s) ds <_ -~o a-(-s)’

-~dx)ds

0

= ~ds

Notice that (2.5)-(2.6) i~nplies a(j) > 0 for j = O, 1, however there are some functions a which satisfy (2.6) even if a(O) = 0 = a(1) a(x) = ~ near 0 a nd ~(x) = ~-~ = a’ (x)). In thi s cas e W(x) = ~ and henc e a(x) W(x) = 2 fo r 0<x<~. Moreover from (2.5)-(2.6) we have

Ix - ] <

=

d~ [ ~(z) J

~

. ~(s)W(s)

~(x)’

<x <1

and d°

[½--~1

d8

<-R(x)...~W(x)~x½

d

0<x<

1 ~(s)W(s)< ~ l ~(~)

~, l ~(x) J

1 -. 2

Then it follows that Q ~ 51(0,

~), Q ~ LI(~, 1), R ~ LI(O, ~), R~ L~(~,

and hence the operators A~ and AT coincide. Notice that d~

~

-~ ~ C~ [0, 1]. and thus W Now, we consider Au := au" + flu’

a(x)W(x) = aW(W-~u~)~ with domain

D(A) := D(T) = D(V). Since according to [2, Theorem 2.3] we know that the operator given by D(A~o):={u e C[0, 1] n C’(0, 1)l W-in’ A~u :=(W-lu’) ’, u E D(A~o)

Cc vii0,

(A~,D(A~o))

11)

generates a Co-contraction semigroup and D(Aoo) = D(T), similar arguments as in the alternative proof of Theorem2.2 imply that (A, D(A)) generates a Co-semigroup on C[0, 1].

Differential OperatorsGeneratingAnalytic Semigroups

117

Moreover,

~(w-~) = a e~p(7 1

If we assume that ~ ¯ C[0, 1], then we can apply [2, Theorem3.7] to deduce that (A~, D(A~)) generates a bounded analytic semigroup on C[0, 1]. By Lemma2.3 we conclude that (A, D(A)) generates a bounded analytic semigroup on C[O, 1]. Thus, .we have proved the following variant of Theorem2.2 THEOREM 2.7 /fa,¢? ¯ C[0,1] satisfy (1.1)-(1.2), (2.1) and (2.5), AT = Av with domain D(V) D(T) ge nerates an analytic sem igroup on C[0, 1]. Using different techniques, as in Theorem 2.1, we have seen that assumption 0 < do <_ aW<_da is, in fact, unnecessary. REMARK 2.8 If v/~ ¯ C~[0, 1], then a ¯ C1

[0,

1] and if (2.3) holds, then

e~.p[~(.~ f~~~Z(8) ds)]a [0,q. Hence D(A~) = {u ¯ C[O, 1]V1CI(O, 1)IW-lu’ ¯

C1[0,11,

lim w-l(x)~t’(x)

~-+0,i

(see [2, Introduction]). On the other hand, since

- a’ a ~(aW) -

~ (aW) C[O, 1],

we deduce that

~rom

ff.~(W-d _lut)

-_

~ w_l,g

t

.~-

w_lutt

:

1 ( ~~,,+~u’ ) -gi-4

0}

Faviniet al.

118 ,are obtain also that

D(T) ={u e C[0, 11V~C2(0, 1)[ au" + f~u’ e C[0, 11,x~0,1 lira a(x)u’(x) 0} ={u e C[0, 1] ~ C~(0, 1)[~u" + ~u’ e C[0, 1], lim u’(x) 0}

¯~o,,. W(z)

:=P(n). Take ANU:= au" + ~u~ with domain D(N). Hence, under assumption (2.3) all the operators Av, AT and A~ coincide. Notice that if O<do ~aW~d~ and

then, according to [15, Section 6], au" + ~3u’ ¯ C[0, 1] implies lim u’(x) _ O,

¯ -~o,~W(z)

that AN= AT holds. On the other hand, since

1

1 l a(x)w(x) W(s)dsl dx =

~(z)w(z)I W(s)dsl

dx

= a(x)W(x) W(s) ds) dx + ~(~)~(~) ( ~(~) we obtain that a(x)W(x) (

W(s) ds) dx

( a(x)W(x)

)W(s) ds

1

> ~ ~W(s) do fo ~ s

Therefore, if a has multiple zeros at x = 0, 1, then a(x) = xmq(x), with m _> 2 and q(x) > 0 on [0, ½], q ¯ el0, ½] implying that ~ ds >_ smq(s)

d

s ~-mds = +o~.

A similar reasoning can be repeated on (½, 1). Summingup, we have shown that, if a,f~ ¯ C[0, 1] verify (1.1),(1.2), (2.1) and (2.4), Av = AT = ANgenerates an analytic semigroup on C[0, 1]. In particular, this holds for a,~ ¯ C[0, 1] which satisfy (1.1),(1.2) and (2.3).

Differential OperatorsGeneratingAnalyticSemigroups

119

REFERENCES 1.

2. 3.

4. 5.

6.

¯ 7.

8.

9.

10. 11.

12. 13. 14. 15. 16. 17.

V.Barbu and A.Favini, The analytic semigroup generated by a second order degenerate differential operator in C[0, 1], Suppl. Rend. Circolo Matem.Palermo 52 (1998), 23-42. M.Campiti, G.Metafune and D.Pallara, Degenerate self adjoint equations on the unit interval, SemigroupForum 57 (1998), 1-36. Ph. Clgment and C.Timmermans, On Co-semigroups generated by differential operators satisfying Ventcel’s boundary conditions, Indag. Math. 89 (1986), 379-387. K.J.Engel and R.Nagel, One-parameter Semigroups for Linear Evolution Equations, Springer Verlag (to appear). A.Favini, J.A.Goldstein and S.Romanelli, An analytic semigroup associated to a degenerate evolution equation, Stochastic Processes and Functional Analysis, (J.A. Goldstein, N.A. Gretsky and J.Uhl, eds.), M.Dekker NewYork, 1997, pp. 85-100. A.Favini, J.A.Goldstein and S.Romanelli, Analytic semigroups on LP~(0, 1) and on LP(O, 1) generated by some classes of second order differential operators, Taiwanese J. Math (1999) (to appear). A.Favini and S.Romanelli, Analytic semigroups on C[0, 1] generated by some classes of second order differential operators, Semigroup Forum 56 (1998), 367-372. A.Favini and S.Romanelli, Degenerate second order operators as generators of analytic semigroups on C[0, +cc] or on n:_ ½ (0, +~c), Approximation and Optimization, Proceedings of International Conference on Approximation and Optimization, Cluj-Napoca, July 29 - August 1, 1996, VolumeII (D.Stancu, G.Coman, W.W.Brecknerand P.Blaga eds.), Transilvania Press, 1997. A.Favini and A.Yagi, Degenerate Differential Equations in Banach Spaces, Pure and Applied Mathematics : A Series of Monographs and Textbooks 215, M.Dekker, NewYork, 1998. W.Fetler, The parabolic differential equations and the associated semi-groups of transformations, Annals of Math. 55, No.3 (1952), 468-519. G.Fichera, On a degenerate evolution problem, Partial Differential Equations with Real Analysis, (H.Begehr and A.Jeffrey, eds), Pitman Research Notes Mathematics Series 263, LongmanScientific and Technical, 1992, pp. 15-42. J.A.Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, Oxford NewYork, 1985. A.Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkh~user Verlag, Basel Boston Berlin, 1995. G.Metafune, Analyticity for some degenerate one-dimensional evolution equations, Studia Math. 127 (3) (1998), 251-276. G.Metafune and D.Pallara, Trace formulas for some singular differential operators (preprint). R.Nagel (ed.), One-parameter Semigroups of Positive Operators, Lecture Notes Math. 1184, Springer-Verlag Berlin Heidelberg NewYork, 1986. C.A.Timmermans, On Co-semigroup in a space of bounded continuous functions in the case of entrance or natural boundary points, Approximation and Optimization, Proceedings of the International Seminar held at the University

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of Havana (Havana, January 12-16, 1987), J.A.G6mez Fernandez et al. (eds), Lecture Notes in h~Iathematics, 1354 Springer-Verlag, Berlin NewYork, 1988, pp. 209-216. 18. H.Triebel, Interpolation Theory, Function Spaces, Differential Operators, Math. Library 18, North-Holland Publ. Co. Amsterdam New York Oxford, 1978.

A Characterization of the Growth Bound of a Semigroup via Fourier Multipliers MATTHIAS HIEBERMathematisches Institut I, Englerstr. 2, Universit~t Karlstube, D-76128 Karlsruhe, Germany, email: [email protected]

1

INTRODUCTION

A classical result of Liapunovasserts that the solution of the initial u’(t)

value problem

= Au(t), u(O)

where A is an n x n matrix, is exponentially stable provided that all eigenvalues of A have negative real part. On the other hand, it is well knownthat, in general, the asymptotic behavior of a C0-semigroup T acting on a Banach space X cannot be described adequately by the location of the spectrum a(A) of its generator A. Indeed, set s(A) := sup{ReA;A E a(A)} w0(T) := inf{w e ll~;sup Ile-~tT(t)ll < c~}. (1.1) Then s(A) <_ wo(T), but strict inequality mayoccur. For examples illustration this fact, we refer to [5],[7] and [9]. For this reason it is interesting to characterize the growth bound to0(T) of T by properties of the resolvent R(A, A) of A. For the of Hilbert spaces H, it was shownby gearhart (see [5], p.95) and Prfiss [8] that too(T) = inf{# > s(A); An example of a semigroup given by Arendt shows that the above characterization of too(T) is restricted to the case of Hilbert spaces. For details, see [2] and [7], Ex.l.4.4. It is the aim of this note to present a generalization of the result described above to semigroups T acting on arbitrary Banach spaces X. In order to do so, we make use of the concept of operator valued Fourier multipliers. For related results we refer to [3] and [10]. 121

122 2

Hieber EXPONENTIAL

STABILITY

AND CONVOLUTIONS

A result due to van Neerven asserts that uniform exponential stability of a Cosemigroup T on a Banach space X, i.e. wo(T) < 0, can be characterized by the LP-boundednessof certain convolution integrals. More precisely the following holds true. PROPOSITION 2.1 (van Neerven [6], Thm.3.1.8, [7], Thm.3.3.1.) Let T be a Cosemigroup on a Banach space X with generator A and let 1 <_ p < oo. Then wo(T) < 0 if and only T*tfeLP(]ll+;X)

for all

f E LV(R+;X).

Here the term T ’l f is defined by T ** f(t)

:= T(s)f(t-

s)ds,

t >_

The following proof of Proposition 2,1 seems to be more direct than the one given in [61 or [7]. PROOFFor f E LP(II~;X)

define

f E LP(~; X) by

( f(t) t ~_ ](t) := ot < and set K(t)

:--~

[ T(t) > o

0 t<0. Assumethat ~o(T) < 0. Then t ~ K(t) L~(ll~; £( X)) an d it fol lows fro m You inequality for Banachspace valued functions (see e.g. [1]) that K * f ff LV(l~; and hence that T ,~ f e LP(tl+ ; X). Con’,’ersely, choose w > max{0,wo(T)}. For x e X and t >_ 0 we then have t

fo

T(t- s)e-~’~T(s)xds :

Define a fl~nction g we have that

1--(1 -

e-~’)T(t)x.

W

~ LP(~.+;X)

by g(s) := e-’"ST(s)x.

(2.1)

By assumt)tion and (2.1)

W

and hence that f~ IlT(t)xll~dt implies now that wo(T) <

3

< ~, Datko’s theorem (see e.g. [7], Thin. 3.1.8) ~

GROWTH BOUND AND FOURIER

MULTIPLIERS

Let 1 _< p _< oc. Wecall m E L°°(I~ ; £(X)) an operator-valued Fourier multiplier " ) if for LP(IR ~--i(m]’) :=.T-l(m)*f, f ~ L’(I~’~;X)

123

GrowthBoundof a Semigroupvia FourierMultipliers

is well defined and belongs to £(LV(IRn; X)). For the precise definition of the Fourier transform and the convolution product in (3.1), we refer to [1] and [4]. The set of all £(X)-valuedmultipliers for LB(I~’~; X) is denoted by A4pc(x) (IR~ ; X). If no confusion c(x) for MpC(X)(IR; seems likely, we also write Mv THEOREM 3.1 Let T be a Co-semigroup on a Banach space X with generator Let 1
A.

wo(T) = inf{# > s(A); I1~( REMARK A result similar due to Theorem 3.1 was obtained by Clark, Latushkin, Montgomery-Smithand Randolph (see [3], Thm. 4.4) using the theory of evolution semigroups. PROOFDefine r0(A) := inf{# > s(A);supo.>_ u IIR(a + i.,A)IIM~¢.~ < oo}. first show that ro(A) <_ wo(T). In fact, assume the contrary and choose a ~ (w0 (T), r0 (A)). Denote by Ta the exponentially bounded semigroup on X generated byA-a. Fort~lRset ~G(t):= 0T~(t) t>O t<0. and for f e L~(1R+;X) and t ~ IR set

](0:= {f(t) o It follows from Young;s inequality for Banach space valued functions that K~ ¯ ] E L;(IR; X) provided ] e LP(]R; X). Taking Fourier transforms we see that

fo

~ eitT~(t)dt

R(i., A - a) ~ £(x)

and that there exists a constant M> 0 such that 1

IIR(~ + i., a)ll,a~.,~< M a - ~o(T)" This contradicts the definition of ro(A). Wenext show that to(A) >_ wo(T). Assumero(A) < wo(T) and choose w~,w~~ IR such that wo(T) < w~ < w~. Then

supII_r¢(~

~>~o(T)

The resolvent equation implies that for s 6 I~ R(co~+ ,is, A) = R(~.~+ .is, ,4) + (w.~ - o0~)R(w~ + is, A)R(cv.~ Therefore

124

tlieber

and letting wi -+ w0(T)it follows that IIR(wo(T) + i., m)ll~¢x) (1 + M(w2 - wo(T)))llR(~.~ + i ., A)II Hence Two(T) *t f E LP(IR+;X) for all f ~ LP(~+;X), where Two(T) denotes the semigroup generated by A-wo(T). Theorem 2.1 implies now that the growth bound of T,~o(T) is strictly negative in contradiction to the construction of T~o(T ). Choosing in particular

p = 2 and X = H a Hilbert space we have .Mec(x)(ll~" ; = L~(n, £(H)

(3.2)

Hence we immediately obtain as a corollary the following well knownresult due to Gearhart (see [5], p.95) and Prfiss [8]. COROLLARY 3.2 Let T be a Co-semigroup on a Hilbert space H with generator A. Then wo(T) = inf{# > s(A); sup IIn(A,A)ll< ~}. Re

REFERENCES 1. AMANN, H.: Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications. Math. Nachr. 186 (1997), 5-56. W.: Spectrum and growth of positive semigroups. In: G.Ferreyra, 2. ARENDT, G.R. Goldstein, F. Neubrander (eds.) : Evolution Equations. Lect. Notes in Pure and Appl. Math. 168, Marcel Dekker, 1995, 21-28. 3. CLARK, S., LATUSHKIN, Y., MONTGOMERY-SMITIt, S., RANDOLPH, T.: Stability radius arid internal versus external stability in Banachspaces: an evolution semigroup approach. Tiibinger Berichte zur Funktionalanalysis 1999’/98 7, 72-102. 4. HIEBER,M.: Operator valued Fourier multipliers. In: J. Escher, G. Simonett (eds.): Topics in Nonlinear Analysis, The Herbert AmannAnniversary Volume. Birkh/iuser, 1999, 363-3S0. 5. NA~aL,R. (ED.): One Parameter Semigroups of Positive Operators. Lecture Notes in Math. 1184, Springer, 1984. VAN NEERVEN, J.M.A.M.: Characterization of exponential stability of astral6. group of operators in terms of its action by convolution on vector-valued function spaces over II¢+. J. Diff. Eqns. 124 (1996), 324-342. 7. VANNEERVEN, J.M.A.M.: The Asymptotic Behavior of a Semigroup of Linear Operators. Operator Theory Adv. Appl., 88, Birkh/iuscr, 1996. 8. PPJJSS, J.: On the spectrum of Co-semigroups. Trans. Amer. Math. Soc. 284, (1984), 847-857. 9. RENARDY, M.: On the linear stability of hyperbolic PDEs and viscoelastic flows. Z. Ange~. Math. Phys. 45 (1994), 854-865. 10. WE~S,L.: Stability theorems for semi-groups via multiplier theorems. In: M. Demuth, B.-W. Schulze (eds.): Differential Equations, Asymptotic Analysis, and Mathematical Physics. Akademie Verlag, 1997, 407-411.

Laplace Transform Theory for Logarithmic Regions PEER. CHRISTIANKUNSTMANN Universit~it Karlsruhe, stitut I, Englerstral3e 2, D-76128 Karlsruhe, Germany e-mail: [email protected]

Mathematisches

In-

ABSTRACT Wedefine new spaces of test functions and distributions admitting a Laplace transform in the classical sense, i.e. by evaluation at exponentials. Weuse these spaces to give a characterization of the Laplace inverses of analytic functions which are polynomially bounded on logarithmic regions and have values in a Banach space. As an illustration we give a conceptually new proof of the characterization of tempered convolution operators that have a distributional fundamental solution. Connections to asymptotic Lapalce transforms and Laplace transforms of (Laplace) hyperfunctions are sketched. AMSSubject Classification: 44 A 10, 46 F 05, 44 A 35. Keywords:Laplace transforrn, logarithmic regions, distributions, convolution equations, fundamental solutions.

1

INTRODUCTION

The classical Laplace transform £f of an Ll-function f on [0, co) is given by if.f (A)

e-~ f(t) dt,

for, at least, Re A >_ O. For those A we have e-~’ E L¢~ = L~. Considering f as a functional on Lo~, we see that £,f(A) = f(e-~’). Thus, the definition can be extended to certain distributions with support in [0, co), and we obtain the Laplace transform of such distributions T by evaluation at exponentials: £T(A) := T(e-~’). 125

(2)

Kunstmann

126

Hence e-x" should belong to the underlying space of test functions or at least to its completion with respect to a suitable seminor~n. The space of test functions which is the basis for classical Laplace transform theory is K:~,+, the space of all C~-functions on [0, c~) which together with all their derivatives tend, as t ~ c~, faster to 0 than any exponential. K~,+ is a Fr6chet space for the family of norms p~,,(~)

:= sup

e~tl~,(Y)(t)l, k ~

t>_O,j=O,...,k

and does not contain any function e-~’, A ~ ¢, but all of its completions(K:~,+, Pk)~, k ~ SV0do. For any Banach space X, the space K:I,+(X) := L(K~,+, X) is the space of X-valued exponentially boundeddistributions on the half line [0, ~). Its elements T have Laplace transforms £T, defined by (2), that are polynomially bounded holomorphic functions on right half planes {Re ~ > 3’} where 3’ and the particular growth bound depend on T. Conversely, to each polynomially bounded holomorphic X-valued function F on a right half plane {Re A > 3’} the inversion formula T := -2~ri

e~F(A) e~=~+~

(3)

defines an element T of K~,+(X) such that £T = F. Hence we have a characterization of polynomially bounded analytic functions on half planes in terms fo the space tQ,+ (see [3], chapter 8, and observe that K:L+ = proj.~ exp(-7.)S + where $+ denotes the space of the restrictions to [0, of the functions in the usual Schwartz-space Leaving the case of half planes we make the following observation: If we have an X-valued holo~norphic function that is polynomially bounded but does only exist on a region of~ that does not contain any right half plane, the formula (3) might - with a suitable change of the contour - still define a distribution T ~ iO~(X). Wegive an example where this phenomenonarises for a special type of regions. The theory of Laplace transform is a suitable tool for dealing with autono~nous differential equations or, more general, with convolution equations of the type P*U= T, T e:D’o(E),P

~S~(L(D,E)),

(4)

where E and D are Banach spaces and the subscript 0 indicates tliat the distributions in these spaces have support contained in [0, c~). For this kind of equation and details about vector-valued distributious we refer to chapter 8 of [3], the case P := ~ ® I - (~ ® A gives the connection to the theory of semigroups of linear operators (see [6]). The solution U of (4) we seek for should belong to :D~(D), and the equation is uniquely solvable for any T if and only if P has a fundamental solution G ~ Z)~o(L( E, D) satisfying P*G=5®I~

and

G*P=6®ID,

(5)

in which case U is obtained by convolution: U := G * T. H.O. Fattorini (see [3], Theorem8.4.8) characterized those P that have a fundamental solution in ter~ns of the Laplace transform t:P of P which has values in L(D, E). To be precise, P has a

LaplaceTransformTheoryfor LogarithmicRegions

127

fundamental solution if and only if £p(,~)-t L(E, D)exi sts and is p olynomially bounded on a logarithmic region A. -1 Formally , applying the Laplace transform to (5) indicates that £G(A) = £P(A) hence - somehow- the Laplace transform exists, even as a function, but in general G is not exponentially bounded, which means that it is not clear what EG(,~) should be (in terms of G). Also, logarithmic regions are far from containing right half planes. There are several ways to circumvent this difficulty. H. Komatsuhas introduced a Laplace transform theory for hyperfunctions via socalled Laplace hyperfunctions ([5]). Since each element of T)~)(X) is also an X-valued hyperfunction, it possesses a Laplace transform which is an equivalence class of germs of analytic X-valued functions defined on a neighbourhood of the "right half circle at infinity" of the radial compactification of the complex plane IT. Another way was developped by B. B~iumer([1]), G. Lumer and F. Neubrander ([8]) using an asymptotic interpretation of (1). The Laplace transforms of LLloc-functions are then also equivalence classes of X-valued analytic functions defined on postsectors. One may show that, in a certain sense, the Laplace transforms of [5] and of [1] and [8] coincide (see [7]). Wewill use another approach here, giving (2) a meaning in the classical sense, in cofitrast to the asymptotic approach in [1], [8], or even [5] (recall that asymptotics enter when passing from hyperfunctions with compact support to hyperfunctions with support in [0, ~c), this phenomenonis reflected by the fact that each hyperfunction induces an equivalence class of Laplace hyperfunctions). Weintroduce suitable test function spaces on [0, ec) containing exponentials such that the corresponding distributions are characterized by having a polynomially bounded Laplace transform on a logarithmic region, i.e. on a set of the form {Re A > max(7, 1 log IAI + The space of those distributions is contained in 79~(X). Comparedwith the classical theory of Laplace transform for exponentially bounded distributions, which is a "right half space theory", our theory is a Laplace transfrom theory for logarithmic regions. Weconcentrate here on logarithmic regions since this seems to be the simplest case. They also appear in [5], [1], [8], and our characterization result is very useful in connections with those theories since it shows exactly what one gets by Laplaceinverting holomorphic functions which are polynomially bounded on logarithmic regions. In particular it shows that having a polynomially bounded Laplace transform on a logarithmic region does not only imply the local property of being a distribution but also a 91obal property involving an interplay between "mere growth" and "evolution of regularity in time". Hence,the - highly desirable - identification of Laplace transforms of distributions or even of Ll,~o~-functions or continuous functions a~nong Laplace transforms of (Laplace) hyperfunctions needs further research: Howcan we read off local regularity properties of (Laplace) hyperfunctions T from their Laplace transforms The paper is organized as follows. Section 2 contains the definition of suitable Banach spaces and their basic properties concerning Laplace transform. In section 3 we define test function and distribution spaces and use the results of section 2 to derive the announced characterization. As an illustration of our theory, we present

128

Kunstmann

in section 4 a conceptually new proof for the result of Fattorini. with some examples and concluding remarks.

2

BANACH SPACES

OF TEST

FUNCTIONS

The paper closes

AND THEIR

DUALS

The spaces introduced in this section will be the basis for the test function and distribution spaces we introduce in the next section. Webegin with an analogue of the spaces Lt(~/) of all measurable functions f [0, cc) -~ (Z’ satisfying IlfllL,(~,):= j’~e~tlf(t)l dt < co. Let c~ > 0 and/3,3’ .e t/-/. We define

We consider here measurable complex-valued functions f defined on [0,~c). The derivatives f(~) are understood in the weaksense. Weoccasionally write I1’ IIw, the constants a, fl, ~/are clear from the context. Clearly, (Wl(a,~,7),l]. I]w~(a,~,~,)) is a Banach space which contains ~functions [0, oo) -~ (Z’ with compactsupport. It is continuously embedded Setting 0. c~ := 0 and r. cc := cc for r > 0 we even have W~(co, 7, 7) = L~ (3). Wenow study the Laplace transform of functions f ~ W~(a, ~, 7). To this purpose, we first give a representation lemma. Let / ~ W, (c~, ~,~). For all Re A >_ -7, I)~1

LEMMA2.1 have

PROOFWe first

>_exp(a(’)’

- fl)),

observe that ¯ lim exp(Tt) ] (t)

(7)

for each k _> 0. Fromthis and integration by parts we get for each k >_ i and s _>_ e’~Sf(k-~)(s) = -

e~tf(~)(t)

dt - 7

(8)

e~tf(~-~)(t)

Hence the expression f(t~-~)(ak) ~nakes sense for k > 1. Wenow split the integral .f~ exp(-At)f(t) at o~ andinte grate by p art s (rec all Re )~ >-~ and ( 7)) to Obtai e-Xtf(t)

dt = e-Xtf(t)

dt e- X’~f(c~) +

e- ~t f’(t) dt.

Splitting the integral at 2a we again integrate by parts. Repeating this (which means splitting at kc~ and integrating by parts) we get that f~ e-~tf(t) dt eqnals

E k=0

e-’Xakf(k-l)(cd~:)

e-Xtf(k)(t) Jo~k

k=l

e-Xtf(")(t) n

LaplaceTransformTheoryfor LogarithmicRegions

129

The assumptions on f and ,~ imply that the remainder term f~,~.., 0 as n -~ oo. This gives the assertion. []

dt/A’" tends to

Weuse this representation, valid for Re A _> -7 and I~1 > exp(a(7-/~)) to extend the Laplace transform of f E Wl(a,/~, 7) beyond {Re ~ >_ -7}. PROPOSITION2.2 Let f ~ W~(a, fl,7). The Laplace transform tended to an element of H(~a,Z,3.) V~ C(f~a,/L~) where

£f can be ex-

1 ~,~,~ := {~ e g’ : Re ~ > -7 or "y < -Re k < - log Ikl +/~}. For ~ ~ f~,,~,y we have the estimate

, u~ > -7 , ReA < -7.

(9)

PROOF Wefirst observe that all coefficients in the series in the representation (6) are analytic functions in ~ ~ 0. Tlms we only have to study convergence of the series for ReA < -7. Weestimate the first series in the representation (6) o’{kq-1)-I~ReXlfl~’)(t)l dt <_ e~tlf{~)(t)l = e -~(~+~)

~~(z-7) ~e-~(n~’x+e)e

e~tlf(k)(t)ldt k

k=0 ~ e-~(Re~+~)(sup(e-~(Re~+~),k~O

’ ~ ~/k)

Using (8) for s = ak we estimate the second series in (6)

Tim sup’s are finite (which is equivalent to _< 1 here) if and only if .k ~ ~,z,-r. This implies the assertion since

for those ~. [] In order to see what the regions fl~,/~,’r can look like we denote, for REMARK2.3 every { >_ 7, by I~ the set

{~ e ~n: -{ + io ¢ ~,e,,} = {~ e ~: { > (2~)-x log({~ + ~) + ~}. Then g’ \ f/a,5,~ is connected if and only if I{ # ~ for all { > 7 if and only if

{ > a-~logl{I +~, {>7.

(10)

130

Kunstmann

For ~ = 3’ = 0 this happens if and only if a >_ 1/e as can be seen by co~nparing the functions ~ ~-~ ~ and ~ ~-~ a-~ log~. For a < 1/e the boundary of ft~,o,o is not connected. Moreover,0f~,~,~. can be parametrized by the imaginary part if and only if ~ ~, I¢ is strictly increasing which is the case if and only if aexp(2c~(~- ~)) > ~, ~ >

(11)

This happens for fl = 3’ = 0 if and only if a > 1/v/~. Of course, the conditions (10) and (11) are for fixed a and .3 always satisfied if is sufficiently large. Weintroduce the space (12) k_>0

where]]f]]w~(~,fi,,) := supk>_o,t>_~k e(’O-’~)a~’+’~tlf(k)(t)], and its closed subspace Wo(a,¢~,~,) := {f ~ ~’~ C0~([~k,~),eT) : lim e(’° -7)~’+~tlf(~)(t)l = O} k~O

k

~ t~k

(13) where C~([a, ~), "~’) denotes t he s ubspace of Ck([a, ~)) c onsisting o f t hose f unctions f that satisfy timt~ e~*[f(J)(t)] = 0 for all j = 0,..., k. The following two propositions showthat, as l,~ (a, ~, ~) plays the role of L~ (~), the spaces ~l~(a, ,~, 7) and Wo(a, ~, 3’) play the roles of the corresponding L~C0-spaces, respectively. PROPOSITION 2.4 Let f ~ Wc~(a,,~,3’). Then £ f has an analytic continuation to ~,;~,~,. and for A ~ f~a..~,~, we have the estimate

-

{1

(1 + ~lAle~(R~ x+~))e~(~-~Id~,~(A)[[f[lw~,

Re A > -7 Re A <_ -7 Re A _< -7 and

(14)

~(~IAI~) >e

~(~~+m_ 1)-~, wh~re d~,~(A):= (IA[~ Observethat, just as Re A + 3’ is the distance between ,~ and the line {Rez = -3’}, the expression d~.,~(~) -~ is a measure for the distance between A and the curved parts of Ofi~,.o.~,,. Observe also that the estimate in the third line of (14) is only interesting for Re A +3’ s~nall. PROOFThe first line in(14) is clear and we turn to the second. Weuse again representation (6) which is nowvalid for Re A > -3’ and I~1> exp(a(3’ - ¢~) as be seen by repeating the proof. Weestimate the first series in (6) for Re ~ <_ -3" by t’ ~a~fa(t~+;) -~(Rex+~,)dt EC~ e_c,t.(~_~)[A[ k=O

LaplaceTransformTheoryfor LogarithmicRegions

~

e-~(~-’)) -’~

131 x+~’i~(~+~)

k=0

k=0

¯=

~llfllw~(~,~,~)e"(~-~)d~,~(a)

Weestimate the second series in (6) [~f[[w~(~,~,~)

~ -~ ~e -~e-~(~-1)(~-’)

k=l k

This proves the second line in (14) and we turn to the third. Weuse the representation (6) and estimate the first series as before nowusing that (~+1) e -(Re -(Re ~+Tlt dt ~ ae

~ k

Hence the first series in (6) is I~fl~w~(~,~,~l~e~(~+~exld~,~(X). estimate of the second series in (6) nowyields the third line in (14). Up to Laplace the set Laplace

The ab ove

now, we have seen that the elements of Wr(c~,/3,7) for 6 {O,l,c~} ha ve transforms on certain supersets of a right half plane, namely essentially on 12~,Z,~. Wenow show that the elements of the duals of these spaces have transforms on logarithmic regions 1 A~,~,-~ := (T \ (-~,2,~.) = {Re z > 7 and Re z > - log

To this purpose we first prove PROPOSITION 2.5 Let it 6 (~. Then exp(-it.) if It 6 A~,~,-r where

belongs to Wl (a, fl,7)

i] and only

and for ~ ~ A~,~,~. we have II exp(-~’)l[w,(.,e,~,) Moreover, the mapping ~ ~ exp(-~.)

1 - Re>- ~(1 is analytic

-’.

(15)

on Aa,B,~ with values in

w~(~,~,~). The ]unction exp(-~.) belongs to W~(~,~,7) ff and only ff ~ ~ A~,~,~, and it belongs to Wo(~,~,7) and only if ~ ~Aa,~,~. In e ith er case we then have

II exp(-,’)llw~(.,~,~) I[ exp(-~.) belongs to one o] the spaces its Laplace trans[orm is given by Cexp(-p.)(£) = (~ + for £ in t heappropriate region.

132

Kunstmann

PROOF We have e-t(n~"-~l dr,

II exp(-p’)llw, (~,3.~) = ~ e~(~°-~)~’lt~l~’ k

k=O

which clearly is = ~ if Re p ~ ~/and otherwise equals

Rep - 7

k=O

wherethe series is finite if and only if Re p > a-~ log ]p~ + ,~, and in this case equals For W~(a, ~, 7) we have ]~exp(-~.)Hw~(a,Z,7)

-~¢n ~’-7}, = e~{~-7)k~t~e

which is = ~ if Re p < 7, and otherwise equals

which is finite if and only if Re ~ 2 a-~ log I~*l+ ~, and in this case = 1. Furthermore, exp(-p.) ~Cg([ak,~),e ~’’) if andonly if Re p >7, a ndin this case lim e ~(~-~)k sup e~t[pl~e-*n~’ = 0 k~

t>ak

if and only if Re p > a-~ log [p[ + ~. For the proof of analyticity we restrict ore’selves to the case W~. By Cauchy’s theorem analyticity is clear if p ~ exp(-p.) is continuous since then integrals hr := fr exp(-p.)dp over closed contours are elements of ~V~(a,~,7) and application of functionals g ~ L~(O,a) and ~t, t > a, yields hr = O. Writing II exp(-A.) - exp(-.’)llw,

= ~(a-~) k=0

k

eXtlAt’e-xt - .ke-"t I dt

we apply dominated convergence twice, first for the integral and then for the sum to get lim~, exp(-A.) = exp(-p.) in Wl(a,~,7). The assertion on £exp(--p.)is then clear by continuation. ~ Denoting by lVr(a, fl,7)’ the dual space of Wr(a,.~,7) immediately get the following.

for r ~ {0,1,o~},

COROLLARY 2.6 Let T ~ W~(a,3,7)’. Then the Laplace transform T(exp(-#.)) is analytic on A~,~,7, and we have the estimate I£T(/~)I <

------~-~(1 -.. IITIt -I#le~(z-~")) Re#- 7

£T(p) :=

(16)

For T ~ I,V~(a, fl,7) ~ the Laplace transform ezists on A~,~,.~, is analytic on and bounded by IITII. The same holds for the Laplace transform oft ~ Wo(~,fl,7)’ with the domain of existence replaced by

133

LaplaceTransformTheoryfor LogarithmicRegions

Compared with Laplace transforms of L~-functions, the factor (Re # - 7) -1 in (16) is natural. Observe that the inverse of the other factor appearing in (16) measure for the distance between # and the arcs of 0Aa,~,v. Wewant to compare the spaces introduced so far and define for r E {0, 1, c~} and Wr~(a, fl,7) := {f ~ Wr(a, fl,7) : f@) ~ Wr(a,~,7) for u : 0,...,k ~ (a, ~3, 7)’. Thenwe have the following inclusion result. and their duals 14Z~ PROPOSITION 2.7 If7 >_ 0 then l’Vit(a,/3,7) have 14~(a,~ +6,7+d ~ Wt(a,~,7).

¢-~ Wo(a,/~,7). For all 6, e > 0 we

PROOF In the first case we have f(~I(t) = -.[~ f(k+~)(s)ds which implies by 7>0 te~(~-’~)~+’~ e~(~-~)~+’~tlf(~)(t)[

I$(~+O(s)lds

The last term tends to 0 as t ~ ~ and is majorized, for t ~ ak, by

k which in return tends to 0 as k ~ ~ and is majorized, for k Thus the first statement is proved. To prove the second statement, observe that

k=0

~(~-~ ~ Ilfllw~(~,~+a,~+,~e k=O

k

and that the sum equals

k=0

Weuse the W~-spacesto formulate the reverse of Corollary 2.6. PROPOSITION2.8 Let ~/ > 0 and F ~ H(A~,Z,~) ~ Cb(A~,~,~). Laplace transform of an element T ~ W3~(a, 8, ~)’. PROOFWe let

Then F is the

F := OA~,~,~ and define S(~)

:=

£~o(-A) dA,

(17)

134

Kunstmann

tbr ~ E Hq(ct,,g4,’~). Here F is oriented such that Ao,:3,~ lies to the right of Since 0 ¢ l’~,~,z,~ and F is bounded we get fi’om Proposition 2.2 that the integrand decays like ~-2 as Ik[ -+ c~ and that (17) defines an element of W1(c.~,/3, ~/)’. the Laplace transform of S we obtain by Proposition 2.5

cs( )

1 fr F(,~)

ea,

which equals F(#)/p a by Cauchy’s formula (recall the orientation of F). If we let T := S’" in the distribution sense, we have T ~ Wi~(a,/3,’7) ’ and T(#) paS(p.) = F(#). fil Wemention the following denseness result which is proved in [7] using ideas from the proof of Nyman’s Lemmain [2]. THEOREM 2.9 Do is dense in W~(a,/~,7). COROLLARY 2.10 The space l)+ of restrictions to [0, oo) of functions from 7) is dense in all spaces W¢’(ct,/~,7) where k ~ *Wo. We(:lose this section with the following remark on X-valued counterparts of the dual spaces. REMARK 2.11 Let X be a complex Banach space. For r ~ {0,1,co} and k ~ /No, we denote by Wf(~,/3, 7)’(X) the space L(Wr~(c~,~,7), X) of bounded operators Wr~ (c~,/~,’),) -~ X supplied with the operator norm. Then the assertions of Corollary 2.6 and Proposition 2.8 remain true where the Laplace transform now has values in X.

3

SPACES OF DISTRIBUTIONS, AND CONVOLUTION

LAPLACE TRANSFORM,

In this section we define spaces of test functions using the basic Banachspaces of the previous section. These test function spaces are nuclear Frfichet spaces. Wealso define the corresponding spaces of distributions and prove the announced characterization (Theorem 3.1) of their Laplace transforms. Wefinally use Theorem3.1 to define the convolution of the distributions we introduced. For any a > 0 and/3, 7 E (-oo, oo] we define 14;~,~,~ := projj<e,5<~,~e~.° W¢’(~,/~, ~) = proj.~0Wc,. All these spaces are clearly Fr6chet spaces which, by standard techniques, can be shown to be nuclear. Note that, recalling our conventions, the space W~is the space/C~,+ defined in the introduction. For any Banach space X we now define W~,.o,.~(X ) := Lb(Wc,,B,.~, X). The following characterization, whose proof is nowi~nmediate, is the main result of this section.

LaplaceTransformTheoryfor LogarithmicRegions

135

THEOREM 3.1 Let X be a Banach space and T E D’o(X) be a distribution. Assume that a > 0 and/3, 7 E (-~, c~]. The following are equivalent: (a) We have T ~ W’~,~,~(X). (b) There are ~ < ~3 and :~ < 7 and a polynomially bounded holomorphic function F: A~,3,~ --> X such that 1 freX.F(X)dA T i = -~-~ in the sense of distributions where F := OA~,~, 5. Moreover, if (a) holds, then (b) holds with £T PROOFThe implication (b) ~ (a) is clear from the inversion result Proposition 2.8 and Remark2.11. The implication (a) -~ (b) is cleat’ from Corollary and Remark 2.11 for /3,7 < ee. For /3 = o¢ or 7 = oe we first have to choose finite /3’ < ,~3 and 7’ < 7 such that T ~ ld,’~,~,#(X). This is possible since W~,~,-r = proj~,<e,~,<.~ld:a.,/~,,v, and 14,’~,~,~ is dense in all the spaces appearing in this projective spectrum. [] Wemay use Theorem 3.1 to define convolution via Laplace transform. If X, Y, and Z are Banach spaces and B : X x Y --+ Z is a continuous bilinear mapping, then the convolution T *~ S with respect to B of elements T ~ I/Y’(X) and S e ]N’(Y) is obtained by Laplace-inverting the Z-valued holomorphic function A ~ B(ET(A), ES(A)) which is polynolnially bounded and defined on a logarithmic region. This gives T *s S E IN’(Z). From the construction it is clear that £(T ,~ S)(A) = B(£T(A), £S(A)). The perhaps most interesting case is X = L(Y, Z) and B the evaluation map. It may be shown (see [7]) that the convolution thus obtained coincides with other definitions of convolution of vector-valued distributions (see [3] and [6]).

4

FUNDAMENTAL SOLUTIONS OF CONVOLUTION ARE EXPONENTIALLY BOUNDED

EQUATIONS

Let D and E be Banach spaces and assume that P ~ 8~(L(D,E)). We show in this section that a fundamental solution G for P is always exponentially bounded in an appropriate sense, i.e. when we compensate growth by regularity. It is well knownthat a fl~ndamental solution need not belong to K.~,+(L(E,D)) even if it has a represeutation as a strongly continuous function (see [4], ExampleVIII.2.1). Hence we cannot obtain exponential boundedness by considering the growth alone. THEOREM 4.1 Let P be as above and assume that G is a fundamental solution for P. Then there are constants a,/3,7 such that G belongs to I,V~,~,.~(L(E,D)). More precisely, for any 0 < a < b < oo we have factorizations G = Go * (~ I~ + HE) = (~ ® ID + HD) * Go where Go ~ g’(L(E,D)) has support [O,b] and Hx e W~(c~,/3,7)(L(X)) for X ¢ Combiningthis theorem and the characterization theorem 3.1 for Laplace transforms in the previous section we obtain a new proof for the following result of H.O. Fattorini ([3], Theorem8.4.8).

136

Kunstmann

COROLLARY 4.2 Let P be as above. Then P has a fundamental solution G if and only if there are constants a, ~,~f such that gP(.)-~ exists and is polynomially bounded on a region A~,~,~. In this case we have gG = ~p(.)-l. PROOFIf P has a fllndamental solution G then, by Theorem 4.1, G belongs to some ~4)’~,Z,.~(L(E, D)). Hence we can apply the Laplace transform to both of the equations (5) and get £G(A) = ~P(A)-1 for A ~ Aa,B,-~ which proves the assertion. If, on the other hand, 79(.) -1 exists and is polynomially bounded on some A~,~,-~ then it is the Laplace transform of some G ~ W~(L(E, D)). Recalling the rernarks concerning convolution in the previous section, we see that (7 is a fundamental solution. ~ The proof of Theorem4.1 will be based on the following lemma. LEMMA 4.3 Let X be a Banach space and F : ff~ -r L(X) be a strongly continuous function such that suppF C [a, oc) for some a > 0 and [[F(t)[ I <_ Me~’t. If m e zVVo and ~ := F(’’0 then ~.~=~ ~*~ belongs to W[(a, fl,7)(L(X)) where := a/m, -llogM, .2:=w+l-a andT:=w+l. PROOFBy induction we get for any k ~ tW that supp F*~’ C [ak, oo) and

IIF*~(t)ll M~ e~ t ~-~ -

(k - 1)--~---~ < M~e(~+~)~’ t >_

Since O,k = (F,k)(k,n)

we have for any ~ ~ 79

This yields absolute convergenceof the series S(~) := ~-~k~__~(I)*t~(~a) fbr any ~/V1 (O~, ~, T) and the estimate I}S(~)I}< ll~ollw,(~,Z,-~). PROOFof Theorem 4.1 Choose a function p ~ 79 such that p = 1 on [0, a] and p = 0 on [b, cx~), and let Go := pG. Since G is a fundamental solution, we have P,Go = P,G- P* (1- p)G = 5®Iu - ~F~, Go * P = G* P- (1 - p)G * P = ~®ID -where supp (I)E U supp 4~ C [a, ¢c) and ~E and (I)D are tempered distributions since this is true for the left hand side of the equations. For X ~ {E,D}, the convolution inverse of 5 ® Ix - (I)x is given by the Neumannseries Y~-k=0(I)~ this series converges. Fix w > 0. Since we can find a strongly continuous function F : K~ -~ L(X) and M> 0, m ~ ~r0 such that suppF C [a, cx~) and [IF(t)[[ M,e~, Lermna4.3 implies the clai~n with Hx := ~=~ (I).~ ~. []

LaplaceTransformTheoryfor LogarithmicRegions 5

CONCLUDING

137

REMARKS

Note that the proof we have given here parallels very closely the usual proof of exponential boundedness for strongly continuous semigroups T = (Tt) in a Banach space E: If G is the induced distribution and we choose p := l[o,a] then we arrive at OE = 5a ® Ta and the factorization in Theorem 4.1 reflects the fact that we can continue T from the interval [0, a] to [0, oc) by the semigroup law. This proves exponential boundedness, and then Laplace transform theory yields existence and boundednessof the resolvents R(A, A) on a half plane in this case, and that we used the same argument in the proofs above, but for the operators To give an illustration of our results we present two examples. EXAMPLE 5.1 Let X := 12 and A(x,) := (anx.n) with maximal domain where a,, := n + i(e n2 - n~)1/~. Letting S(t)(xn) := (etan /an ¯ x,), the distribution S’ is a fundamental solution for P := £ ® I - 6 ® A. Hence S has a Laplace transform existing on a logarithmic region which is given by S(e-~’) after having extended S to an element of 14;~(X). The function t ~-~ S(t) is strongly continuous with IIS(t)ll = sup,, et’-"~ t~/4. ~,, e The function S belongs to ~4;~(L(X)) since it is a fundamental solution. It would be interesting to characterize those X-valued functions that belong to 14;*(X) terms of the function itself. The second example illustrates the remarks concerning Laplace transform of (Laplace) hyperfunctions and asymptotic Laplace transforms in the introduction. EXAMPLE 5.2 The distribution T := ~=o 5k does not belong to 142’. If A ~ F(~) is the Laplace transform of a corresponding Laplace hypeffunction (see [5]) an asymptotic Laplace transform of T (see [1], [8]) then either F does not extend to an analytic function on a logarithmic region or, if it does, this extension is not polynomially bounded on any logarithmic region. Wewant to close with the general rule that can be drawn from Theorem3.1: For a polynomially bounded holomorphic function on a logarithmic region the growth determines the initial 0r)regularity of its Laplace inverse and the shape of the region determines the evolution of (it)regularity in time where, as we have seen, "regularity" is not a local property here, and has to be understood in the sense that overexponential growth has to be compensated by a loss of regularity.

REFERENCES 1. 2.

3.

B B~umer. A Vector-valued Operational Calculus and Abstract Cauchy Problems. PhD dissertation. LSU, Baton Rouge, 1997. HGDales. Convolution algebras on the real line. In: JM Bachar, WGBade, PC Curtis Jr, HGDales, MPThomas, eds. Radical Banach Algebras and Automatic Continuity. Lect Notes Math 975. Springer, 1983, pp 180-209. HOFattorini. The Cauchy Problem. Reading: Addison-Wesley, 1983.

138 4. 5.

6. 7. 8.

Kunstmann HOFattorini. Second Order Linear Differential Equations in Banach Spaces. Amsterdam: North-Holland, 1985. H Komatsu. Operational Calculus and Semi-groups of Operators. In: HI Komatsu, ed. Functional Analysis and Related Topics, 1991. Lect Notes Math 1540. Springer, 1993, pp 213-234. PC Kunstmann. Distribution semigroups and abstract Cauchy problems. Trans AmMath Soc 351:837-856, 1999. PC Kunstmann. Laplace transformation of generalized functions and evolution of regularity, in preparat, ion. G Lu~ner, F Neubrander. Asymptotic Laplace transforms and evolution equations. Preprint, 1998.

Exact Boundary Controllability of Maxwell’s Equations in Heterogeneous Media SERGENICAISE Universit~ de Valenciennes et du Hainaut Cambr~sis, MACSB.P. 311, 59304 Valenciennes Cedex, FRANCE

1

ISTV,

INTRODUCTION

Let ~ be a Lipschitz polyhedron, in the sense that fl is a bounded, simply connected Lipschitz domain with piecewise plane boundary. Wedenote by F the boundary of gt and by p the unit outer normal vector on F. Wesuppose that fi is occupied by an electromagnetic mediumof piecewise constant electric permittivity ¢ and piecewise constant magnetic permeability #. More precisely, we assume that there exists a partition 7) of f~ in a finite set of Lipschitz polyhedra ~1,’" ¯, ~J such that on each ~’~j, ~" : ~j and # = #j, where Cj and #j are positive constants. For any T > 0, we denote by QT the cylinder ~×]0, T[ and by ~T : F×]0, T[ its lateral boundary. Whenno confusion is possible we drop the index T. Wenow consider (non-stationary) Maxwell’s equations: OE ¢~- eurlH = 0 in QT, OH #-~+ curl E = 0 in QT, div(¢E) = div(#H) = 0 QT, H×~=J,E.~=0on E(0) = Eo, H(0) = H0 in

(1)

Here we suppose that the time evolution of the electric field E and magnetic field H is driven by an externally applied density of current J flowing tangentially on F. In that paper we want to find sufficient conditions (on fl, 7), ¢, #), which guarantee that the next exact controllability problem is solvable: Given a time T > 0 and initial data {E0,H0}, find a surface density of current J in appropriate function spaces such that the solution of (1) satisfies E(T) = H(T) = 0 in This exact controllability problem was already investigated in [10, 12, 14, 9] but in the homogeneouscase, i.e., for ¢ and # constant on the whole of ~ with a smooth boundary F. Our goal is then to extend some of their results to the inhomegeneous case and for nonsmooth boundaries, which requires some nontrivial adaptations. 139

140

Nicaise

The first step is to introduce and analyse adapted function spaces. Existence results for equations like (1) with homogeneousboundary conditions follow from semi-group theory. Afterwards we shall attack the exact controllability proble~n by means of the Hilbert Uniqueness Methodof J.-L. Lions [13] as in [12]. In that case, the exact controllability problemis equivalent to the unique solvability of the adjoint problem of (1), which is usually obtained with the help of energy estimates, consequence an identity with multiplier. In our case, they are available under someassumptions ensuring regularity properties of somefunction spaces [3, 4] as well as a geometrical condition (namely the condition (9)) in order to avoid internal control. In homogeneouscase, the first assumptions reduce to the convexity of f~, while the geometrical condition (9) disappear. Note that the identity with multiplier is proved with the help of some integratious by parts, which require a careful analysis due to the loss of regularity. ACKNOWLEDGEMENTS We thank proof of Lemma2.2.

2

FUNCTIONS

Dr. F. Jochmann

to have

suggested

the

SPACES

For any s _> 0, H*(fl) denotes the usual Sobolev space on f~ [8]. In the sequel, /)(fl) is the space of all ~¢ functions with c ompact support i n Ft while C m(~) i the space of restrictions to f~ of functions from D(Ra). For our future uses, we also need the space PH~ (f~, )) of p iecewise H~ functions r elatively t o t he partitiou 7 more precisely pHI(fl,P) = {(p ~ L~(~)I~ ~ g~(fl~),Vj = 1,... For all j = 1,...,J, we denote by Fjk,k = 1,...,kj, the open faces of the boundaryof f~j. Let J:’,,t = {F~k[Fj~C f~} be the set of interior faces and let, be the set of exterior faces. Let us nowintroduce the following spaces (compare with [11, 12]) J(fl,¢) = {X e n2(fl)a[div(gx) J(gt, e) = {X~ J(fl,¢)[X’ u = 0 on

(2) (3)

Accordingto the definition of the spaces J(fl) and 3(i2) in [11, 12] corresponding to e constant, we first prove the LEMMA2.1

L~(a)~ of

The space J(fl, ¢) and J(f~, ¢) are respectively equal to the closure X = {~o ~ L~(i2)~[~o e C~(~) and div(s~o) = 0} and f( = {~o ~ L~(12)’ale~o ~ I)(f~) div(~o) = 0}

PROOF Let us first assume that ~ = 1. In that case, the second density result is proved in Theorem1.2.8 of [7]. For the first one, let us fix u ~ J(f/, 1). Then Theorem1.3.4 of [7], there exists ~bo ~ H~ (12) a such that u = curl¢o. Since C~’(~) is dense in H~(f~), there exists a sequence of .~b~ 6 C~(~)3 such that curl¢,~ -~ curl¢0 in L:(i2) 3, as n ~ ~.

141

Exact Boundary Controllability of Maxwell’sEquations Consequently curl~pn belongs to C~(~) is divergence-flee and converge to u 3. L2(ft) For an arbitrary ¢, we simply use the equivalence ~ E J(ft,~) (resp. j(Ft,¢)) and only if ~ E J(ft, 1) (resp. ~](12, 1)). For the different fbllowing spaces. J~(ft,#) J~l(ft,

formulations of our Maxwellequation (1), we further need the = {X e Y(ft,#) I curlx e L2(f~)3}, e) = {~(e J(ft, e)]curl~eL:(ft)~

andxxu=Oonr},

= J~*(f~,c,#)

and curlx x u = 0 on a= {X e J¢(ft,e)]curl(#-’

curlx) e L2(ft)

and curl~(. ~ = 0 on F}. Whene and # are constant in ft and 12 has a C1’~ boundary, the above spaces coincide with those introduced in [11, 12] owing to Theorems 2.9 and 2.12 of [1] and the results from [3]. Whene = # = 1 and ft is a polyhedral domain, these spaces no more coincide, in general, with those of [11, 12] according to the results of [3] (see also [2, 6]; note that if ft is convex,then the spaces J~ (ft, 1) and J)(ft, are embeddedinto Hl(ft)). Indeed in these papers, it is shown that any function in the above spaces ad~nits a decomposition into a regular part with the optimal regularity and a singular part induces by somesingularities of the Laplace operator with Dirichlet and Neumannconditions in ft. The same kind of results were recently extended to the case when e and # are piecewise constant in [4]. LEMMA 2.2 The space J~(~,e)

is dense in J(~,e),

while J~(ft,#)

is dense

PROOFEndow L~(fl) a with the inner product (X,~) f~ ¢x’~ dx , and le t P be the orthogonal projection on J(ft,~) in L~(ft) 3. As T)(fl) is dense in L2(ft), subspace P/)(ft)a is clearly dense in J(ft,¢). The first density result follows by Green’s formula, we easily show that P79(ft) 3 C J~(ft,~). The second density result is similarly proved.

3

SOLUTIONS

OF THE ADJOINT

¯

PROBLEM

The homogeneousadjoint problem to (1) is (see §5 for a justification) # o°-~t - curl ¢ = 0 in Q, e°o~t + curl~o = 0 in Q, div(#~) = div(e¢) = 0 in (p × u =O,~b.u=Oon E, ~(0) = ~o, ¢(0) = ¢0 in

(4)

Contrary to [12] where a vector potential is used leading to a second order evolution equation, we shall prove existence of (4) by keeping the first order system

142

Nicaise

(compare with [5]). For that reason, let us introduce the Hilbert space g = Y(f~,#) x j(~,~), equipped with the inner product ((~,

’¢),

(~,,

-[-

~’1 ))H = j~]{[t(k9~l

~¢~1 }

dx.

Nowdefine the operator A as

~)(A) =J~’(~,#)×J~’(~, A(~, ¢) = (#-1 curl ¢, -1 curl ~) Wethen see that formally problem (4) is equivalent o_q_¢ =

o~ 4(0) = ¢0,

(5)

when¢I, = (~, ¢) and 40 = (~0, ¢0). Weshall prove that this problem (5) has a unique solntion using Lumer-Phillips’ theorem [15] by showing the LEMMA 3.1 A and -A are maximal dissipative PROOFWestart that

with the dissipativeness

operators.

of +A, in other words we need to show

N(A4,4)~! O,V4 E D(A). With the above notation

(6)

we have

(A¢,4). = f~{~,~rl¢~- ~url~} But the definition of D(A) implies that ~ belongs to Ho(curl, f~) (see [7, 1]). Moreover by section 1.2.3 of [7], the space ~)(gt) 3 is dense in Ho(curl, ~t), which Green’s formula (I.2.22) of [7], yields

/~curl¢~dx=/~curl~(~dx. The real part of this identity yields (6). Weprove the maximality by the Lax-Milgram lemma.

¯

Since Lemma2.2 guarantees the density of D(A) into H, by Lumer-Phi~lips’ theorem (see for instance Theorem 1.4.3 of [15]), we conclude that A generates Co-group of contraction T(t). Therefore we have the following existence result. THEOREM 3.2 For all 4o E H, the problem (5) has a weak solution 4 C([0, cx3),H) given by 4 = T(t)4o. If moreover 4o D(A~), with k ~ N*, th e problem (5) has a strong solution 4 C([0, ~x 3), D( Ak)) ~ ~ ([0 , cx~ ), D(A~-~)). To finish this section, we establish the conservation of energy.

143

Exact Boundary Controllability of Maxweli’sEquations LEMMA 3.3 If q~ = (~,~) is a weak soiution of problem (5) (or equivalently define the energy at time t by E(t) = ~ {~tl~(x,t)[ 2 + el~b(x,t)l 2} dx. Then we have E(t) = E(O), for all > O. PROOFSince D(A) is dense in H it suffices to prove the result for (~o,¢o) D(A). For such a initial datum, ~ and ,~b are differentiable and owingto (6), we

= ~/~{~L-~T~+ -f[, dz = ~/a{curl~b~-

curl~}

dx = O.

¯

Similarly, we prove the LEMMA 3.4 If ¯ = (~, ~) is a strong solution of problem (5) with initial in D(A), define the modified energy at time t by

datum

/~(t) = ~ {p-~lcurlV(x,t)l~ +z-~lcurl~(x,t)l~} dx. Then ~(t) = ~(0), for all > O.

4

ESTIMATION

OF THE ENERGY

As in [12], the estimation of the energy is based on an identity with multiplier; which, in our case, will be permitted under the following regularity assumptions: ~(~, 7))3. J~ (~, #) ~-~ ~ (f l, 7) 3, J~ (~/, ~) PH

(~)

Owingto Theorein 3.5 of [4], these inclusions hold if the operators -iX Dir arid -A~. have no edge singular exponents in (0, 1] and no corner singular exponents in (0, 1/2] (see [4] for the right definitions). Before going on, let us remark that the above assumptions do not guarantee (see Theorem7.1 of [4]) that the spaces J~*(f/, #, ¢) and J~*(fl, #, ¢) are embeddedinto the space of piecewise 2 functions (as in the case ¢, # constant and F smooth [12]). As already mentioned, when ¢ and # are constant, the above inclusions hold if i2 is convex or if fl has a C1,1 boundary. Here are two examples when (7) holds. EXAMPLE 4.1 If F/ is a parallelepiped divided into two subdomains separated by a plane parallel to two faces, then by the results from section 7.a of [4], the assumption (7) holds. EXAMPLE 4.2 Assume that ~ is convex and that any edge of any ~j is an edge of g/. Denote by ¢mi, = minj=~,...,j £j and ¢max= maxj=h...,J ¢j. If the ratios and ~ are sufficiently close to 1 then (7) holds.

144

Nicaise

Weintroduce a vector field m defined by re(x) = x-Xo, where x0 is a fixed point. Wenow give the identity with multiplier. LEMMA4.3 Let ~o ~ J;(~, P, e) and ~o ~ J~(i~, #, e) and let ~, ¢ be the solution of (~). Thenit holds

(8)

1_2/Q(e-~[ curl ~1~ + P-~I curl ¢l~-)dxdt = _ m.u(e-~leurl~ol~-~-~lcurl~l~)dadt+Xt+X~+~I .2 ~

i=l

where we have set X~ = - £ e¢’m.

VCdx]~,X~

= £ p~.~’dx~ V¢]~dadt,

F ~i,,~J F

x (O,T)

Fe~i,,,

__1 ~JFf I3 = 2

x(O,T)

p-lm" UF[]curl’¢l~]Fdadt.

x(O,T)

PROOFBased on Green’s formula and the assumption (7). At this stage to obtain the so-called estimate of the energy, we need a geometrical assumption in order to cancel the interface terms Ii,i = 1, 2, 3 (to avoid internal control!). Namely we assume that xo may be chosen such that m. ~]F

:

0 on

F,

VF

~

~int.

(9)

LEMMA 4.4 Under the assumption (9), we have Ii = O, Vi 1, 2,3. PROOF The nullity of I: and Ia is direct. For I~, the fact that ¢ x ~F is continuous through any interior face F allows to prove that

¯ (~’×~,~)[.~.v’¢,],~.=(v’~,~).~. ~, ~-[~],~. onF, where ¢T = Y) -- (¢

’ I]F)PF

is the tangent part of ¢.

Weare now ready to formulate the desired estimates of the energy. THEOREM 4.5 Let ~,¢ be the solution of (~) with ~o ~ J*~(~,#,e) and J~(~,#,~). Assume that (7) and (9) hold. Then there exists a minimal time such that

1 f~ (T- To)~(O)< ~ m./:(E-11

~ - #-~l dadt" curl~[ curl¢l~)

(10)

Exact Boundary Controllability of Maxwell’sEquations

145

3PROOFThe assumption (7) and the compact embedding of J~(12, it) into L2(12) [16] guarantee the existence of two positive constants C1, C2 such that I[el/~VXIIL2(f~)9 < Cxll~-1/2 curlX[[L2(f~)~,VX J~(~,e), 11~,~/2xll~=(~)~ C21IE-1/2 curl xll~(~)~ These estimates and Cauchy-Schwarz’s inequality yield IX~I ~ 2CtR(xo)~(0),

IX~l 5 2C~(0),

where we have set R(xo) = max~eo Ix - Xo[. Starting from the identity (8), recalling that the Ii are zero and using the above estimates, we obtain (10) with T0 = max(C~R(xo), C~ In addition to the above hypotheses, if we assume that F is star-shaped respect to somepoint xo (for which (9) holds), i.e., m. ~ ~ 0 on F,

with (11)

then the estimate (10) simplifies (T - To)~(0)

m. ~e--l

l curl

~12dadt.

(12)

~omthat estimate, we deduce an additional one for initial data satisfying J) (~, ~), ¢o ~ J)(~, e). As in [12], the trick consists in introducing the auxilliary functions

0 = ~(s)ds + 0o, ~ = ~(s)ds where 0o ~ J~(~,~,e) and Xo ~ Jg(~,p,e) satisfy eurl0o = ~o and eurl~o -e~o. Their existence follows from Lemmas3.1 and ~.2 of [4]. Once 0o, Xo are determined, since O, ~ are solution of (4) with initial data 0o, ~o, applying the estimate (12), we obtain (r - To)E(O) ~ ~ m ~s-~l cu rl xl ~ da dt ~ C l¢ 12 da dt,

(13)

because curlx = -~¢. Consequently, the expression

= ( I¢l a dt) define a normon J~ (~, ~) x J~ (~, ~). If we define F~ ~ the closure of this space the norm (14), from (13), the algebraic and topological inclusion F~ C J(~,p) J(~, ~) holds. If the star-shaped condition (11) does not hold, we deduce from (10) (see [12] the details) that I[{~0,¢0}llF~ = (fz(l

curl~l 2 + I curl~l~)dadt) ~/2,

(15)

is a norm on J~(~,~,~) x J~(~,p,e). Wethen define F2 as the closure of space for the norm (15) and we have the algebraic and topological inclusion F2

146 5

Nicaise EXACT

CONTROLLABILITY

Followingsection 4 of [12], we deduce that tile energy estimates of the above section and the use of HUM allow to prove exact controllability results for our Maxwell’s equations. Here the main difficulty is to make precise the transposition method (the remainder is made exactly as in [12] and is therefore omitted). Let us now assume that a solution (E, H} of (1) exists such that E e C([0, T], J~ (ft, ¢)) 1 ([0, T], 0~(F t, ~)),

He C([O,TI,J~(a,.))c~C~([O,Tl,J(a,~d), whichis the case if Eo E J~ (ft, ¢), HoE J~ (ft, #) and J is sufficiently regular. Fix also a solution {~o,¢} of (4) ~vith initial data {~0,¢o} in J¢(ft,p) x J~(fl,e). Then we may write 0 =

[e(E’ - -1 curl g ). ¢ - ~(H’ - #-1 cu rl E) . ~o dxds.

Applying integration by parts in s and Green’s formula in x (allowed thanks to the assumptions (7)), we ~ (#H(t)~o(t)

- sE(t)~(t))dx

=/n(#Ho~o

Werewrite this identity as < {H(t),-E(t)},

{~o(t),¢(t)}

=< {Ho,-E0},

{~o,¢o} > -

(16) J.¢dads.

In our two applications J is fixed such that the mapping j: {~20,¢o}--¢

J. ¢dads

is a continuous linear form on F~, k = 1 or 2. Consequently the arguments of section 4 of [12] guarantee the existence of a solution {H, -E} ~ C([0, T], F~.) of (16) all {~o,¢} solution of (4) with initial data {~Po,¢o} in F~ (this solution {H,-E} then called the weak solution of (1)) with the property E(T) = H(T) = once {Ho,-Eo}e F~. is fixed such that (which is always possible) < {No,-Eo}, {~Oo,¢o} >= In the case k = 1, J = -¢ ~ L2(E)a and we directly

conclude the

THEOREM 5.1 Assume that (7), (9) and (11) hold. Then for {Ho,-Eo} e H, there exists To > 0 (given in Theorem ~.5) such that for all T > To, the control J = -¢ ~ L~(E)~ drives our system (I) to rest at time

Exact Boundary Controllability of Maxwell’sEquations

147

In the case k = 2, as in [12], we impose that

since the continuity of j is then guaranteed. This yields THEOREM 5.2 Assume that (7) and (9) hold. Then ]or {Ho,-E0} E D(A) there exists To > 0 (given in Theorem4.5) such that ]or all T > To, there exists control J driving our system (1) to rest at time EXAMPLE 5.3 If e and # are constant, then the above exact controllability results hold if ~t is convexor if it has a C1’1 boundary. They hold in the setting of Example 4.1. Finally, in the case of Example4.2, they hold if, in addition to the hypotheses prescribed in Example 4.2, the assumption (9) holds.

REFERENCES 1.

2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

12. 13. 14.

C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional nonsmooth domains, Math. Meth. Applied Sc., 21, 1998, 823-864. M. Birman and M. Solomyak, L2-theory of the Maxwell operator in arbitrary domains, Russian Math. Surv., 42, 1987, 75-96. M. Costabel and M. Dauge, Singularities of Maxwell’s equations on polyhedral domains, Preprint IRMAR97-19, Univ. Rennes 1, 1997. M. Costabel, M. Daugeand S. Nicaise, Singularities o] Maxwellinterlace problems, RAIROMod~l. Math. Anal. Num., to appear. G. Duvaut and J.-L. Lions, Les indquations en mgcanique et en physique, Dunod, Paris, 1972. N. Filonov, Syst~me de Maxwell dans des domaines singuliers, Thesis, Universit~ de Bordeaux1, 1996. V. Girault and P.-A. Raviart, Finite element methods ]or Navier-Stokes equations, S. Comp.Math. 5, Springer-Verlag, 1986. P. Grisvard, Elliptic problems in nonsmooth Domains, Monographsand Studies in Mathematics 21, Pitman, Boston, 1985. V. Komornik, Boundary stabilization, observation and control o] Maxwell’s equations, PanAm.Math. J., 4, 1994, 47-61. K.A. Kime, Boundary exact controllability of Maxwell’s equations in a spherical region, SIAMJ. Control Optim., 28, 1990, 294-319. O.A. Ladyzhenskaya and V. A. Solonikov, The linearization principle and invariant mani]olds ]or problems o] magnetohydrodynamics, J. Soviet Math., 8, 1977, 384-422. J.E. Lagnese, Exact controllability o] Maxwell’s equations in a general region, SIAMJ. Control Optim., 27, 1989, 374-388. J.-L. Lions, Contr~labilitg exacte, perturbations et stabilisation de syst~mes distribuds, tome 1, RMA8, Masson, Paris, 1988. O. Nalin, ContrSlabilitd exacte sur une pattie du bord des dquations de Maxwell, C. R. Acad. Sc. Paris, S~rie I, 309, 1989, 811-815.

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15. A. Pazy, Semigroupsof linear’ operators and applications to partial differential equations, Appl. Math. Sc. 44, Springer Verlag, 1983. 16. Ch. Weber, A local compactness theorem for Maxwell’s equations, Math. Meth. in the Appl. Sci., 2, 1980, 12-25.

A Sufficient of Parabolic

Condition for Exponential Evolution Equations

ROLAND SCHNAUBELT Tiibingen, Germany

1

Mathematisches Institut,

Dichotomy

Universit~t Tiibingen,

INTRODUCTION

In this note we establish the exponential dichotomy of the abstract Cauchyproblem

assuming that A(t), t E 1~, generates an analytic semigroup (eTA(t))r>_O -1 and A(-) satisfies a certain HSlder condition (see (P) below). Moreover, each semigroup (eTA(t))r>_0 is supposed to have exponential dichotomy and the HSlder constant A(.) -~ must be sufficiently small. Already simple matrix examples, [6, p.3], [15, Ex.3.4], showthat one cannot omit this smallness condition. Similar sufficient conditions for exponential dichotomy are knownfor matrices A(t), [6], bounded operators A(t), [4], and delay equations, [12]. In [15] we have shown the main Theorem 5 of the present paper for the special case of time independent domains D(A(t)) =_ D(A(O)). This seems to be the first result of this kind for parabolic evolution equations; but the hypothesis of constant domains is rather restrictive because it excludes, for instance, Neumanntype boundary conditions in the context of parabolic partial differential equations, see [17, §4]. The results in [4] and [15] are further improved by Proposition 6 which determines the rank of the splitting projections for (CP). Our approach is based on a characterization of the exponential dichotomy of (CP) by meansof the associated evolution semigroupdefined in (10), cf. [5] or [7, §VI.9]. In order to prove our main result in the third section we use ideas from [4] and [15], but we supplement them by the theory of inter- and extrapolation spaces as studied in, e.g., [3], [7], [13]. The necessary tools are introduced in the next section. 149

150

Schnaubelt

2

PREPARATIONS

An operator ,4 on a Banachspace X is called sectorial of type (¢, K, w) if its domain D(A) is dense, E C_ p(A), and IIR(~,A(t))ll K for~ E:= {~v}~0{~: [arg(A- w)[ < ¢} and constants ¢ e (y,~r ]7r and K,w >_ O. Recall that a sectorial operator generates an analytic semigroup. Given an invertible sectorial operator A and 0 < (~ < 1, we define the interpolation space A X~A := D(A)llll~ with norm [[x[[~ A := sup [[r~AR(r,A)x[I.

By X~m we denote D(A) endowed with the norm IIAxl[, and X~I := X. Wefurther introduce the completion X__A~of X with respect to the norm [[x[[m_~ := Then A has a unique continuous extension ,4-1 : X ~ X~ which is sectorial of A~ (X~)~-’, 0 < ~ < 1, equipped with the type (¢, K, w). So we can define X~_ ~orm A ~lZ [[~-~ := sup ~lroR(r,A_~)x[[. r>w

The continuous and dense embeddings X ~ ~-~X ~ ~-~ D( ( w - ~ ) ~-~ X A~~-+X ~+ X ~A_ ~ ~-~ x A_ ~ hold for 0 <_ ~ < f~ <_ 1, where the domain of the fractional power is endowedwith the graph norm and the norms of the embeddings depend only on the type of A and [].4-]]], c~, /~. Moreover, the operator A can be extended to an isomorphism A~ X~A_~ A~_~ : X~ for0 <_a _ <1 which coincides withthepartofA_~ in X~_~. The semigroup e *A restricts to an analytic semigroup on X~A and extends to an ~A analytic semigroup on Xa_ ~ generated by A~_~, 0 _< a < 1, and it satisfies (1) yA 0 < t < 1, for k = 0,1, 0<~< 1, xfi..,_~, _ and a constant C depending only. on the type of .4 and I]A-~I[. These facts follow from, e.g., [3, §V.1, V.2.1], [7, §II.5], [13, ~1, 2.2]. The next assumption was introduced by P. Acquistapace and B. Terreni, [21], in a slightly different form.

(P)

A(t), t ~ II~, is a sectorial operator of type (¢, K, w), with a uniformly bounded inverse, and IIA(t) -~ - A(s)-~llc(x,x,~) <_ Lit - holds for" t,s ~ IR and constants L _> 0 and #,u ~ (0, 1] with # + u > 1.

Here we set X~ := X~A(0 and [[. [1~ := I[" [[~A(0. Assumption (P) implies that there is a unique evolution family (U(t, s))~_>, in the space L:(X) of boundedlinear operators, i.e., U(t, s) = U(t, r)U(r, s), U(s, s) = I, (t, s) ~-~ is st ronglycontinuous for t >_ r >_ s, t,r,s ~ IR, such that U(.,s)x C~([s, oo ),X) so lves (C P) if O(A(s)),see (1, Thm.2.3,Prop.3.2], and also [2], [31, [13], [16], [17]. Moreover,

151

ExponentialDichotomyof Evolution Equations U(t, s)X C D(A(t)) and

II(w- A(t))~U(t,s)ll<_C’(t - -°,

(2) (3) (4)

IIA(t)U(t, s)(w A(s))-~ll < C’(t - s ~-~ , II(w -A(t)) ~’ U(t, s)(w - A(s))-~ll < C’, ~ C IIU(t, s)(w- A(s))axll <_ ~ (t -~Ilxll ,

(5)

~)in (5),0_ 0 only depending on the constants in (P), see [17, Thin.2.1, (2.13)]. Thus U(.,.) exponentially bou nded, i.e ., the re are constants M >_1 and ~/¯ I~ such that

~(t-s)fort >_s. IIu(t,s)ll_<M~

(6)

By [1, Thm.2.3] and [17, Thm.2.3] the mappings

{(r, a) ~’~ :~- > a}~ (t,s) ~U(t, s) e C(X ) (~) {(T,a)

¯ ~ : r>_a}9(t,s)~(w-A(t))aU(t,s)(w-A(s))-~xeX,

( S)

x ¯ X, are continuous for 0 <_ a <_ 1. Using the estimates (5) and II(w A(s))°-~xll <_ co.~llxllS~_~ for 0 <_ 0 < ~ <_ 1, we can uniquely extend U(t,s), t > s, to a bounded operator O(t,s) : X~_~-~ X for 1 > ~ > 1 - #. Finally note that (P) i~nplies II(w - A(t)) ~-~ - (w - A(s))~-~ I] ~< L, It - ~’

(9)

for t, s ¯ 11~, 0 _< a < u, and a constant L(~ depending only on a and the constants in (P). Let E := C0(l~, X) be endowed with the sup-norm. For a given evolution family U(., .) on X satisfying (6), we define evol ution semi group T(.) on E by (T(t)f)

:= U(s, s - t )f (s - t),

s ¯ I~ > O,f ¯ E,

(10)

compare[51, [7, §VI.9], [101, [111, [141, [151, and the references therein. Clearly, T(.) is a strongly continuous semigroup on E. Its generator is denoted by G. It is easy to see that the resolvent of G is given by (R(~,G)f)

(t)

= e-X(t-s)U(t,s)f(s)ds,

(11)

for Re A > % To obtain a representation of G, we introduce the space C~ (Ii~, X) {f ¯ C~ (I~, X) : f, f’ ¯ E} and the multiplication operator A(.) on E with domain D(A(.)) := {f ¯ E : f(s) e D(A(s)) : A(.)f(.) Notice that A( .) is secto ria of type (¢, K, w) with resolvent R(A,A(.)) for ,~ ¯ E I.j LEMMA 1 Let (P) hold. Then G extends the operator C~ (~, X) fq D(A(.)). Moreover, D(A(.)) ~ D(G) C C~ (~,

-~ + A(.) defined

152

Schnaubelt

PROOF For f E C0i (]~, X) ¢] D(A(.)), t > 0, and s E il~, we compute ! (T(t)f(s)

- f(s)) + f’(s)

t

=

1 (U(s,s - t)f(s

- t) - f(s))

- A(s)f(s)

where we have integrated by parts and used °+ U(t, s)x = -U(t, s)A(s)x for t > s Os and x ~ A(s), [1, Thm.2.3]. This shows the first assertion. The second one can established exactly as [15, Lem.3.6] using (4) and (8). Wefurther need the inter- and extrapolation spaces Ea := E~A() and E~_~ := E’~(’) 0 < a < 1, of the operator A(.) on E. By [8, Thm.4.7], the extrapolated operator A(.)-i : E -~ E_I is given by the multiplication operator A-l(.) f e E_~ can be identified with an element of ~te~ X~1. Note that E~_~ is the closure of E~ in E-1 with respect to the norm

IIfllo-1:=r>u, sup sup liteR(r, A(s))f(s)ll, sER As-I(’) : E~ -~ Ea-1 is an isomorphism for 0 _< a _< 1, and E¢ ~ D((wA(.)) ~) ’-~ Ea for 0 _< a % given by ([{~f) (t) =

e-~(t-~)~I(t,

s)f(s)

ds, t e IR,

oo

The following lemmawill be crucial for the proof of our main result. LEMMA 2 Let (P) hold, A > 7, and f ~ C~(I~,X) rq D((w -A(.)) a) for ~ ~ (1 - it, u). Then we have R(A,G)f’ = f - AR(A,G)f + [{~da_~(.)f. PROOFBy rescaling we may assume that A = 0 > 7 and w = 0. First, and 0 < h _< l, we write Dh,I(s) := (-A(s + h)) ~-~ -~ [U(s + h, s) - Ill(s) = ~ [(-A(s

for s E l~

+ (-A(s))af(s)

+ h)) ~-’ --(-A(s+~-))=-~]A(s+~-)U(s+%s)f(s)d~

h [(-A(s + r)) ~ U(s + r, s)(-A(s)) -° - I] (-A(s))~l(s)

dr.

Using (9), (a) for the first integral and (4), (8) for the second one, we lim IIDh’f[l~ = O.

h-t0

(12)

153

ExponentialDichotomy of EvolutiOnEquations Next, we compute

. ao

U(t, s)f’(s)

=

l im _t-h

h-~Og_~ U(t,s+h) (f(s+h)-f(s))ds -h 1 l] = lim (U(t, s + h)I(s + h) - U(t, s)f(s)) h -~ O -~

~x~

U(t, s +h)[U(s+ h, s) - Ill(s) ds

+ lim 1 h-~Oh = f(t)

li m 1 ~f~-h h-~O-~a-

U(t, s + h)[U(s + h, s) - Ill(s)

To determine the remaining limit, we note that U(t, s)(-A(s)) 1-~, t > s, has a unique bounded extension V(t, s) : X -~ satisfying IIV(t,

8)llE(x

)

~ C’

(,

-}-

o~ - 1) -1

°~(t 1- s)

(13)

for 0 < t- s <_ 1 by (5). Hence, U(t, s + h)[g(s + h, s) - I]f(s)

-~

t-h

= J--]

-

s)A~_~ (s)f(s)

V(t,s + h) {(-d(s + h)) a-~ ~ [U(s + h,s) - Ill(s)

+ (-A(s))af(s)}

l/(t, s + h) [(-A(s))~](s) - (-A(s h))~](s +h)] ds.

As consequence of (la), (12), and (-A(.))c~f(.) in X, and the assertion follows.

~ E, the right hand side tends

Wesay that an evolution family U(.,.) has exponential dichotomy if there are projections P(t) ~ ~(X).. t ~ and constants N, ~ > 0suchthat P( .) is stron gly continuous and (a)

P(t)U(t, s) = U(t, s)P(s) =: Up(t, s) for t > s,

(b)

the restriction UQ(t, s) : Q(s)X -~ O(t)X is invertible for t _> s (and we set UQ(s,t) := UQ(t,s)-l), (c) IIUp(t,s)P(s)l I <_Ne-5(t-*) and IIUo.(s,t)V(t)ll < We-5(t-*) for t >_ s.

Here and below, we let Q = I-P for a projection P. Weremark that the dichotomy projections P(t) are uniquely determined, see [15, Cor.3.3]. The Green’s ]unction of an exponentially dichotomic evolution family is given by Up(t,s)P(s), r(t, s):= -uo.(t,

> s,

Nowwe can formulate our second main assumption supposing that (P) is valid. (ED) Each semigroup (erA(t))r>_O, t ~ IR, has exponential dichotomy with constants N, a > 0 and projection Pt.

154

Schnaubelt

Observe that (ED) implies [I,4(t) -~ll _< :~. It is not difficult to showthat (ED) holds if [-~, ~] + iN C p(A(t)) and R(A, A(t)) is uniformly boundedfor A E +~ + cf. [7, §V.l.c], [13, §2.3]. Assuming(P) and (ED), we Ft(s)

{

esA’~(~)Pt, S >_ O, t ~ := -cAo(t}Qt, s < O, t ~ l~,

whereAp (t) and AQ(t) denote the restrictions of A (t) to Pl X and Q t X, respectively. One sees as in [15, Lem.3.5] that the mappings t ~ Pt ~ £(X) and (t,s)

~ Ft(s)

~ £(X) are continuous

(14)

for t ~ If{ and s E N\ {0}. Clearly, Pt and Ft(s) leave Xt~ invariant and have unique extensions to Xa_t, 0 _< a <_ 1, t E I~. The following estimate easily follows from (1) (where the second integral in the first line is understood as an upper integral). LEMMA 3 Assume that

3

MAIN

(P) and (ED) hold.

RESULTS

~Vestart with an auxiliary result which is an interesting refinement of the following fact: An exponentially bounded evolution family has exponential dichotomy if and only if the generator G of the associated evolution semigroup T(.) on E = C0 (ll~, is bijective. In this case the unit circle 3" belongs to p(T(1)), P(.)

1/,

= ~ R(A,T(1))dA,

and G-~y = - F(.,s)i(s)ds,

(15)

see e.g. [10] [11], [14], and also [5], [7, §VI.9] for further references. LEMMA 4 Let U(., .) be an evolution family on a Banach space X with I[U(t, s)][ (t-~) and let G be the generator of the associated evolution semigroup on MC E = Co(II~,X). If G is bijective and IIG-~II <_ 1/rl, then U(.,.) has exponential dichotomy and the dichotomy constants depend only on r/> 0. PROOF From a(G) = a(G) p(G). Using rescaling and the above cited results, we see that U(., .) has exponential dichotomy with exponent 6’ Fix 5’ ~ (0, r~). Suppose the dichotomy constant ~ were n ot u niform i n r /. Then there would exist evolution families U,~(., .) on Banach spaces X,~ such that [[Un(t,s)l[ <_ 7(t-s) and [[ G~[[ _<1/~1 for the generator G~ o f t he associated

ExponentialDichotomyof Evolution Equations evolution semigroup on Co(lI~,X.,~), I[x.~ll = 1 and lim egltn-s’l

155

but there ale xn Xnandtn,s n ~ II ~ with

IIr~(t,~,s.,~)x,,ll=

whereF,~(., .) is the Green’s function of U~(., Set f,~(s) := ~,~(s)e +~’(’-’") U.,~(s,s,~)z,, for s >_ sn and f~(s) := 0for s < s,~, where ~,~ ~ C(I~) has compactsupport in (s,~,s,~ + 2), 0 _< ~o,~ _< 1, and f~o,~(s)ds = and we take +6’ if t,~ >_s~ and -g if t, < sn. As a result, R(~:~’, a,,)f~(t,~) = ~ e±e’(*°-’)rn(t,~, s)f~(s)ds = e~’lt"-~"l ~. This contradicts

r.~(t~,s~)z..

the uniform estimate of G2

THEOREM 5 Assume that (P) and (ED) hold and let q be given by Lemma If qL < 1, then the evolution family (U(t,s))t>_s solving (CP) has exponential dichotomy with constants N~, ~ > 0 depending only on q and the constants in (P) and (ED). PROOF(a) If Gf = 0 for some f e D(G), then f(t) = (T(1)f)(t) 1)f(t - 1) for t e N. Thus (2) and [1, Thin.2.3] yield f e D(A(.)) so that C~ (II{, X) by Lemma1. Wedefine on E a bounded operator L (Lg)(t)

:=

Ft(t - s)g(s)ds, oo

Using Lemma1 and integration 0 = LGf = -Lf’

by parts,

we obtain

+ LA(.)f LA(.)f- A_

~(.)Lf- f

=:V f

where we have set (V f)(t) := ./~

F,(t -

s)(A(s) - A_I (t))y(s)

= £ A(t)F~(t s) (A(t) -~ - A(s)-~)A(s)f(s) ds Here the integral in the first line is understood in the topology of Xt_~. Then (P) and Lemma3 imply IIA(t) Vf(t)ll < qL IIA(’)fll~ for each t ~ tlL Since Vf = f and A(.) is invertible, this estimate gives f = 0. HenceG is injective. (b) Define on E a bounded operator R (Rf)(t)

:=

F~(t - s)f(s)

Notice that due to (I) R extends to an operator/~ : Ea-1 --~ E for a e (0, 1) norm

-+

-

=:~..

(~6)

156

Schnaubelt

For f E D(A(.)), one easily sees that Rf ~ Cd (1R, X) and ~Rf a = f +/L4(.)f. let (Sf)(t)

:= £(A_~(t)

A(s))rs(t

= A_I (t) ~(A(s) -1 -

- s) f(s)as

A(t)-~)r,(t s)A(s)f(s)

ds,

where first integral is understood in the topology of X~1. This leads to ~Rf := -~ Rf + A-I(.)Rf

= (S- I)f.

(17)

Wefurther need the Banach space Fa-1 := {f 6 E_l : [Ifll~-I < oo} endowed with the norm [1’ [[~-1 for 0 < a < 1. Wenote that E is in general not dense in F~_l and that F~_I ~ E~_I for 0 < a < v by [7, Prop.II.5.14]. Using (P) and Lemma we can extend S to a bounded operator S : F._~ -+ Fv-i with norm less than qL. Fix a ~ (1 - #, v). Since (P) holds with v replaced by c~, S is also continuous respect to For a given g ~ E, we have f := (~- I)-~g 6 Fv-1 C E~_~. There are f,~ ~ D(A(.)) converging to f in II’ [la-~- From (17) we derive ORf,~ = (S - I),f,~ ---+ inI}"I]~- t as n -+ oo. Moreover, A_~(.)Rf~ ~ Ea-1 so that Rf, ~ E~ C D((w A(.)) ~) for 1 - p < fl < a < u. Then Lemma2 yields

from which we deduce [:l:~g = R()~, G)g = AR(A, G)~.: Therefore ~f ~ D(G) and G~f = g; i.e., G is also surjective. Considering (~-I) -~ as an operator from E to F,_~, we obtain G-~ -~ =/~.(~-I) and IIG-~[[ < ~ ~vhere p := p, is given by (16) and c is the norm of the embedding o}--E i~ F,,_I. So Lemma4 yields the assertion. [] Wefurther prove that the dimensi’ons of the stable and unstable subspaces of U(t, s) and e*A(*) coincide, cf. [6], [12]. PROPOSITION6 Under the assumptions of Theorem 5 we have dimP, X = dim P(t)X and dim ker P8 = dim ker P(t) for t, s ~ IR. PROOF (a) By [9, p.298] the dimensions of the kernel and the range of two pro.jections P and P’ coincide whenever lIP- P’II < 1. So (14) shows that the dimensions of ker Pt and PtX do not depend on t. (b) Weset Ae(t) := A(et) for 0 < ~ _< 1 and t E IR, and Ao(t) = A(0) for t ~ IR. Observe that Ae(.) satisfies (P) and (ED) with the same constants and the same So the corresponding evolution families U.~(., .) have a uniform exponential bound and exponential dichotomywith uniform constants 6’, N’ > 0 and projections P~ (t),

ExponentialDichotomyof Evolution Equations

157

where Pl(t) = P(t) and P0(~) = P0 for ~ E [¢. Due to (a) it remains to show e ~ P_~(t) E L:(X) is continuous for eacht Eli. Clearly, U~(t,s) = (]~(et, es), where ~r.~ (t, s) solves the Cauchyproblemcorrespondingto the operators ~iA(t), 0 < ~ <_ Wenote that ~_A(t) fulfills (P) with the same constants except for w replaced w/~. For 0 < ~,r] <_ 1 we compute Ue(s,s - 1) - Un(s,s - 1) = ~e(~s,e(s - 1)) - ~re(,?s, r/(s +

O~_(~s,r)(~ - ~)A(r)Ov(r,,I(s

1))dr

(where the existence of the integral is verified in the next step). The first summand tends to 0 in L~(X) uniformly for s in compact sets as r/ -~ e by (7). The second summandis equal to ~-~1 L~s ~J~(rls,

r)~-"(w-

A(r))"rl~-~(w-

A(r))~-"~J,(r,

- 1))d

where 0 < ct < #. Using (5) and (2), we see that this expression tends to operator normuniformly in s E IR as r~ ~ ~. In the case r] -~ 0, we write Un(s, 8 - 1) eA(O) = ~.o(~8,~(8 - 1)) - en~A(~(s-1)) + e A(n(s-~)) -- CA(0). The right hand side converges to 0 in E(X) uniformly for s in compact subsets due to [1, (2.6), Lem.2.2] and (P). (c) Let ~(.) be the evolution semigroup on E = Co(~, X) associated with U,(., Notice that n(A,T,(1))f

= ~ A-("+’)Te(n)P~(.)f

- ~ An-l(~(n)Q,(.))-i

n=O

n=l

= ~~.. x-(,~+~)~r _(. ,. - ~)~(.- n) - ~ ~"-’U ~,~, ,,Q~.,.+ n)y(.+ ~

for f ~ E. In particular, I]R(X,T~(1))[[ 2N ~ ~ =: cfor each I~1 = 1, 0 < ~ Given t ~ ~ and x ~ X, we set f = ~(.)x for a function ~ ~ C(~) with support (t - 1,t + 1), 9(t) = 1, and 0 ~ ~ ~ 1. Then (15) and (6)

IlPo(t)~- e,(t)zll 5 llP,(.)y - P~(.)YI~ 5 c supIf(T,(1) -T~(1))n(~, I~1=~

5cma~{sup ~lY~(s,~-~)-U,(s,~-~)lll~(~,T,(~))y(s-~)~l, ~up 2M~lln(~,~(~))](~5 m~x(~~ ~.~ IIV~(~,~ - ~) - V,,(~,~-

~)~1, ~MN’~’ ~ ~-* ¯

ibr each r > max{0, 1 - t, t + 1}, where N(r) is the maximumof the integer parts of r - t - 1 and r + t - 1. Nowpart (b) easily implies the assertion. REMARK 7 Theorem 5 and Proposition 6 can be generalized to operators satisfying (P) and (ED) for t ~ t0 by setting A(t) := A(to) for t < t0, cf. [15, ~3].

158

Schnaubeit

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3. 4. 5. 6. 7. 8. 9. 10. 11.

12. 13. 14.

15. 16. 17.

P. Acquistapace. Evolution operators and strong solutions of abstract linear parabolic equations. Differential Integral Equations 1:433-457, 1988. P. Acquistapace, B. Terreni. A unified approach to abstract linear nonautonomous parabolic equations. Rend. Semin. Mat. Univ. Padova 78:47-107, 1987. H. Amann. Linear and Quasilinear Parabolic Problems. Volume 1: Abstract Linear Theory. Birkh~user, 1995. A.G. Baskakov.Someconditions for invertibility of linear differential and difference operators. Russian Acad. Sci. Dokl. Math. 48:498-501, 1994. C. Chicone, Y. Latushkin. Evolution Semigroups in Dynamical Systems and Differential Equations. Amer. Math. Soc. (to appear). W.A. Coppel. Dichotomies in Stability Theory. Springer-Verlag, 1978. K. Engel, R. Nagel. One Parameter Semigroups for Linear Evolution Equations. Springer-Verlag (to appear). T. Graser. Operator multipliers, strongly continuous semigroups, and abstract extrapolation spaces. Semigroup Forum 55:68-79, 1997. S.G. Krein. Linear Differential Equations in Banach Spaces. Transl. Amer. Math. Soc., 1971. Y. Latushkin, S. Montgomery-Smith. Evolutionary semigroups and Lyapunov theorems in Banach spaces. J. Funct. Anal. 127:173-197, 1995. Y. Latushkin, T. Randolph. Dichotomy of differential equations on Banach spaces and an algebra of weighted translation operators. Integral Equations Operator Theory 23:472-500, 1995. M.P. Lizana. Exponential dichotomy of singularly perturbed linear functional differential equations with small delays. Appl. Anal. 47:213-225, 1992. A. Lunardi. Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkh~iuser, 1995. F. R~ibiger, R. Schnaubelt. The spectral mapping theorem for evolution semigroups on spaces of vector-valued functions. Semigroup Forum 52:225.-239, 1996. R. Schnaubelt. Sufficient conditions for exponential stability and dichotomyof evolution equations. Forum Math. (to appear). A. Yagi. Parabolic equations in which the coefficients are generators of infinitely differentiable semigroupsII. Funkcial. Ekvac. 33:139-150, 1990. A. Yagi. Abstract quasilinear evolution equations of parabolic type in Banach spaces. Bolletino U.M.I. 5-B:351-368, 1991.

Edge-Degenerate on Cones

Boundary Value Problems

E. SCHROHE and B.-W. SCHULZEInstitut dam, 14415 Potsdam, Germany

ftir

Mathematik, Universitgt

Pots-

ABSTRACT Weconsider edge-degenerate families of pseudodifferential boundary value problems on a semi-infinite cylinder and study the behavior of their pushforwards as the cylinder is blown up to a cone near infinity. Weshow that the transformed symbols belong to a particularly convenient symbol class. This result has applications in the Fredhohn theory of boundary value problems on manifolds with edges.

INTRODUCTION Pseudodifferential boundary value problems on manifolds with edges form an algebra in which parametrices to elliptic elements can be constructed by inverting the components of an associated symbol hierarchy. Operators of this kind arise not only in concrete edge situtations, but also in mixed problems, crack theory, and in the solvability theory of boundary value problems in corner domains or in configurations with higher order singularities. Pseudodifferential calculi for edge operators in the boundaryless case have been developed by Mazzeo[4] and Schulze [12, 2]. A crucial step is the inversion of the operator-valued edge symbol. Essentially, this a family of pseudodifferential boundaryvalue problems on an infinite cone with boundary, parametrized by the edge covariable which degenerates in a typical way. Requiring its invertibility is an analog of the Lopatinskij-Shapiro condition for boundary value problems; the inner normal is here replaced by the cone. Similarly as in the theory of boundary value problems, invertibility can only be achieved by imposing additional trace and potential conditions. They have to be included in the full calculus, similarly as this is done in Boutet de Monvel’s concept. In order to come close to an inverse for the edge-degenerate family, one has to perform a careful analysis both near the tip of the cone and on the infinite part. 159

160

SchroheandSchulze

The analysis near the tip relics on tile Mellin calculus for manifolds with conical singularities; it is a central topic in the paper [10] by the authors. Here, however, we shall deal with the problems arising from the non-compactness of the cone at infinity. Weshowthat the special structure of these symbolsallows us to treat them within the framework of the SG-calculus of boundary value problems introduced by the first author; of. the papers by Shubin [13] and Parcnti [5] for earlier use of this symbolclass in the boundaryless case. The main result of this note is Theorem 3.10: Under a blow-up of the cylinder, the pseudodifferential boundary value problems stemming from edge-degenerate families transform into SG-boundary value problems. The blow-up preserves the grading of the algebra so that invertibility of the principal usual symbol will allow us an SG-parametrix construction on the infinite part of the cone. Together with the analysis near the tip this will provide a Fredholminverse to the edge-degenerate family arising from an elliptic edge sytnbol and thus an essential step in the inversion process for elliptic edge symbols. This in turn is crucial for the parametrix construction iu the edge boundary calculus, of. [11].

1

OPERATOR-VALUED

SYMBOLS.

SOBOLEV

SPACES

Throughout this article let X be an n-dimensional C~ manifold with boundary Y = OX, embedded in an n-dimensional manifold f~ without boundary. El, E~,... are vector bundles over f/and F~, F2,... vector bundles over Y. On ft we fix a Riemannian metric; moreover we endow the vector bundles with Hermitian structures so that we can speak of L~-sections. By 0~ we denote an operator coinciding with the normal derivative in a neighborhood of the boundary and vanishing outside a slightly larger neighborhood. Wealso fix a function [.] : ]Rn ~ I N+such that Ix] > 0 for all x and [x] = [x for Ix I _> 1. In connection with pseudodifferential operators, one often uses (x) (1 + Ix[~) ~/~. Clearly, there are constants c~,c~ such that c~(x) <_ Ix] _< c~(x), so that both are comparable. This allows us to obtain the usual estimates like Peetre’s inequality or the fact that [t(] _< ct[(] for t _> 1 and suitable c > 0. H"(~n) is the usual Sobolev space on I~n, and HS(Ii~_) = {ult~,~ : u ~ H~(l~’~)} the space of all restrictions (in the sense of distributions) of elements in Hs(lI{’~) to the open set R~. /-/0(+) is the set of all u 6 H’(II~’~) whose support is contained in l~+. Fora= (~l,a~) e ~, we let H~(IR~)= {[x]-~u : u e H~’~I~n)}, and H~(~’~), = {[x]-#=u : u e Ho (IR+)}; here x is the variable in IR+. Finally, S(liR~ ) = {ula; : u ~ S(ll{" )}. Wehave $(N~) = proj - lim,ea= H#(II(~_) and 8 (I1¢+) = ind - limae~= H~ (IR+).

Operator-valued

SG-symbols

1.1 Group actions A strongly continuous group action on a Banach space E is a family n = {n~ : A ~ 1R+ } of isomorphisms in £(E) such that n~n, = ~, and t, mappingA ~ ~,~e is continuous for all e ~ E.

Edge-DegenerateBoundaryValue Problemson Cones

161

For all the above Sobolev~paces on II~’~ and ll~_, x~e will use the group action (~c~,f) (x): An/2f(,~x).

(1.1)

This action extends to distributions by tc,xu(qo) = u(~,-~ ~o). On E = C~, I E N, we use the trivial group action ~ = id. Sumsof spaces of this kind will be endowedwith the sum of the group actions. 1.2 Operator-valued SG-symbols Let E, F be Banach spaces with strongly continuous group actions ~, g, let a ~ C~ (l~ ’~ x ~’~ x l~k , £(E, F)), and tt ~. WeshM1write a ~ SG’(~~ x ~, ~k;E, F) provided that, for all multi-indices a,3, 7, there is a constant C = C(a,3,7) with

"

(1.2)

Weshall simply write SG", if the arguments are obvious from the context. The choice of the best constants C provides a ~chet topology for SG". In c~e a is independent of ~ we shall write a ~ SG"(~~, ~k;E, F) with ~ 6 Extension ~o projective and inductive limits: Let ~,# be Banach spaces with group actions. If F~ ~ F2 ~ ... and E1 ~ E2 ~ ... arc sequences of Banach spaces with the same group action, and F = proj-limFk, E = ind-limEk, then let SG~’(~" x ~, ~};~, F) = proj - limkSG~’(~~ x ~, ~;~, Fk) and define SG"(~~ x ~", ~; E, ~) as well as SG~’(~n x ~n, ~k ; E, F) similarly as projective limits. Weshall use this concept particularly with E = S’(~) and F Pseudodifferential opera~ors:For a ~ SG~’(~" x ~n , ~ x ~t ; E, F) , the parameterdependent pseudodifferential operator op a with parameter space ~ is the operator family {op a(A) :A ~ ~ } defined (1.3) This reduces to (opa($)f)(v) ei Vna(v,~, A)]( ~)d~ for symbols ~hat are i ndependent of #. Here, ](~) = ~v~f(q) = f e-iV"f(v)dv is ~he vector-valued Fourier ~ransform of f, and dq = (2~)-qd~. One obtains a full symbolic calculus for operators with SG-symbols. In par~icular: (i) There is asymptotic summation; (ii) an opera~or wi~h ’double’ bol in SG(tt~’v2’ga) has a ’left’ symbol in SG(v~’v=+~’a)and a ’right’ symbol SG(v~,o,v~+~,~), (iii) ~he composition of two SG-pseudodifferen~ial operators is SG-operator; i~s order is the sum of ~he orders. 1.3 General operator-valued symbols Weuse the above notation. shall say that a ~ S~’(I~’~ x ll~", II~; E, F) if

For tt ~ ~, we

Ilgin]-, D~D~D~a(y, ~, ~/)t;in]ll£(~,~, ) < Wethen have the above properties replacing SG 1.4 DEFINITIONLet E, F be Fr6chet spaces and suppose both are continuously embedded in the same Hausdorff vector space. The exterior direct sum E @ F is Frfchet and has the closed subspace A: = {(a, -a) : a ~ E ¢3 F}. The non-direct sum of E and F then is the Fr6chet space E + F := E @F/A~.

162 2

Schroheand Schulze A BOUTET WEIGHTED

DE MONVEL TYPE SYMBOLS

CALCULUS

WITH

Wereview the concept of Boutet de Monvel’s calculus with weighted symbols introduced in [6]. More details can be found in [7, 8]. Westart with a review of the relevant spaces and terminology. 2.1 DEFINITION(a) Given a function f on I~_ we denote by e+f the function on ll~ n which is equal to f on l~ and zero otherwise. In the same way we let e+g be the extension by zero of a function 9 on X to a fimction on Ft. By r+f we denote the restriction of a function f on lI~" to I~_. Similarly we denote the restriction of a function 9 on f~ to X by r+g.

(b) Let + ={(e+f)~: f e 8(II¢+)},H~- = {(e-f)^: f e S(l~_)}; th indicates the Fourier transform on IlL H’ denotes the space of all polynomials. Then define ’. H=H+@H~@H (c) A symbol p SG~t(I~ n x It ~n, lI ~’~ x II ~t) has th e SG-transmission pr operty at , xn = ~n = 0 if, for all k,k~ ~ N, D~D~.v(x ,O,x,O,~’,[~’,~]¢~,~)

e SG"(N~7’ x ~, ,~,

The subscripts indicate the variables with respect to which the corresponding properties hold. Write p ~ 8G~,.(N" x N~,N~ x NI). The above definition also makes sense if p is independent of &. Weshall then say that p has the SG-transmission property (with respect to xn = 0) and write p 2.2 Weighted parameter-dependent operators and symbols A parameter-dependent SG-operator of order p ~ l~: and type d ~ N in Boutet de Monvel’s calculus on ~ is a family of operators

s(a~) ~,

~ s(s~) ~ -+

A(~) :

@

(2.1)

, A ~ IN

of the form opk(A) A(~)

= (P+(A)0)(~=oOpgj(A)oJr 0 0 + E~=o op t~(A)O~J

ops(A)

where O~ is the normal derivative and (i) P(.) = op p(.) p ~ S G~’, .(W~,II~" × ~t) ,p + = r+pe+ (ii) The symbols9~, t j, k, and s belong to the following spaces: gj e SGU-(J’°)(N"-’,~ n-’ x

~ SG.-(~,o)(~’,-~,~’-~ x ~;S’(~)~,,C’~), C SGu(~n-I,~

n-1

N ~l;cm~,s(~)nz),

m’ ,Cm~). s ~ SG.(N"-~ ,N"-~ x N~;C

and

)

(2.2)

Edge-DegenerateBoundaryValue Problemson Cones n

163 -c~,d

12l~,d [~ . ]l~l Weshall write A E ,~sa~+, ~ ); moreover, we write A regularizing parameter-dependent operator of gype d, i.e. for an opera~or in the ~d~ I intersection ~zea~ Bs~(~+; ~ ). The decomposition P+ + G is not unique. The topology on n~"a~n’ ~) ~nd -~,d{

~sa

~n ,

~+, ~t) is that of a non-direct sum of ~chet spaces.

2.3 DEFINITION(a) (Erkip & Schrohe [3]) Let f~ be an n-dimensional manifold without boundary. Call 12 SG-compatible if conditions (SG1) - (SG3) hold. (SGI) (SG2) (SG3)

J f~ has a finite covering by coordinate charts: fl = (Jj=xfij. This cover has a good shrinking. All the changes of coordinates X satisfy Oax(x) = O([x]~-a).

The existencc of a good shrinking in (SG2) means that fl may also be written the union of sets f~ _C flj, such that there is an e > 0 with B(x,e[x]) C_ aj(flj) for every x ~ aj(f~). Here, aj : flj ~ ’~ ar e th e coordinate maps. Clearly, ll~ ~ is SG-compatiblewith its standard coordinates and so is every co~npact manifold. A more elaborate example will be given in 2.5, below. Let X be an n-dimensional submanifold of 12 with boundary OX = Y, where Y is an (n - 1)-dimensional submanifold without boundary. Assumeadditionally that (SG4) The functions ~j : f~j ~ ll~ n xnap X ~ f~j to ~, Y f’l f/j to 0ll(~, and

~ ~ (~\x) ~_. (SGS)

There is a Riemannianmetric g on ~t whosetensor in local coordinates, gij, satisfies the estimates O~g~(x)= O([x]-~), (gij)(x) -~ = O(1).

Wethen call the quadruple (~, X, Y, g) an SG-manifold with boundary or simply SG-compatible. A simple exampleis given by (I~ n, ~_, II~ n-~, Euclidean metric). It is easy to see that pull-backs of the Euclidean metric on aj(fl~) can be patched together using a partition of unity to yield a metric with property (SG5). (b) 3(9/) S(X)denote the s paces of al l s mooth funct ions on 9/ andX, respectively, that satisfy the estimates for rapidly decreasing functions in all local coordinates. All notions are justified by (SG3). Given a vector bundle E over f~ with all transition functions satisfying SG-estimates of order zero (SG-vector bundles), we define $(12, E), S(X, E), H~(9/, E), etc. in the obvious way. 2.4 Boutet de Monvel’s algebra on an SG-manifold Let 12, {12~}, X, Y be as above, let E~, Ez be SG-vector bundles over f~, and let F~, F~ be SG-vector bundles over Y. Weshall say that a family A = {A(~)

s(x, E~)

S(X, E.2)

A(~): 8(Y,F~)

$(Y,F~)

is an element of~s~(X, p,d .1~l ), # ~ , d ~ 1~0, provided that (i)

For every choice of functions ~, ¢ supported in the same coordinate neighborhood and satisfying zero order SG estimates, the push-forward of (bA~ is an element of/~(~_~; l~ t ). Here (I) and ¯ denote the operators of multiplication by diag {~, ~]y} and diag {¢, ¢[~}, respectively.

164

Schroheand Schulze

(ii)

If ~o, ~b are as before, but the coordinate chart does not intersect the boundary, then all entries in the matrix (M~oA(A)M¢), - except for the pseudodifferential part - are parameter-dependent regularizing. Given two functions ~, ¢ which satisfy the esti~nates for an SG°-function in all local coordinates and additionally have disjoint support, the operator ¢Aq~, defined as above, is a p,arameter-dependent regularizing operator.

(iii)

Recall that a regularizing operator of type 0 is an integral operator with a kernel section in ($(X, E2) @S(Y, F2))~,~ (S(X, E~ ) (3 F~ )). Aregularizing operat of type d is a sum R = ~j=o R~ with all R~ regularizing of type zero. The parameter-dependent regularizing operators of type d, denoted by B-°°’d(x; ~), are Schwartz functions on ll{ t with values in the regularizing elements of type d. 2.5 ExampleLet f~, {f~.i }, X, Y be as before; assume additionally that it is compact, hence so are X and Y. By n~ : 12j -~ Uj C_ I~’~ denote the coordinate maps. Wethen define the manifold it× by introducing on it x It coordinate maps X~ : it~ x 11¢ -4 ~’~+~, X~(x,t) = ([tl~(x),t). Topologically, f~× ~ it x II¢; intuitively, these coordinates makef~× look like an outgoing cone with two ends. It is easily checked that it × is an SG-compatiblemanifold. Restricting the coordinate maps to X we obtain X×. This.is an SG-manifold with boundary Y× when endowedwith the metric induced from Euclidean space via the )~y. 2.6 The standard version of Boutet de Monvel’s calculus Weobtain the usual version of the calculus, if we ask for uniform boundednessof all x-derivatives instead of estimating by Ix] ~:-I~l. Moreprecisely, we shall write A ~ Bt,,a(g~; Rt) if A of the form (2.2) with P ~ S~tr~tll~’~’,~’~x ~) ((n~x n~)-matrix-valued); S’~(Nn-~,N n-~ xN~;S’(~)n’,N(~)n~), j=0,... gj e n-~ n-1 m~) j = 0,... ,d S*’(N , N X Nl;S’(~)n’,c , tj 6

k s

e e

S.(~"-~,~ "-~ x ~;C",S(~)"’) S"(~"-l,~~-~ x ~;Cm’,C"~) .

Wethen can define operators on open manifolds; we ask that, in local coordinates, all operators be of the above form.

3

BEHAVIOR

UNDER BLOW-UP

3.1 Relation to edge-degenerate boundary value problems In the Fredholm theory of pseudodifferential boundary value problems on manifolds with edges, we have analog of the classical Lopatinskij-Shapiro condition, namelythe invertibility of the principal edge symbol. Apart from the trace and potential conditions (which will be taken care of later) this is a family of boundary value problems in Boutet de Monvel’s algebra on an infinite cone, the so-called model cone. It is parametrized by the edge-variable (which is omitted here, since it plays a minor role) and the edge-covm’iable,r/.

Edge-DegenerateBoundaryValue Problemson Cones

165

In what follows, t is the axial variable of this cone, ~- the Correspondingcovariable; x is the variable along the base X of the cone, ~ its covariable. Geometrically, the use of the variables (t, x) gives the picture of an infinite cylinder. The fact that are dealing with a cone is reflected in the degeneracyof the axial covariable: ~- only appears as tv. Similarly, the edge covariable only comes up in the degenerate form try. As in the case of classical boundary value problems, the t-variable is ’frozen’ at the edge, i.e. t = 0, and the edge-covariable r/¢ 0 is fixed. The boundary value problem therefore depends on t only implicitly, via tr and In order to establish the invertibility of the full edge symbol (including trace and potential conditions) one needs the Fredholm property of this edge-degenerate family of boundary value problems for each r/¢ 0 on the Sobolev spaces over the infinite cone. Establishing it splits into two tasks: The analysis near the tip, which will be performed in [10], and the analysis on the cone near infinity. There the cone over X coincides with the manifold X×, which is a particularly simple SG-manifold. What we shall establish in this section is the following: The blow-up which makes the cylinder X × ll~ the SG ’double cone’ X× induces a push-forward on the level of operators which transforms these edge-degenerate families of boundary value problems into SG-operators, cf. Theorem 3.14. The mapping properties on the cone Sobolev spaces are therefore immediate. Moreover, the natural grading of the algebras is preserved: The push-forward of lower order terms is of lower order in the SG sense. This will play an important role in the construction of a Fredholm inverse. Westart by computing the behavior of the symbol under the push-forward; the main technical result is Theorem3.8, which allows us to deal with all components in Boutet de Monvel’s calculus at the same time. 3.2 DEFINITION and LEMMA We shall

use the diffeomorphism

x: ~nx ~+-~ ~ x ~+,x(x, t) = ([fix, t). Its inverse is given by X-~(y, t) (y /[t], t) We have DX-~(Y’t) We define

=

1 (It] -11 2-YOt[t]/[t] )

-~-n + 1

~n+l

the function

M on N+ x + by

M(y,t,9,{)

= D;~-t(y+a(9-y),t+a({-t))da;

Mis an invertible (n + 1) x (n

M(y, t, ~?,~) =

1)-matrix. For t,~" >_ 1,

~’~~-~-’~ ~[~-~0 - f°l(~+~(~-~))~+ ~ (,9-~)1da)

by (3.1). Its determinant only depends on t and ~. Using the abbreviation T(t,~)

(3.1)

lnt - ln~

(3.2)

166

Schroheand Schulze

we see that, for t, ~ _> 1, the inverse of the adjoint is M-~r Tf~ .,~+~(~-~) (t+~(~-t))~

o)

(3.3)

anddet M-T ( t, ~) =T ( t, ~)’~ Let p E C~(I1~+ x I~+, S~’(I~’~ x I~’~, l~’~+l+q; E, F)). Fix ~1 # O, and 3.3 LEMMA definep( t, x, i, ~., 7-, ~, ~7)=f~( t, ~;x, ~, ~, tT-, trl). The symbol of the push-forward X, (op p) is given by (t,~;y/[t],~j/[t~,M -y -’. (~r), t~)T(t, ~)n[~]

q(y,t,f/,~,,,7-)=[~

For t, ~ ~ 1 this expression simpl1~es to ~ (t, ~; y/t, 9/~, T~, c~ +tT, t~) T(t, ~)"~-’~. Here c = c(y, t, ~, ~) = T(t, ~) (t+a(~_t))= f~ ~+~(~-~)~ a~d T is as Proo]. Let ] ~ 3(~’+~) and (y, t) e ~+~. (X.op p) f (y, t) = op p(x* f)(x-’ (y, t)) " Here we made the substitution (y/[t]--ff/[~),

(3.4) Y ~ tr,

t~)

~ = ~/[~. Nowwe note that

+ (t--~)T

= = ((y,t)-(~,~))M~(y,t,~,~)(~r).

The integral (3.4) therefore equals

Nowthe first

assertion follows from (3.3). The second is immediate.

3.4 PROPOSITION Let ~, ~ ~ C~ (~n) with disjoir~t support and w, ~ ~ C~ (~ with w(t) = &(t) = 1 for t ~ 1. Moreover let ~ e C~(~+ x ~+,S"(~" x ~",~" ~+q; E, F)). Fix 0 ~ ~ ~ ~q, and define p(t, x, ~, $, r, ~) = ~(t, ~; z, ~, ~, tr, Then the push-forward

x. @(z)(1

-

is an integral operator with a rapidly decreasing kernel taking valucs in £( E, F).

Edge-DegenerateBoundaryValue Problemson Cones Proof. Weabbreviate F(x, ~, t, ~) ~o(2)~5(i)(1 -w(t))(1 K¯No with lt-K<-n-1 wehave

167 5~({)). Fo ’~+1),

(~(1 - w)[opp](1 &)~)u(x, t) .~f[ e~(~-~)~+~(t-~)~ g (x, ~, t, ~)~( t, ~; x, ~, ~, tT, t~)’U(~, ~) d~

JJJJ

e" .... ,¢

" -’

....

A~}(t, ~; x, ~, ~, tr, tn)u(&, i)d~d~d(dr.

The in~egrM exi~s ~nd we m~y rewrite ~(1 -~)(opp)(1 opera, or wi~h ~he kernel H = H(z, t, ~, ~) given

Using integration t

-~)~ ~ ~he integral

by parts we conclude that

]k(x,t,2,~)l ~ c~ ff[~,T]"-~[t~]-g-2Nd~dTF(x,2,

t,~)~x

~ c~ [t~]-~-~F(x, ~, t, -~. ~)lz - ~1 For t > 1 we may esti~nate [t~]-e~’ by t2N[t - ~]-2N. Moreover, for ~ ¢ 0 we have Its] -~-2N = O([tl-K-2N). Applying Peetre’s inequality and using that ~, ~ have co~npact disjoint support we deduce that, for arbitrary L, we have

-~. Ik(~, t, ~, ~)1~ cM-~[~l-~[t]-~[O Differentiation under the integral sign yields the same estimate for the derivatives of k. The push-forward of k under X is the integral kernel l(y,t,~,~) [t]-~k(y/[t], t,$/[~,~). It is obviously also rapidly decreasing with respect to all variables. In the same way as Proposition 3.4 we can show the following. 3.5 PROPOSITIONLet p,~,w,& be as in 3.4, ~,~ e C~(N~).Fix e > O. Suppose¢ is a function in C~(NxN), with ~(t, ~) = 0 tbr It-~l ~ e and ~(t, ~) = for [t - f] ~ Then the push-forward (3.5) is an integral operator with a rapidly decreasing integral kernel taldng values in £(E, F). In the following, we shall often restrict the variables to the set S = {(y,t,~,~,~,T)

:t,~ l, ltfi-

Notethat, onS, wehaveIt]

~l < U2,1y/tl ~ C,l~fil

~ }.

(3.6)

168

Schroheand Schulze

3.6 LEMMA Let c, T be as in 3.2 and 3.3. On the set S, we have (a) ID~D~T(t,~)l = ’-~’-t) andT(t, ~) >_ Cot for someconstant Co > O. (b)]DyaDtkDf~BD[c(y,t t, fl, ~)] = O(t-I~l-k-IBl-/). Theleft handside is zero for [,~[ + SG°-estimates and T those for {°,1,°1. SG

On S, the function c hence satisfies

Proof. (a) Write T(t, ~) = ts(~/t), where s(r) = (r- 1)/ln r. Clearly, T(t, ~) = O(t) and T(t, ~) >_ cot. Moreover, (tOt)T(t,~) = t(s(r) - (rO,.)s(r))l~=u ~ = O(t); in analogous way, one sees that, (~Oi)T = O(t). Since t~D~ can be written as a linear combination of terms of the form (tDt) ~, 1 ~_ j ~_ k, we see that t~D~tD~T(t, [,)

o(t). (b) Ic(y, t, ~), ~’)[ _< max{[y[,[~)[} f~ (t +~(~ - t))-~da : max{ly[,I/)l}(~’t) -~. Since y/t and gift both are bounded and since T(t, ~) = O(t), we conclude that c = 0(1). Next we observe that 1

~o

~ t, 9, ~) = c~,~t Dt kD~c(y,

(1 - a)~a ~ Y +a(~-y) ’2+k+t (t + a(~ - t)) da.

The same estimate as above shows that this term is _< C max{ly[, I~)l}t-:-~-~. Now (a) and Leibniz’ rule imply the assertion for a = fl = 0. In case lal + I~l = 1, the integrand is (1 - a)~+l~ta~+l~l(t + a(~- t)) -2-k-t and we conclude as before. < 3.7 LEMMA Let ~ ~t 0 be fixed. There exist constants C, ~ > O, such that

at[{,,-]_<[T¢,c{+ t~-,t~]_
(a.7)

Proof If t{,TI < 1, then the middle term side is _> [t~] > c~t >_ c2t[&T]. So we may assume that I~,rl > 1. Welet 7 = sup Icl + 1 and distinguish the cases ~vhere Irl _< 2~l~l and ITI > 23’1~1, respectively. Note that in the former, we have I~1 -> (23’ + 1)-1, in the latter Ivl _> 1/2. For Ivt _< 23’1~1, the middleterm in (3.7) is _> c31T~t>_c4tl~[ >_c5t[~,v]. The last inequality is due to the fact that I~[ -> (23’ + 1)-~. For ITI _> 271~1, we note that

Ic{+ trl> Itrl-Ic{l> ~lrl+ ~lrl-3’1{I> ½1rl>c~[{,r]. Wethus obtain the estimate fi’om below. For the second inequality we first note that T < cot and therefore [T{] _< crt[{] _< cst[{, r]. Similarly [c{ + tv] _< c9([c{] + [t7]) _< Oot[&T]. This yields the estimate from above. ~ 3.8 THEOREM Let~S~(R’~x~",R’~xR~+q;E,F),p~,let~,~C~(~ and w,D e C~°(IR+) with w(t) = D(t) = 1 for < 1. Fix~ # 0 and defi ne p(t, x, r, {) =~(x, re, {, tr, trl). Then the push-forward X.(~o(x)(1 - w(t))(op,,tp)(1 - &(t))~a(x)) symbol in SGIt’,",°I(IR ’~+1 x I~’’+1, IR"+t ;E, F). Its symbol semi-norms can be estimated by those for/~.

Edge-DegenerateBoundaryValue Problemson Cones

169

Proof. Since w and & vanish for t _< li we deduce from Lemma3.3 that the symbol q = q(y, t, ~l, ~, ~, T) of the push-forwardis given by -’~. ~ (y/t, ~ll~, T~, c~ + t~-, t~7) (1 - w(t))(1 - &(D)~o(ylt)~(~llDT’~[~] The compactness of supp ~o and supp ~5 shows that y/t and ~7/~ may be assumed to be bounded. Moreover, we may suppose that it- [I is small, since we may multiply by a function supported near t = t, at the expense of an error whose push-forward is an integral operator with a rapidly decreasing kernel, as we saw in 3.5. In fact, since t,~ >_ 1, we may assume in particular that It/~- 11 < 1/2. We therefore only have to establish the symbol estimates on the set S of (3.6), for vanishes on the complement. Wenext note the following identities. In order to save space we shall write (...) insteadof (y/t, ~11~,T(t, ~)~, c(y, t, ~1, ~)~tr , t~) -l+(Drp)(...) D~,{iS(...)}

E0ujck{~, = (D~jp)(...)t k=l

Dr{iS(...)}

= E {(D~5)(...)(-y~/t

~’) + (D~)(...)OtT~

(3.S)

k=l

D~{~5(...)} D~{~5(...)}

+ (D~)(...)(Otck~ + ~’) + (D,~)(...)~}, = (D~,iS)(...)+(DrO)(...)cj, (Dr~)(...)t.

Here c~ and {~ denote the components of c and {, respectively. The derivatives with respect to 9 and ~ are easily deduced from those. If we restrict the attention to S, then O~sck{ksatisfies the estimates for an SG(l’-l)-function, Otc~{~ + r and -1)OtT~ those for a SGO,°)-function, while -1 and Y~lt ~ satisfy t hose for a (°, n SG function. Also (t, y) ~ ~(ylt) and (~, ~)) ~ ~(~)/~} SG° functions on S . According to Lemma3.7 we may estimate DvjS(...) by [t]"[{,r] ’ while D~IS,and D,,~/~ are O([t]~-I [{, r] ’-I ). Since [y, t] ,~ It] ~ T ~ [~] ~ [9, ~] on S, we obtain the estimate

~r~en~’’n~m’-,,,~ D<~D ~,~.~ ._.,._.~ ~,~,, t,~7, ~,,¢,r)=0

([~, t]~’-i~ll-~{~,, i’]-IJ’l-~’ [r,{].-I<~1-.)

provided the total numberof derivatives is _< 1. The form of the derivatives in (3.8), however, together with the above observations on the functions O~sc~{~,... ,Y~lt shows that the general result follows in the same way. We shall now apply this to boundary value problems in Boutet de Monvel’s 1 {(t,x~,... calculus. We consider the half-space ~n+ += ,x,~) : t,x~,... ,x,_~ II~, x,~ e ~+). Weshall see that the blow-up induced by X transforms an ordinary t-independent operator in Boutet de Monvel’s calculus into an SG operator. 3.9 The situation (3.9)

170

SchroheandSchulze

We suppose that/5 = op.~iS,~ = op.¢~), ~ = op.~,~ : op~ and ~ = op~g. Now we fix r/¢ 0 and define A e B~’~(~_+~) by A= ( op+p+opg opt

opk) op s ’

wherep(t, x, 7, ~) =~(x, ~, t~-, t~),.q(t, x’, T, ~’) = ~(X’, ~’, tr, t,I),... ~(:c’,~’,t%t’rl). Here, x’ = (z~,... ,xn-~), ~’ = (~,...~n-t).

, s(t, x’,

3.10 THEOREM Let A be as in 3.9, w~,w,z ~ C~(~+),w~(t) = w~(t) = 1 for (2<) --~ t _< I; let ~,~o~ 6 C~ (I~+), and ~j,j : 1,2, the operator of multiplication by diag { ~oj, ~oj }~.- ~ × { o} }. Thenthe push-forward

(3.10)

B := X,((1 - ~)¢:A~(1 is an element of,,s~+

~.

Proof. For each entry of B we may apply Theorem 3.8. Westart with the pseudodifferential part. Here we choose E = F = C. The push-forward then is an SG-symbol. Westill have to check the transmission property. Denoting as in 3.8 the symbol of the push-forward by q, we have on S k k’ Du,.D~q(y,t,~,~,~’,[~

~ T

: t-~-~’((D~o D:~:,~)(~/t, ~/~,T~’,T[~’, ~]~,,,~. (~’,[~’,~]~.+t~, ~’(~/~) ~(~/~)~, ,[~.:~o:o T"[~-~(~ - ~, (~))(1Wenote that cn(y, t, 9, ~) = 0 for Yn= ~n = 0, so that c. (~’, [~’, r]~n) = c" ~’, where d = (c~,..., c,~_~)[u,,=O.,=o. Since i5 satisfies the transmission property

where{,~j} (~ ~1, {pj} is a null sequencein S’(~:’~-~ x II~’~-t ,II~ ’~-~ x II~~+q)and hj is a null sequence in H. Therefore D~D~’~(y/t,~)/~, T(’, T[(’, ~-](,~,c. ((’, [~’, T](n)+ tr, t~)l~-----=~.----O [’

’’

"

\[T~’, c’(’ +tr, t~]])

According to 3.8 the functions

belong to SGO~’~’°); they form a null sequence, since we saw in 3.8 that the mapping is continuous.

Edge-DegenerateBoundaryValue Problemson Cones

171

Wefinally deduce from Lernmata 3.11 and 3.12, below, that hi(... ) E SG°+~H. This shows the SG-transmission property for q. Next consider the push-forward associated with the singular Green sy~nbol ~ = d ~j=0 t}j0~. Focusfirst on one of the (lj e ~’-j (~n-1, ~n+q; S’(lI~_ ), $(IR+ )).We l E run over the scale of spaces H~(~+), a fi 2, an d F over th e sc ale H~(]R~_), an deduce that the symbol of the corresponding push-forward in (3.10) is an element of SG(U-J’~-J)(~n × ll~n; ~n+l+q,~q’(ll~-),8(l~)). Since the push-forward of normal derivative is tot, the argument is complete. <

It remains to establish the two following technical lemmata used above. 3.11 LEMMA Let h E H and f e SG° with If(x,~)l

>_ c > O. Ttmn

9(z, ~, u) = h(f(x, ~)u) e SG°~b~g. Proof. Write h = p+ h0 with p polynomial and h0 e H0. Then the assertion is trivial for p. Without loss of generality we maytherefore treat the case where h ~ H+. The topology of H+ is that inherited fl’om S(IR+) via the Fourier transform. Writing h = (e+u) ~, we have g(x,¢, .) = (~s~.(e+u)(sff(x, ~)))/f(x, Since f(x,~) -~ ~ 8G° it suffices to check that (x,~,s) ~ (e+u)(s/f(x,~)) SG°~,8(~). This space coincides with $(~,SG°). Hence we have to check that for all a, ~, k, k’ and s > 0 the estimate ~--xo ~ {u(s/f(x,~))}

-I ~1) O([~]-Ial[x]

holds. Weobserve that s~O~’ {u(s/f(x, ~))} f( x, ~) ~-k’ (s kO~’u)(s/f(x, ~)). It therefore sufficient to treat the case k = k’ = 0. Now

=

y(x,

An analogous relation holds for x-derivatives. Since f(x, ~)-20~ f(x, ~) (- ~’°) the case of higher order derivatives presents no further difficulty. 3.12 LEMMA On the set S, the function G(y,t,~,{,~,T)

= T[~,T]/[T~,c~ + tT, t~]

IS an dement of SG°. As before ~ ¢ 0 is ~xed. Proof. In view of Lemmata3.6 and 3.7 it is sufficient to showthat ITS, c~ SG(Ll’°). Since we are only interested in the case where t + 1~, v[ is large, we may

172

SchroheandSchuize

replace [...] by I... I. Writing...

instead of TGc~ + tT, tTI, we have

(., .)

(OTTO,Otc~+ T, *1) = O([G7]),

:! @T¢, a~c,o) =o([~,~]), (. 0)=O([~, I. :.~(0,O~e~, (. ’ ) (T~j,c~,0)=O([t]), GI.o.I = (. OTI...I- I. ’)(0,t,0) = O([tl), By Lemma3.7 again all first order derivatives satisfy the desired estimates; the task for checking higher derivatives is the same as before. ,~ 3.13 The manifold case Let X be closed compact with boundary, and let .~ ~ Btt’d(x, I~ l+q) have entries as in (3.9). Wefix 0 # r~ ~ ~a and define A ~ B’,d(X x ~) by the process in 3.9. From Theorem3.10 and Proposition 3.4 we immediately get the following result: 3.14 THEOREM Let ~1, W2 ~ C~<~(~q-), as above. Then the push-forward

o~l (t)

we(t) = 1 fo r t <_1, let X, Abe

X,((1 - w~)A(1 -w~)) × ~’SG k ~ ~,x~ is an element of ~(~’’~’)’d~ ~" Here, X is as in 2.5. REFERENCES 1. 2. 3. 4. 5. 6.

7.

Boutet de Monvel, L.: Boundary problems for pseudo-differential operators, Acta Math. 126 (1971), 11 - 51. Egorov, Yu., and Schulze, B.-W.: Pseudo-Differential Operators, Singularities, Applications. Birkh~iuser, Basel 1997. Erkip, A., and Schrohe, E.: Normal solvability of boundary value problems on asymptotically flat manifolds, 3. Funct. Anal. 109 (1992), 22 - 51. Mazzeo, R.: Elliptic theory of differential edge operators I, Comm.Partial Differential Equations 16, 1615 - 1664 (1991) Parenti, C.: Operatori pseudodifferenziali in ll~ ~ e applicazioni, Ann. Mat. Pura Appl. 93, 359 - 389 (1972) Schrohe, E.: A Pseudodifferential Calculus for Weighted Symbols and a Fredholm Criterion for Boundary Value Problems on Noncompact Manifolds, Habilitationsschrift, FB Mathematik, Universit~it Mainz 1991. Schrohe, E.: Fr6chet algebra techniques for boundary value problems on noncompact manifolds: Fredholm criteria and functional calculus via spectral invariance. Math. Nachr. 199 (1999), 145 - 185.

Edge-DegenerateBoundaryValue Problemson Cones 8. 9.

10. 11. 12.

13.

173

Schrohe, E.: Fr6chet Algebras of Pseudodifferential Operators and Boundary Value Problems, Birkh~user, Boston, Basel. To appear. Schrohe, E., and Schulze, B.-W.: Mellin and Green symbols for boundary value problems on manifolds with edges. Integral Equations Operator Theory. To appear. Schrohe, E., and Schulze, B.-W.: Pseudodifferential boundary value problems on cones. In preparation. Schrohe, E., and Schulze, B.-W.: Pseudodifferential boundary value problems on manifolds with edges. In preparation. Schulze, B.-W.: Pseudo-differential operators on manifolds with edges, Symp. ’Partial Diff. Equations’, Holzhau 1988, Teubner Texte zur Mathematik 112, 259- 288, Leipzig 1989. Shubin, M.A.: Pseudodifferential operators in ~’~, Sov. Math. Dok|. 12 (1971), set. N1, 147- 151.

A Theorem on Products of Non-Commuting Sectorial Operators ~ELJKO~TRKALJInstitute of Mathematics I, University strasse 2, D-76128 Karlsruhe, Germany email: ze:[j ko. strka:[j ©math. uni-karlsruhe, de

1

THE

of Karlsruhe,

Engler-

MAIN RESULT

In [5] and [12] Kalton and Weis present a new theorem about the sum of two closed operators. It is similar in spirit of the Dore-Venni theorem (see [4]). These two theorems claim that the resolvents of both operators have to commute. For the case of non-commutingresolvents, the work [9] gave an extension of the Dore-Venni result. There the authors replace the commutativity by a special commutator condition from Aquistapace and Terreni (see [1]). In [14] it is shown, that the same commutator condition allows also to extend the theorem from [5] and [12]. Recently, the work [11] gave a new contribution for products of sectorial operators with non-commutingresolvents. This result assumes, like the Dore-Venni theorem, that both operators have bounded imaginary powers and fulfill the usual parabolicity condition. Beyondthat, another commutator condition, which first appeared in [3], is necessary. In this case the author proves that the product is again a sectorial operator. The goal of this work is to show, that the same result for products of sectorial operators (under the identical commutator assumption) given in Theorem 1.1 of [11] remains also valid, if the operators fulfill the conditions of the Kalton-Weistheorem instead of assuming that they have bounded imaginary powers. Besides its generality, that allows to use the theorem on all Banach spaces, another advantage of this result is the fact that it, in particular, contains implicitly the product-theorem of [11] in Hilbert spaces (see Theorem1.3 in [11]), for which the author needed special proof. In this paper we will state and prove the main theorem only for B-convex Banach lattices. For general Banachspaces we refer to [14]. 175

176

~trka~

THEOREM 1 Let X be a B-convex Banach lattice, A and B be closed, linear, densely defined operators in X which are sectorial and fulfill the following properties

(a)

A has a bounded inverse and OH~(A) + Or(B) < ~r, where OHm(A):= inf{~ e (O,~r) : A has a H°°(E(~))-calculus} O.r(B) := inf{~ e (O,~r) : B is R-sectorial of type ~};

(b) R(p,B)(D(A)) C_ D(A) for all# (c)

There exist constants a, fl,cAB >_ 0 with a + ~ < 1 and I[[AR(z,-B)

- R(z,-B)A]R(#,-A)[

CAB 1+~ I <_ (1 + ]pll-~)lzl

for all # E E(~r - OH~(A)) and z ~ E(~- - O~(B)). I] these conditions are ]ulfilled, operator

then there exists a constant v >_O, such that the L:= v + AB,

defined on D(AB) := {x ~ D(B) : Bx D(A)}, is sec torial of

typ e ~ f orall

¢ (A) REMARKS a) Wewould like to mention, that the operator L is also sectorial, we change the assumption (a) (a’)

if

A has a bounded inverse and Or(A) + OH~(B) < ~r, where Or(A) := inf{~ ~ (0, ~r) : A is R-sectorial of type ~} OH~(B) :-- inf{~ E (0, ~) : B has a H~(E(~))-calculus}.

This statement can be proved in the same was as Theorem1. b) If we compare Theorem1 with Theorem1.1 in [11] we see that the assumption on the operator A is stronger, since the existence of an H°° functional c~lculus implies the existence of bounded imaginary powers. On the other hand the assumption on B is weaker. In [12] the reader can find the connections between the class of operators with bounded imaginary powers and the class of R-sectorial operators. This fact in particular guarantees that Theorem1 is stronger in case A is a well knownoperator like d~ or -A. Since this is the case in the applications of paragraph 2 in [11], we are in position to derive them also from Theorem1, but for this special setting we have the advantage to take a bigger class of multiplicative perturbations B. The proof of Theorem1 will combine techniques developed in [12] with the ideas from [11]. In [11] it is shownhowto construct the inverse of AB + ~. The construction method, given in that work, depends heavily on the operator 1 fr

S ~ x = 2 ~r--~~,

~R(A-

-B)AR(z,A)xdz,

x ~ D(A)

z’

(for the exact definition see section 2) and Lemma 1 (see Lemma1.10 in [11] as well). The author was able to prove the assertion of this lemma under the assumption that both operators have bounded imaginary powers and fulfill condition (b) and

177

Products of Non-Commuting Sectorial Operators

(c). Only for this statement the author needed the bounded imaginary powers assumption. The remaining parts of his constructions used the fact, that Lemma1 holds, that both operators are sectorial (with the parabolicity condition O~(A) On(B) < n) and that (b) and (c) were given. Since the operators in Theorem also sectorial (notice that OH¢O (A)+ Or(B) < ~r implies the parabolicity condition), we are allowed to use the same constructions as well, if we are in position to show Lemma1. This is the reason, why the paper is primarily concerned in proving the mentioned lemma. The actual construction of (AB + ~)-1 will be omitted in this paper. The reader can find it in [11].

2

NOTATIONS

AND PRELIMINARIES

In this section we collect some notations which shall be used throughout the paper. Moreoverwe will prove some results which will be relevant to verify the mentioned result. For a Banach space X, the set of all bounded linnet operators T : X -~ X is denoted by E(X). If ~ E (0,~r), the open sector (z E C\ (0} : largzl < 8} is denoted Z(~). The abbreviations Fe and F~ will stand for the contours

r~ := -(-~,0]e i° -i°, u[0,~)e F~ := -(-o%-r]e iO U re -~[-0’0] -i UO [r, , oo)e where 0 e (0, ~) and r > 0. Weparametrize the boundary of E(0) with the function To(t) = tei° for t _> 0 and To(t) -= -re -i° for t < 0. Wecall a closed, linear, densely defined operator A in a Banachspace X sectorial, if A is injective, has dense range A((D(A)) in X and the positive real axis (0, oc) is contained in the resolvent p(-A) with (R(A, A) := (A -~)

IItR(t,-A)ll <_

~’t > O.

For those operators we define O~(d) := inf{~ e (0,~r) : E(~r - ~) C p(-A),

sup IIzR(z,-A)ll ze~(rr-~)

<_ MA(~)}.

For a sectorial operator we will use the definition of a bounded H~-calculus in the same way as it was introduced in [8] and later developed in the work [2]. The reader can find examplesof such operators in the cited two works and the references therein. In the article [12] the reader will find references whythe applications of paragraph 2 in Ill] can be obtained by Theorem1. The definition of R-sectorial operators is given in section 3 of [12]. Since X is order continuous, X and its dual X* can be represented as Banach function spaces on the same measure space (12,#) (see [7]). Nowthe B-convexity assumption (in this case X and X* have finite cotype !) allows to characterize R-boundedness via the boundedness of pointwise multipliers; see [12] section 4 a) and the remarks about R-boundedness in section 1 f). This means that assumption (a) in Theorem 1 implies

I (f:t’tR(7~(t)’B)f(t)’~

dt) ~/~l x <- C I (/~[f(t)[~

~/~ lx

178

~trka~

for all qo > O,(B) and all f E ~(~, D(B)). Here L’(IR, D(B)) is the set of all finite step functions ] : IR ~ D(B). Thus, by a density argument, for each measurable function f : 1R x fl ~ G with [[fll(g~)x := I(f~ If(t’ .)[2dt)l/~ [x < ~, we have [(~[tR(~/~(t),B)f(t)[

~ dt) I/2l x <_ C [[fII(L2)X

(1)

Wewill call strongly measurable operator-valued functions t ~ ~ ~ N(t) £: (X), that fulfill such estimates I(fa[N(t)f(t)12

1/~

x < C Ilfll
also (Le)X-multipliers. With this definition (1) means that the function t ~ ~ ~ tR(7~(t),B)

L( X)

is a (L2)X-multiplier. From now on we assume that all conditions of Theorem1 are fulfilled. Condition (a) enables us to fix constants ~ OH ~(A),~2 > O~(B) wi th ~ + ~2< ~. If take y ~ (0,~ - ~ - ~2) and ~ ~ (~,~- ~2 - Y), [arg(~)[

~ ~+y < w-~z

holds for all A ~ ~(~) and p e P~ ~ {0}. Thus, for A e E(y), we are in position define the operator S~x := ~1 ~a ~R(~ zz

z’-B)AR(z’A)xdz’

xeD(A).

Here PA:= F~"t is the contour with a fixed constant rA e (0, I[A-1]I-~). Obviously Sx is bounded from the graph norm space XA into X. To apply the mentioned construction technique, given in [11], it is enoughto verify the following result LEMMA 1 S~ has a unique bounded extension

on X, which satisfies

The proof of this statement will use similar ideas already presented in section 3 of [12]. For them we need some preliminary results. THEOREM2 Let t e ~ ~ N(t) ~ E(X) and t ~ ~ ~ N*(t) ~ £(X*) be measurable operator-valued functions. Then N(.) is a (L~)X-multiplier if and if N*(.) is a (L~)X*-multiplier.

Productsof Non-Commuting Sectorial Operators

179

This is obviously true, since the canonical duality mappingis given by

for f e (L:)X, g (L2)X*. THEOREM 3 Let t e ~ ~ N (t) e E( X) be an operator-valued function which an integrable Frdchet derivative on l~. Then N(.) is a (L2)X-multiplier. This is a consequence of Proposition 2.5 in [13] and the ch~racterization-Lemma from section 4 a) in [12]. The next result provides a link between the Banach sp~ce X and the double norm space (L2)X. LEMMA 2 For each 6 ~ (0, 1) the square function G~.,(x)

:= II(£,~(t)~-~/~[Al-~R(~(t),A)x](.)l~

X

defines a norm on X with 1

The proof of this statement can be found in [14]. It is based on square function estimates for operators with ~ bounded H~-calculus. (see for example Corollary 6.7 in [2] or Lemm~ 5.3 in [6]) The next lemma, which we will need in the proof of Lemma1 is a result about singular integrals. LEMMA 3 For every ~ ~ (-1/2, (~.~[f])(,)

1/2) the operator PV- ~ ,)f(z)dz

is bounded on the scalar space L~(F~). This result in particular leads to the following (for the proof we refer again to [14]) COROLLARY. 1 There exists

a constant

C = C(~,~),

such that

for all

f

(n~(r~))X

A further tool in the proof of Lemma1 is a consequence of condition (c). If introduce the commutator bracket

[U;V] := uv-vu

180

~;trkalj

for bounded operators U and V, one can see that the following identities [R(#,

,A);

R(z,-B)]

[R(z,-B); From the first

= R(#,-A)[AR(z,-B)

hold

- R(z,-B)A]R(#,

AR(#, -A)] = #[ROt,-A);

R(z,-B)].

equation and (c) we get

C(~I)eAB

[[[R(#,-A)~R(z,-B)][[

(3)

for all # E E(~ - ~1) and z E E(r - ~o~). Since a + fl < 1, we can find constants al,a~ with 0 _< a < a~ < ae _< a2 + fl < 1. Nowthe inequality (2) in particular guarantees the following estimates (B(, 0 :=

B+ ~I)

LEMMA 4 For 5 with al < ~ < a2 there exists a constant C, which only depends on 6, qo~ and ~2, such that for all A ~ E(r/) and all e ~ (0, 1) the inequalities hold (a) ]I[A~_~R(#,A);R(A_,_B(~))]{ I < C’CAB[I-t{

Ior ~lt p e~,with {~{ e(0, ~], (b) <

[[[A~-6R(#, A); R(~,

C. CAB{#[

for all # ~ F~ with [#[ 6 (1, ~). PROOFFrom Theorem II.6.9

in [10] we know

A~-~ = sin/~r/1--

~)}

t-~AR(-t,

A)z dt,

x ~ D(A).

Using the resolvent equation we obtain [A~-~R(#, A); R(~, -B(¢))] AR(#, sin(~r(l~r- 5))[ + sin(~r(1rr

A); R( , -B(O)] t6(# + t) dt ~ ~

~)) fo°° t~(/z + ~[R(~ ,-B(~)); AR(-t, A)] dr.

Since

~

f0t~(~+t~)dtl <_ {ul Mi(cfli,992)

~o

t~l# [

+t

~)

1

t~+~ dt <_ -~, M~(~,qal,~2){#l

181

Productsof Non-Commuting Sectorial Operators using (3), the first term can b~ estimated c~

f0

1 dtl[ [I sin(Tr(17r - 6))JAR(#, A); R(~,-B(~))] t~(/~ -

<

2+’-~ M4(& ~, ~o2)cA~ Iml IAII+~(1+ M)(1+ I~1’-")" Thusthe first

term obtains the required estimations. For the secondterm we have

sin(~r(1-~)) II ;

fo~ t~(#+t)[R( ~,-B(~));An(-t,A)]dt[[

M~(~)ca~ ~ fo ~ I~ +~+~1

M6(~91 ’ ~)CABI#II+~

t dt < + 1 t~l# ~1( + 0(1 + ~-~)

t

dr+

[I.II~1(1+ t)(1 ~tl -~)t {~o

~+~ I~1

t f ~ I (1 + t)(1 -t- tl-a)tl+a dt }. 1. case I#l e (0,1]: If we use 1 + 1-a >t ~-a~ and (1 + t ) >_ 1, thefoll owing hold t~1

~o

t

t

lulO + t)(1 t~-~)t ~ dt + (1 i;

I + t )( 1 + t ~- ~)t TM dt <_

lul "~ t

fo

lUl ~ dt

+fl~ ~ ta~ ~ dt. < M70)I#I case I~1~ (1,~):Because of 1 + t ~-~ > t ~-~=, we have I"1

foI#1(1+

t t)(1 t~-")t ~

~ dt+ (~ i~

t + 0( 1+ t~ _~)t~+~

dt <

Nowthe proof is complete.

3

PROOF OF LEMMA 1

For e 6 (0, 1) we set 1 fr

1R(~ -B(~))AR(z,A)xdz,

x e D(A).

182

,~trkalj

Using Lebesgue’s theorem on majorized convergence one obtains S~,~x -+ S~x as ~ -+ 0+ for all x E D(A). Therefore, it is sufficient to prove the assertion of the lemmafor all S~,~, uniformly in ~ E (0, 1). To verify this, we first derive another representation of S:~,s. A usual holomorphyargument yields to (r ~ (0, rA) arbitrary~)

= 2~i 1’ ~ ZR(~ 2~i

=

z

a

1" ~ ~R(~,_B(e,)R(z,A)xdz

~ z ~ z -B~))xdz

+ ~

~

z

z

B(~) has a bounded inverse, thus the first term is zero. Therefore we get

Since 7 ,~, -B(~))R(z, A)[] is bounded in a neighbourhood of zero, we obtain for all x e D(A) Sx,ex

=

1 2~i~ ~~R(A z -B(~))R(z,A)xdz

= ~1~~R(z’A)R(~’-B(e))XdZz z

+

~, ~ ~ IR( = Tx,~x

+

Thanks go the commutator condition (a), Ux,~ defines a bounded operator with

c(~, ~) Using the resolvent equation, for p e F~ ~ {0} and A ~ g(~) we have

~-~/~A~-~R(~,A)

PV ~ (z - ~) " z z -B(~))xdz

f

~5-~/~)z~l/~

A)R(~,

dz.

Productsof Non-Commuting Sectorial Operators

183

Applying Lemma2, (2), assumption (a) and the techniques given in the proof [12] we get (x E D(A))

Ci -~1N-(~ R(-~, -B(~))(.)~-~/2 A~-~R(., A)x][(L~)x +

Intheremainin6 partitsuffices to showthat

1(/2 ’7~(t)~-3/~[R(~

t) B(~));A~-~R(%(t)’A)]x’~dt)

~/~l

holds for all ¯ ~ D(A) (with Ca independent of e and I). argument we can restrict ourselves ~

A~-~R(%(t), (£~l%(t)~-an{R(%(O, B(~));

A)]xl:dt )

Nowby a symmetry

C4 1/2

~x~ ~llzllx.

.

So, using the Minkowskiinequality, this shall be true, if

(~l I~(t)~-al~[R(

B(~)); AI-~R(%(~), A)]xi~dt) ll~ ~x ~

C5

and

A (~l%(t)~-al~[R(~t),

A~-~(%(t), A)]~l~dt B(~)); ~) 1/~ x

for all For each ¯ ~ D(~) there exists a ~ ~ (D~(1, ~))~’ ~ (~’~(’)~-31~[R(~,-B(e));AI-~R(7~(t),A)]x’

Ilvll (~i)x. = 1s~ck that k 1/~ii 2d’) ix ~

2~ < ta_al2[R(@, 2llzllx.

The last inequality is a consequence of the H61der inequNity, where

~~ll~llxC~

184

~trkalj

with ¢ ~ (0, 1 - a2 - fl). An analogous argument guarantees for each x E D(A) h e (L~(0, 1))X* with Ilhll(t~) x. = 1 such that

with D2 = where we take a E (0,al).

dt

If we set

A r+~-I := t [R(,~(t),

N~,~_,~(t)

,

(t >0),

B(~));A~-~R(7~(t),A)]

so far we have shown

and I ( fo’ I’~(t)~-31~[R( 7~’ B(~)); Ai-~ n( ~,~(t),A)]xi~ dt ) ’l x <- 2D2llxilx " II N* x,~,-~(’)g(’)il(L:(o,1))X.. Now,using Theorem2, the proof shall be finished, N.x,.~,-, (.) are (L2)X-multipliers with

if we show that N~,e,¢(.) and

e

e

(4)

(c independent of A and e). Differentiating N)~,e,r(t) one N’~,~,~(t)

= (r + ~ 1)t~+~-~[R( A---~ -B(e));A~-aR(te iv,A)] + + t-~A R(t~, ~ -B(~))tr+~-~[R(~, ~ + t r+~--2 [R(tei~,-B(e)),

.)

-B(~));

A~-~R(te ~, A)]

A1-sR(tEi~,A)]~R(~,_,(e)

- teiVR(te i~, A)t~+~-2[R(~, -B(~)); A~-~R(te iv, A)] -

- ~-~ [R(~, -B(~)); ~, A)]~%~ ~-aR(~e~, A A). case Nx,e,~: Because of Lemma4, the first t¢+8_ 2 Clt ~-~

four terms can be estimated by

_ C~

-

185

Productsof Non-Commuting Sectorial Operators with C1 independent of ~ and )~. For the last term we use the momentinequality I[t~e~Al-~R(te~,A)[[

<_ C2 Vt >_

and obtain therefore by (3) the same type of inequality as for the first four terms. Thus we get Since a2 +/~ + ¢ < 1, Theorem3 states,

that N~,~,~(.) is a (L2)X-multipliers

(4). 2. case N;~,~,_~: With the same arguments as for the previous ease one obtains

IlAl+~N~,,~,_~(t)[I<_ C4taa+o-a-1 ~’t ~ (0, 1). Again Theorem3 is applicable (a~ +/3 - a > 0) and leads to the desired assertion (4). This completes the proof of Lemma1 and the theorem as well.

REFERENCES 1.

2.

3. 4. 5. 6.

7. 8.

9. 10. 11. 12. 13. 14.

P. Aquistapace and B. Terreni, A unified approach to abstract linearnonautonomous parabolic equations, Rend. Sere. Mat. Univ. Padova 78 (1987), 47107. M. Cowling, I. Doust, A. McIntosh and A. Yagi, Banach space operators with a bounded H~ functional calculus, J. Austral. Math. Soc. Ser. A60 (1996), 51-89. G. Da Prato and P. Grisvard, Sommesd’ opdrateurs lingaires et ~quations diffgrentielles opdrationelles, J. Math. Pures Appl. 54 (1975), 305-387. G. Dote and A. Venni, On the closedness of the sum of two closed operators, Math. Z. 196 (1987), 189-201. N. Kalton and L. Weis, The article is in preparation. F. Lancien, G. Lancien and Ch. Le Merdy, A joint functional calculus for sectorial operators with commuting resolvents, Proc. Lon. Soc. 3 77 (1998), 387-414. J. Lindenstrauss and L. Tzafriri, Classical Banachspaces II, Springer (1979). A. McIntosh, Operators which have an H~functional calculus, Miniconference on Operator Theory and Partial Differential Equations, Proceedings of the Centre for Mathematical Analysis 14 (Australian National University, Canberra, (1986), 210-231. S. Monniauxand J. Priiss, A theorem of the Dote- Venni type for noncommuting operators, Trans. A.M.S. 349 (1997), 4787-4814. A. Pazy, Semigroups of Linear Operators and applications to Partial Differential Equations, Springer-Verlag (1983). F. Weber, On products of Non-Commutingsectorial operators, to appear in Ann. Scoula Norm. Sup. Pisa C1. Sci 4. L. Weis, A new approach to maximal Lp-regularity, this volume. L. Weis, Operator-valued Fourier Multiplier theorems and Maximal Lvregularity, preprint. :~. ~trkalj, Sums of noncommutingsectorial operators, in preparation.

The Spectral Radius, Hyperbolic and Lyapunov’s Theorem

Operators

VU QUOCPHONGDepartment of Mathematics, Ohio University, 45701, E-mail address: [email protected]

1

Athens,

OH

INTRODUCTION

Let A be the generator of a C0-semigroup T(t),t >_ 0, on a Hilbert space H. It is the well knownLyapunovTheoremthat T(t) is exponentially stable if and only if for some (or for any) positive definite operator Q, the operator equation AX + XA* = -Q has a positive definite solution, see [4,5]. Thus, there exists a close relationship between the asymptotic behaviour of C0-semigroups and the solutions of differential equations, from one side, and the solvability of the operator equations of the Lyapunov form, AX + XA* = -Q, from the other side. It is natural to ask whether some suitable variant of the LyapunovTheoremremains true for semigroups on Banach spaces. In our recent papers [15,16], one variant of Lyapunov Theorem has been prerented. Namely, we shown that a semigroup T(t) on a Banachspace E is exponentially stable if and only if it is uniformly bounded and the operator equation AX - XD = 6o has a unique bounded solution, where D is the differentiation operator on F := BUC(R,E) (the Banach space of bounded uniformly continuous E-valued functions on R) and 50 : F ~ E is the Dirac operator defiaed by 50f = f(0). The discrete case of the LyapunovTheoremalso holds (see e.g. [11]): namely, T is a boundedlinear operator on a Hilbert space, then ]IT’~II ~ 0 (or, equivalently, r(T) < 1, where r(T) is the spectral radius of T) if and only if for some (or any) postive definite operator Q, the operator equation X -T*XT = Q has a positive definite solution. Lyapunov Theorems motivated motivated the study of the operator equations AX - X B = C and X - AX B = C on Banach spaces, see [1,4-8,10-13]. In Section 2 of this paper, we give further results on the relationship between the spectral radius of the operators A and B and the solvability of the operator equation X - AXB = C. Weshow that if T is a bounded linear operator on a Banach space then r(T) < 1 if and only if the sequence Y~.=o AnTnx is 187

188

Vu

bounded for every x E E and for every A,[A[ = 1. From this we deduce that a n is sufficient boundedness condition for the geometric series with elements AnCB for the solvability of the equation X - AXB= C. In Section 3, we consider hyperbolic operators, i.e. operators whose spectrum does not intersect with the unit circle, and extend results of the papers [15,16]. Namely, we give various equivalent characterizations of hyperbolicity of an operator T (in particular, of the condition r(T) < 1) in terms of solvability of someoperator equations of the Lyapunov type X - TXSo = 50, where So is the shift operator on the Banach space/~(Z, E) of bounded E-valued sequences and 50 : F -~ E the discrete Dirac operator defined by 5o : x := x,~ ~ x_~, as well as in terms of solutions of a difference equation x,~+~ = Tx,~ + y,~, n E Z in the class of boundedor almost periodic sequences. It should be emphasizedthat the results on hyperbolicity and on difference equations present independent interest, although the methods of this section are similar to those presented in [15,16]. Our notations throughout the paper are standard. Thus, E denotes a Banach space, T - a bounded linear operator on E. The spectrmn, resoIvent set, and spectral radius of T are denoted by a(T), p(T), and r(T), respectively. As usual, Z is the set of integers and F is the unit circle.

2

GEOMETRIC SERIES

Let E be a Banach space and T be a bounded linear operator on E. It is easy to see that if the series ZT~=S k----O

converges (in the strong or unifbrm operator topology), then 1 p(T) and S = (I - T) -*. The following simple proposition shows that one can make the same conclusion under a weaker conditiou of boundedness of the geometric series. THEOREM 1 Suppose that,

for every x ~ E, the series

Z T~x k=0

is bounded. Then 1 ~ p(T). Proof. By tim Uniform Boundedness Principle, sup,>o []T"[[ = M< co, so that k-1 )~ ~ p(T) for all A > 1. Let Sk = ~i=o Ti, k >_ 1, So =- O. From the following equality which is valid for any sequence ak, bk ak(bk+~-- bk) = anb,,+l aobo - Y~bk(ak -- a~:_~) k=0

k:l

Spectral Radius, Hyperbolic Operators, Lyapunov’s Theorem

189

it follows that ~ A-(k+I)T ~’ = ~ A-(~+I)(S~+I k:0 k:0

- S~)

n

~-("+I)S,~+~ - ~-’~(~-(~+~) A-~’)Sk= k=l

A-(’~+IlS,~+~ - (1 - A)-(~’+~~S~ . Since S~ are uniformly bounded, the series converges (in the uniform operator ogy) to some operator R. It follows that R = (A - -~ and

IIRll = II(A -T)-lll ~ (A- 1)M~A-(k+’)

topol-

< M.

1

Therefore

dist(A,a(T)) _> ][(A- T)-’]l > ,VA> 1, which implies

that

1 E p(T) and dist(1,

a(T)) >_

1

(1)

Remark: It is easy to see that if 1 E p(T) and {Tn : n >_ 0} is bounded, then the series ~=0 Tk also is bounded. In fact, ~=o T~ = (I - T)-~(I ’’ +~) (se e e.g [2, Ch. 2, Exercise 9]). As an immediate consequence of Theorem 1 we obtain the following zation of the class of operators whose spectral radius is < 1.

characteri-

COROLLARY 2 Suppose that T is a bounded operator on a Banach space E. Then r(T) < 1 if and only i~ for every x ~ E and for every A ~

sup ~-~A’*T"x < ~o.

(2)

n>0 - k-=--0

For the proof we need to apply Theorem i to the operator

AT to see that F C p(T).

Remarks: 1. It is easy to see that r(T) < 1 if and only

~ llTX’[I< ~c, k:0

hence Corollary 2 implies that conditions

(2) and (3) are in fact equivalent.

(3)

190

Vu

2. There is the following knownchacraterization of operators with spectral radius r(T) < 1 which is due to Weiss [17]: an operator T has spectral radius r(T) < 1 and only if ~1{ Tkx,¢*)f

< ~

k=0

for some 1 ~p < ~ and for allx that r(T) < 1 if and only

e E,x* ~ E*. Earlier,

McCabe [9] h~shown

k=O

for some 1 ~ p < ~ and for all z ~ E. Corollary 2 implies McCabe’scharacterization but neither does it imply Weiss’ characterization, nor vice versa. 3. If n~O

~hen from (1) we obtain ~he following inequality for ~l~e spectral radius of M-1

Consider now the operator equation X-AXB=C,

(4)

where A is a bounded operator on a Banach space E, B is a bounded operator on a Banach space F, C is a bounded operator fi’om F to E (and so is the unknown solution X). It is easy to see that if the series ~ A~’CB ~ = X

(5)

converges (in the strong or uniform operator topology), then X satisfies equation (4). If we assume that the series (5) converges for every C, then Eq.(4) has solution for every C and, moreover, the solution is unique (see [11]). It is convenient introduce the following operator T: L(F, E) ~ L(F, by TC = ACB. Then. the mfique solvability of the Eq.(3) for every C E L(F, E) is equivalent to 1 E p(T). Applying Theorem1, we obtain the following result. THEOREM 3 Suppose that for every C ~ L(F, E) and for every y ~ sup n>O --

E A~CBky
Then the operator equation has a unique solution for every C. From Theorem3 we obtain the following corollary.

(6)

191

Spectral Radius, HyperbolicOperators,Lyapunov’sTheorem COROLLARY 4 Let A E L(E), B ~ L(F).

Then r(A)r(B)

sup At~At’CBt:y n>_OEk=O

< 1 if and (7)


for every C ~ L(F,E), y ~ F, and A ~ For the proof we apply Corollary 2 to the operator T, and use a result of Lumer and Rosenblum [8] that r(T) = r(A)r(B). It is easy to see that the following conditions are equivalent (and by Corollary each of them is equivalent to (7)): (a) IIA’"II[[B’~I[ < (b) r(A)r(U) < 1; (c) r(T) < 1;

(d) 27’=0 IIAkllllBkll < and any of these conditions implies the uniform convergence of the series (5) for every C (which in turn implies that the equation X - ABX = C has a unique solution for every C). Conversely, if the series (5) converges (strongly or uniformly) for every C, then r(A)r(B) <_ 1 (see [11]), the latter inequality is the best possible (see [3]).

3

HYPERBOLIC

OPERATORS

A boundedlinear operator T on a Banachspace E is called hyperbolic, if there exists a projection operator P on E and positive numbers r < 1, p > 1, M, N >_ i, such that (i) IlT~xll < Mrnllxll, Vx e R(P),n >_ 0; (ii) [[Tnx][ >_NpnnHx[],Vx e ker(P),n >_ Equivalently, T is hyperbolic if a(T) f3 F = Consider the difference equation x,~+~ = Tx., + y,,,

n=0,±l,±2,...,

x,,,yn~E.

(8)

Let /°°(Z,E) denote the Banach space of bounded E-valued sequences x (xn),~°°___~ (we also denote (X)n == Xn), and consider the shift operator S: /°¢(Z,E) -+ /~¢(Z,E): S: (x,~),z~=oo (Xn_~),~m___~ (th us, (Sx )~ = Xn-~). Clearly, S is an invertible isometry. Assumethat Mis a closed subspace of/o°(Z, E) which satisfies properties: (A1) SM = (A2) If C : M+ E is a bounded linear operator, x ~ M, and Yn = CSnx, then Y = (Yn)n=-~o ~ (A3) a(SIM) = {~ e C: IAI = 1}. Below we denote So = S[M. As an example of spaces Msatisfying (A1) (Aa), we can take the space of all almost periodic sequences. Then the statement (vii) in Theorem5 gives a characterization of hyperbolicity of T in terms of existence

192

Vt~

of a unique almost periodic solution {xn } of (8) for every ahnost periodic sequence

THEOREM 5 Let T E L(E) and M be as above. The ]ollowing statements arc equivalent: (i) T is hyperbolic; (ii) For every invertible operator ]7 : F -4 F such that a(V) C F, and for every C e L(F, E), the operator equation X-TXV=C

(9)

has a unique solution; (iii) For every isometric operator V : F -4 F and ]or every C ~ L(F, E), the operator equation (9) has a unique solution; (iv) For every isometric operator U : M -4 M and every C e L(M,E), operator equation

x - TXU= C

(10)

has a unique solution; (v) For every C ~ L(M, E), the operator equation X - TX~o = C

(11)

X - TXSo : 6~’,

(12)

has a unique solution; (vi) The operator equation

where ~oM: M-4 E, 8oMX= X_l, has a unique solution; (vii) For every sequence y (Yn)n=-~o in M there exi sts exa ctly one sequence x = (x,)~°°___oo in Mwhich is solution of the difference equation (8); (viii) For every sequence y = (Yn)~=-~o /c° (Z, E) the re exi sts exa ctly one sequence x = (Xa)n°°=_oo in/°°(Z, E) which is solution of the difference equation

(s). Proo]. (i) ~ (ii). Let P be the corresponding projcction, E~ = PE, E~ = (I-P)E, T~ = TIE~,T2 = T[E~,C~ = PIC, C2 = -P~C. Since r(T~)r(V) < 1, there exists a unique solution X~ : F -4 E~ to the equation X~ - T~X~ V = C~.

(13)

Analogously, since r(T(~)r(V -~) < 1, there exists a unique solution X~ to the operator equation Xz - T~X2V-~ = -C2,

or X2 - T2XzV = P~_C.

(14)

Let X = X~ + X:, then X- TXV= C. Conversely, if X is a solution of Eq.(9), then X~ = P~Xand X~ = P.~X, being the solutions of Eq.(13) and Eq.(14), respectively, are uniquely determined, hence the solution X is unique. The implications (ii) ~ (iii) ~ (iv) ~ (v) ar e tri vial .

193

Spectral Radius, HyperbolicOperators,Lyapunov’sTheorem (vi) ~ (v). Let X be the Unique solution of Eq.(12). It follows that a solution Eq.(ll), if exists, also is unique. Define an operator Y : M-4 E Yx = X~, where

(~),~ = CS~’~-lx. It follows that Y - TYSo = C. In fact,

we have (SoX)n CS~’*x, hence

~ (CS~-"*x) Yx - TYSox = X(CS~"-lx) - TXSoS~ ~y(CS~-~x) = Cx. (vi) =~ (i). It is kn~vonthat if 7Z is a boundedlinear operator on L(M,E) deftned R(X) = X TXSo, th en a( ~) = {1- A# : A a( T)p a(So) } (s ee [8, Theor 10]). Assumethat (v) holds. This implies that the operator ~ is invertibe, i.e. 1 # Ap, A ~ a(T), p e a(So). Since a(So) = 1", it follows that a(T) ~ F = (vi) ~ (vii). Let X be the solution of Eq.(12) and xn = XSg~y. Then from (A2) we have - (x n)n~°=_c~ ~ M,and z,~+l

= XS~I(S~ny)

= (TX + ~oSgl)S~ny

= Tzn

that is x is a solution to Eq.(8). To show the uniqueness, assume that Xn+l = Tx,,,n ¯ Z. Then x _= { ~}.,~=_~ X is a bounded complete trajectory of T. Since a(T) f? F = ~, it follows, by [14, Proposition 5.3], that the spectrum of x, Sp(x), empty, so that x = 0. (vii) ~ (vi). Assumethat (vii) holds. Then we can define an operator G : M-~ Mby Gy = x, where x is the unique solution of Eq.(8). It is easy to see that G linear and closed, so by the Closed Graph TheoremG is a bounded linear operator on M. It is also easy to see that G commutes with So. Let Xy = (Gy)0. Then from (Gy),~+, = T(Gy)n it follows (with n = -1) that (Gy)o = T(Gy)_~ + = T( GSoY)o + yi.e. Xy - TXSoy = ~oMy. From the equivalence of (vii) and (i) it follows that (viii) lent.

and (i) are equiva-

Remarks.1. If, in addition to one of the equivalent conditions (i)-(viii), T is power-bounded(i.e. supn>_o [[T~[[ < o~), then r(T) <

the operator

oo 2. If T is a hyperbolic operator and {Y,~},~=o is a bounded one-sided sequence in E, then Theorem 5 implies that there exists xo such that the sequence {x~},~°°= o defined by

x,,

= Tnxo + ~ Tkyn_~_~ k=0

(15)

194

Vu

is bounded. It is an open question whether the converse is true, i.e. assuming that for each bounded sequence {Yn}~=0E E there exists at least one Xo E E such that (X) ¯ the sequence {x,,},~= o defined by (15) is bounded, does it imply that the operator T is hyperbolic?

REFERENCES 1.

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

14.

15. 16.

17.

W. Arendt, F. R~biger and A. Sourour, Spectral properties o] the operator equation AX + XB = Y, Quart. J. Math. Oxford, Set. (2) 45 (1994), 149. B. Beauzamy,Introduction to Operator Theory and Invariant Subspaces, NothHolland, Amsterdam-NewYork, 1988. B.E.Cain and S.A. Nelson, Solutions to x - axb = w, Linear Algebra App]. 165 (1992), 229-232. J.L. Daleckii and M.G. Krein, Stability of Solutions of Differential Equations in Banach spaces, Amer. Math. Soc., Providence, R.I., 1974. R. Datko, Extending a theorem of A.M. Lyapunov to Hilbert space, J. Math. Anal. Appl.yr1970 32,610-616. 3. Goldstein, On the operator equation AX + XB = Q, Proc. Amer. Math. Soc. 78 (1978), 31--34. S.C. Lin and S.Y. Shaw, On the operator equation Ax = q and SX - XT := Q, J. Functional Anal. 77 (1988), 352-363. G. Lumer and M. Rosenblum, Linear operator equations, Pro(:. Amer. Math. Soc. 10 (1959), 32-41. T.F. McCabe,A note on iterates that are contractions, J. Math¯ Anal. Appl. 104 (1984), 64-66. C.R. Putnam, CommutationProperties of Hilbert Space Operators and Re~!ated Topics, Springer, NewYork, 1967. R. Redheffer and R. Redlinger, The spectral radius and Lyapunov’s theorem, Linear Algebra and Appl. 128 (1990), 169-180. M. Rosenblum, On the operator equation BX - XA = Q, Duke Math. J. 23 (1956), 263-269. Vu Quoc Phong, The operator equation AX - XB = C with unbounded operator A and B and related abstract Cauchy problems, Math. Z. 208 (1991), 567-588. Vfi Qu6c Ph6ng, On the spectrum, complete trajectories, and asymptotic stability o] linear semi-dynamical systems, J. Differential Equations 105 (1993), 30-45¯ Vfi Qu6c Ph6ng, On the exponential stability and dichotomy o,f Co-semigroups, Studia Mathematica 132 (1999), 141-149. Vfi Qu6c Ph6ng and E. Schiller, The operator equation AX - XB = C, admissibility, and asymptotic behaviour o] differential equations, J. Differential Equations 145 (1998), 394-419. G. Weiss, Weakly lP-stable linear operators are power stable, Int. J. Systems Sci. 20 (1989), 2323-2328.

A New Approach to Maximal Lp-Regularity LUTZWEISUniversittit Karlsruhe, Mathematisches Institut D- 76128 Karlsruhe, Germany

I, Englerstrat~e

2,

In this article we describe a new characterization of maximalLp-regularity for the Cauchy problem of semigroup generators in terms of square function estimates, or more generally in terms of R-boundedness (see Section 1). In Section 2 we use Fourier multiplier theorems for operator-valued multiplier functions as in [50], in Section 3 we give a second proof using a new variant of the Dore-’v~nni-Theoremas in [25]. Proofs are sketched for the case of spaces Lq(~l, 1 < q < oc. Furthermore, we give some applications of our method to further Cauchy-problems(Section 2), to perturbation theorems (Section 3), and to Gaussian bounds and contraction semigroups on Lq(12, #), 1 < q < ~c (see Section

1

A SHORT SURVEY OF MAXIMAL Lp-REGULARITY

The aotion of maximalL~-regularity plays an important role in the fl~nctional analytic approach to parabolic partial differential equations. Manyinitial and boundary value problems for elliptic partial differential equations can be reduced to an abstract Cauchy problem of the form y’(t) = Ay(t) + f(t), where ,4 generates a bounded analytic arid y are X-valued functions on [0, T).

y(0) = 0, 0 <_ t <

(1)

semigroup T, on a Banach space X and f

a) Definition of maximal Lp-regularity It is well known (see [35], Sect. 4.2), that (1) has a strong solution for all locally Bochner integrable f, but in many applications one needs that y~ has the same "smoothness" as f, which is not always the case. In particular, one says that A has maximal Lp-regularity, 1 < p < ~, on [0, T], 0 < T _< co, if for every f ~ L~([0, T), X) the solution y is almost everywhere differentiable, has values in D(A) and there is a constant C < oc with Iiy’IIr,~[o.:r},x} + IIAyllL,([o,:r),x} <_CllfllL,,([O,T),X) 195

(2)

196

Weis

This definition is slightly weaker than the usual one, which also requires that y E Lp([O,T),X). But for T = cx~ this additional condition implies already that s(A) su p{ReA : A E a( A)} < 0. On the othe r hand , for manyopera tors A, e.g. the Laplace operator on L,(II~"), we have 0 a(A), and sowe rat her use (2). The two definitions are equivalent if s(A) < 0 or T < oc (cf. [14]). It is also well knownthat maximalLp-regularity for one p ~ (1, oc) implies (2) for all p E (1, co), see e.g. [14]. b) Someapplications Estimate (2) is an important tool in treating evolution equations more complex than the basic Cauchy problem (1), e.g. ¯ non-autonomous equations y’(t) = A(t)y(t) + where A(t) may be an e lliptic differential operator with time dependentcoefficients (see e.g. [1], [30], [38], [19]). ¯ quasi-linear equations y’(t) + A(y(t)) = g(~,y(t)) ](t) with Li pschitz co ntinuous functions A(.), g(., .) (see [1], [30], [47], [10]). Here maximalregularity usually enters in setting up a fix point argument. * Volterra equations y(t) = f~ A(t - "r)y(r)d7 + with an op era tor-valued kernel A(t) (see [37]). * Second order equations -y"(t) + By(t) = f(t) with a sectorial operator B (see e.g. [91, [391). ¯ Uniqueness of mild solutions of the Navier-Stokes-equation (cf. [33]). c) Examples There is a rich literature on sufficient conditions for maximal regularity, which implies that most classical differential operators that ~nay be of interest in the problems just listed, do have maximalLp-regularity. i) Every generator A of a bounded analytic semigroup on a Hilbert space X has maximalLv-regularity for 1 < p < ~ ([16], [47]). ii) Let fl be a measurable subset of a space fl~ equipped with a quasi-metric and a a-finite measure # such that I~(B(x, 2r)) _< C/~(B(x, r)) for all x ~ fl~ and r > 0, where B(x,r) = {y fi f~ d(x,y) < r} andC is a fix edconsta nt. Put V(x, r) = #(B(x, r)). The above doubling property implies the strong homogeneity property V(x,~r) _< C~"V(x,r) for all x ~ 12~,~ _> 1,r > 0 and a fixed n > 0. Let Tt, t > 0, be a bounded analytic semigroup on L.z(12,p). Assumethat Tt has a kernel pt(x, y) which satisfies a "Poisson estimate" [pt(x,y)l

1 <_ min V(x,t~/r,),

1 ] [d(x,y)~ V(y,t~/m) ]p( t~/m

(3)

for all t > 0 and x,y ~ ~ #-a.-e., where p is a bounded decreasing function with lim,._~ r~’n+~p(’r) = 0 for some5 > 0 and m is a positive constaut. Then T~ extends to analytic semigroups on Lq(~l, #) for all 1 < q < oc, and the generators of these semigroups have maximalL~-regularity (cf. [201, [7]). Of particular interest is the case of Gaussian estimates, where # is the Lebesgue measure on f/~ C ~ and p(r) = a exp(-br 1/(m-l}) for suitable constants a, b > 0. It is knownthat manydifferential operators satisfy such estimates, e.g. (see [20] and the references there): Elliptic differential operators of order m on 1Rn or on a bounded subset of with s~nooth boundary with standard boundary conditions, elliptic operators in divergence form with general boundary conditions, the Laplace-Belbrami operators

MaximalL;Regularity

197

on a complete Riemannian manifold with non-negative Ricci-curvature, SchrSdinger operators A + V, where the potential V belongs to the Kate class. iii) If A generates a contraction semigroup Tt on all Lq(~, #), I ~_ q ~_ oo, which is analytic for all q E (1, ~), then A has maximal Lp-regularity on Lr(fl, ~) 1 < r,p < oo (cf. [26]). iv) Let A generate a bounded analytic semigroup on a Banach space X with the UMD-property. If the imaginary powers Ait,t E R, of A define bounded operators on X, so that [[Ait[[ _< e girl for some d < ~ and all t ~ I~, then A has maximalLv-regularity ([17]). A Banach space X has the UMD-property, whenever the Hilbert transform Hf(t) = ~PV- t_ ~ f( s)ds ex tends to a b ounded ope rator on Lp(I~, X) for some (all) p ~ (1, ec). It is well knownthat all subspaces and quotient spaces Lq(f~, #) with 1 < q < oc have this property. v) Every bounded analytic semigroup on X with generator A defines semigroups with maximal Lp-regularity on the real interpolation spaces (X,D(A))o,~ 0 < 0 < 1 (cf. [13]). In application these interpolation spaces are often Sobolev spaces. d) Counterexamples The Poisson semigroup on L~(l~) and on Lp(iC, X) not have maximal Lp-regularity if X is not an UMD-space(cf. [32], [14]). Hence the assumptions 1 < q < c¢ and UMD for X are necessary in c.ii), c.iii) and c.iv). But it was an open problem for 15 years, whether every generator of an analytic semigroup on Lq(f~, #), 1 < q < oc, has maximal Lp-regularity. Recently Kalton and Lancien ([22]) gave a strong negative answer to this question, by proving converse to c.i): If every bounded analytic semigroup on a Banach function space X has maximal Lv-regularity, then X is isomorphic to a Hilbert space. The results of c) and d) make it desirable to have a characterization of maximal Lp-regularity for an individual operator A. The characterization we present next will be used later to derive the results of c) in an unified wayand - in somecases to improve on them. e) A characterization of maximal L~-regularity If A generates a bounded analytic semigroup T~, I arg(z)[ _< ~, on a Banachspace X, then the following three sets are bounded in the operator norm

i) ii)

- A) : e i t, ¢ 0} {Tt,tATt:t

> 0}

iii) {T~.: l a.rg z[ ~ 5}. In Hilbert spaces this already implies maximalLp-regularity (by c.i)), but only Hilbert spaces (cf. d)). The additional assumption that we need in more general Banach spaces will be R-boundedness, a notion implicit in [5], and studied in more detail in [2], [12] and [48]. For Banach spaces X and Y denote by B(X,Y) the space of all bounded linear operators from X to Y.

198

Weis

A set r C B(X, Y) is called R-bounded, if there is a constant C < c~), such that for all Tt,...,Tn E r and xl,...,xn ~ X,n ~ N rj(u)Tj(xj

du ~ C

ry(u)xj

Y

"=

(4) "

"j=0

where (rj) is a sequence of independent symmetric {-1, 1}-v~lued random v~riables, e.g. the R~demacherfunctions rj(t) = sign(sin(~J~t)) on [0, The smallest C, so ~h~t (4) is fulfilled, is cMled the R-boundednessconstant of, and is denoted by R(7). The following theorem is due independently to N. K~lton and L. Weis. THEOREM ([48]) Let A generate a bounded analytic semigroup T, on a UMDspace X. Then A has maximal Lp-regularity i] and only if one o] the sets i), ii) iii) above is R-bounded. The equivalence of R-boundednessfor the sets i), ii) and iii) follows from Bourgain’s observation, that the closure ~ of absco(v) in the strong operator topology is R-bounded,if ~- is R-bounded([12], 3.2.). Then

f~ w(t)N(t)dt

if f~ Iw( t)[ dt <_ 1,

N(t

This can be applied to the Laplace transform and Cauchy’s formula )~R()~, A) _)~R()~,A)

)~e-~tTtdt,

Re ~ > 0

__1 fr ~ )~) iR(#’A)dp’ ~ =27ri (~)

)~ e i~

with F(~) = {p e C :ltt - ~1 = alAI} with a constant a (see [48] for details). Before we discuss the mait~ part of the proof in Section 2 and 3 and implications of the theorem (mainly in Section 4), we look at the special case X La(~,tt), where R-boundedness takes a more classical form. f) R-boundedness in L~(fl,#) For a Hilbert space X L2(~,#) weobt ain from Kahane’s inequality and the orthogonality of u -~ rd(u)xj in L2((O, 1), X) that

Hence every bounded set v C B(X) is R-bounded if X is a Hilbert space and we recover c.i) from our theorem. For q G (1, c~),q # 2, we obtain by Kahane’s inequality, Fubini’s theorem and Khintchine’s inequality that

~(u)x~d~,~ ,

r~(u)x~ d~) Lq

/

~

(6)

[~(.)~ Lq

199

MaximalLp-Regularity Now(4) takes the form

(7) and R-boundedness is recognized as a square function estimate familiar in Harmonic Analysis (see e.g. [40, 41]). Using methods from Harmonic Analysis it sometimes easier to work with condition (7) rather than with (4) and the corresponding methods from Banach space theory. Therefore we indicate the proofs in the next two sections mostly for Lq-spaces. Also note that if the theorem in e) holds for X = Lq(~, ~), it also holds for all spaces X isomorphic to complemented subspaces of Lq, 1 < q < co, i. e. for most reflexive spaces of interest in applications such as reflexive Sobolev or Hardy spaces. (If X @Y ~ Lq, consider the operator A0 = A @(-idy) on Lq.) (6) holds more generally for Banach lattices X which are q-concave for some q < oc (cf. [31], Theorem1.d.e). Therefore our proofs also work for these spaces. In this setting, the author presented Theorems1.e and 3.b at the University Delft in December98, and in July 99 at the "Accademia Nationale dei Lincei" in Rome.

2

MAXIMAL Lp-REGULARITY

VIA

MULTIPLIER

a) A reformulation of maximal Lp-regularity bounded analytic semigroup, the Cauchy problem y’(t)

= Ay(t)

+ ](t),

THEOREMS

For the generator

A of

y(O) = O,

(1)

for a locally integrable f : R~_-~ X has the mild solution (cf. [35], Sect. 4.2) y(t) Formally differentiating

Tt-8(f(s))ds,

(2)

(2) leads

y’(t) = ATt-s(I(s))ds

(3)

and we see that A has ~naximal L~-regularity if and only if the operator

KI(t) = AT~_s(I(s))ds, I e Co(ll~_,

(4)

extends to a bounded operator on L~(II~,X), 1 < p < oe. Since IIArtll ~ ~ for t --+ 0+, we can think of K as a convolution operator with a singular operatorvaluedkeruel k(t)=ATtfort>0 and k(t)=0fort<_0. Applying the Fourier transform ~’(t) = e-~I(s)ds to(4)gives

K~"](s)=M(s)[]’(s)l, s e {0},

(5)

200

Weis

where the "Fourier multiplier function" Mis given by M(s) = (ATt)A(s) = AR(is, for s ~ 0. Since Tt is analytic we have that the functions M(s) = AR(is,

A), sM’(s) = -isAR(is,

’~, s E ~ \ {0}

(6)

are bounded. Hence we obtain the boundedness of K on Lp(E,X), 1 < and therefore the maximal Lp-regularity of A whenever we can apply Mihlins multiplier theorem to operator-valued multiplier functions. In particular, Theorem1.e is nowa consequence of the following version of Mihlin’s multiplier theorem, where boundedness of the functions in (6) is replaced by R-boundedness. b) Mihlin’s multiplier theorem for operator-valued multiplier functions. Let M: ll~ \ {0} -~ B(X, Y) be differentiable and define for f E ~(~, X) the function Ka4f ~ ,~’(~, X) by KMf(s) = /~(s)[]’(s)], THEOREM ([48])

s e I~ \

(7)

If X and Y have the UMD-property (see 1.c.iv))

{M(s) : s e I~ \ {0}}, {sM’(s):s

e ll~ {0}} are R- bounded,

(8)

then KMextends to a bounded operator from Lp(IR, X) to Lp(IR, Y). REMARK i) If X is a Hilbert space, then the norm boundedness of the sets (8) is sufficient, but G. Pisier has shownthat this is only true for Hilbert spaces. ii) Since the Hilbert transform is itself a Fourier multiplier with M(s) i sign(s)Ix, the UMD-assumption is necessary. One can also show that the R-boundedness of {M(s) : s ~ 0} is a necessary condition ([48], [11]). For tiplier functions that extend analytically into a cone C = {A : I arg(q-A)l < ~}, the R-boundedness of {M(~) : ~ E C} is equivalent to the Lp-boundedness of the multiplier operators (g~f)^(s) = M(eids)[~f(s)] for all Idl < ~ (cf. [48]). In particular, this applies to resolvent functions. iii) A multi-dimensional version of the theorem and analogues of the Marcinkiewicz multiplier theorem are contained in [45]. c) Sketch of proof forX =Y = L~(f~,#), 1
Xj]’,

or Sjf

= Xj * f.

Then we have f = ~j Sjf in the distributional sense and by the Paley-Littlewood decomposition for L~(II~) (see [40]) and Fubini’s theorem there is a constant Ca such that (9)

201

MaximalLp-Regularity For a multiplier Mof the special form M(t)

= E)o(t)Mj,

{Mj}

C B(Lq)

(10)

R-bounded

we have [(SjgM)(f)]^(s) = Xj(s)Mj(]’(s)) = Mj(Sj-~(s)), so that by (10) and the R-boundedness of {Mj}:

or SjKM(f)

and KMis bounded on Lq (~, Lq). This esti~nate can be extended to multipliers of the special form M(t) = EX(aj,b~)(t)Mj,

(aj,bj)

C Jj,

R-bounded.

(1

1)

jEZ

Indeed, M(t) = ra(t)(~~jXj(t)Mj), where the scalar multiplier operator K,~, re(t) = Ej )~(aj,b~)(t), is bounded on L~(II~) by the Marcinkiewicz theorem on E by Fubini’s theorem. It remains to show that multipliers M with (8) can be approximated by convex combinations of multipliers of the special form (11). To simplify our notation may assume that M(t) = 0 for t < For j ~ Z,k ~ N we put Xk,j,l(t)

~--- X(2/+/2.~-k,2J+l](t),

k, I = 1,... ,2

and define multiplier functions Mo(t) = E xJ(t)M(2J)’ jez M~,~(t)=EX~,~d(t)2JM’(2J+(l-1)2J-k),

l=1,...,2

k, k=l,2,...

By assumption (8) these functions are of the form (11) and we obtain

Iig~.,Ylle<_CllYll~,l = 1,...,2 ~, k = 1,2.,... with a constant C independent of 1 and k, since it only depends on the R-boundedness constant of the sets in (8). Then we obtain for the linear combinations 2k /:l

that

[]KM~f[IE

<_ 2C[]fllE.

NOWthe

claim follows from the fact that

I[M~(t)[] _< 2suplltM’(t)l [,

lim M~(t) = M(t).

202

Weis

Indeed, e.g. for t E [2J,2 j+l) we have tbr k -~ ~ t

f2

Mk(t) -~ M(2j) + M’(s)ds

= M(t).

In the case of general Banach spaces with the UMD-property one uses Bourgain’s extension of the Littlewood-Paley theory to vector-valued functions in Lp(X) (see [5], [48]). [] d) Equations of second order The multiplier method described in a) is flexible enough to be applied to further evolution equations. As a first examplewe consider second order equations on the line y"(t) = Ay(t) + f(t),

(12)

with a sectorial operator A on X. For f ~ 2~(11~, X) we take the Fourier transform and obtain formally

=

+

or

=2, A).[(s).

Hence the mappingf -~ y" is a Fourier multiplier operator with M(s) = R(-s ~,A),

~,A) 2. = 2s2R(-s sM’(s)

NowTheorem 1.e implies COROLLARY (see [39]) Let A be a sectorial operator on an UMD-spaceX. Then (12) satisfies the maximalregularity estimate (13) if and only if {sR(s, A) : s < ()} is R-boundedin B(X).

e)

Volterra

equations

Now we consider

the equation

y(t) = A(t - r)y(r)dr + I(t), where A(t) : tI~_ ~ B(Y,X) is locally integrable Hence the Laplace transform of A(t) exists: ~(A) =

e-~tA(t)dt,

(14)

and of subexponential growth.

Re A > 0

Weassume further that Y is continuously embedded into X, and that (I - ~(A)) -~ exists

(15)

in B(X) for Re A > 0.

Taking the Laplace transform of (14) we see that y must satisfy ~’(A) = .~(A)[~(~)] + 97(A), or ~(A) = (I - ~(A))-’f(A),

Re A >0.

MaximalL~-Regularity

203

Hence the solution operator y = T(f) of (14) is given by T = lim~_~0 KM~,where /(Me is given by the Fourier multiplier M~(s) = (I - ,~(~ + -1, ~ > 0. Note that ~(I - ~()~))-I = (I -,’~()~))-1,’~()~)’ (I Hence the Inul tiplie r th eorem in b) gives the following extension of Theorem8.9 in [37]: COROLLARY Let X and Y be UMD-spaces. Assume (15)

and that

{(~ - ~(~))-~ : Re~ > 0}, {~X(~)(I- ~(~))-1 : Re~ are R-bounded in B(X). Then Lp(~+, X) for all If (15) and (16) also hold in B(Y), then in. addition T e B(Lp(~+, EXAMPLE Let A be a generator of an analytic semigroup on an UMD-space X with R({AR(A,A) : Re A > 0}) < c~. For a e Ll,~o(.(]l~-) of subexponential consider the equation y(t) = a(t - ~’)Ay(r)d~" f( t), t

>_

(17)

Then (17) has a solution y ~ L~(~_, D(A)) for every f ~ L~(IR~_, D(A)), if the Laplace trans~brm ~ satisfies i) ~(A) ~ 0 and larg~(A)l > w~ for some w~ > w aud all Re A

ii) I~a’(~)l_ Indeed, we may apply the corollary with Y = D(A) and A(t) = a(t)A, (I- ,~’(A))-~ = ~(A)-~(~(A)-~ -~ and(15) and ( 16) are s atis fied. f) Non-autonomous equations y’(t)

Consider

= A(t)y(t)

since

the equation

+ f(t),

0

(18)

with a fmnily A(t) of uniformly analytic generators on a Banach space X satisfying the Aquistapace-Terrini condition (see [19], [44]). The solution operator of (18) not a convolution operator any more and the multiplier theorem does not apply directly. But it is shownin [19] that the maximalregularity of (18) on [0, T] can reduced to the boundedness on L~([0, T], X) of a pseudo-differential operator (K f)(t) = / eit~ a(t, s)f(s)ds, with an operator valued symbol a(t,s) = is(is - A(t)) -~ for t e [0,T]. If R({~R()~,A(t)) : ~ ~ iI~}) <_ C < ~ for all t ~ [0,T], then one can apply boundedness theorems for such pseudo-differential operators, where the classical boundedness assumptions on the symbol a are replaced by R-boundedness, e.g. (see [44]) for some 0 _< a
R({(1Isl)~D~a(t,s): s e l~}) < C~ IIDg~(u, ~)- DJ~a(v,s)ll ~ Cylu- vl~(1+-p÷~ [sl)

for allt~l~, for u,v,s ~ ~.

jEN (19)

205

MaximalL~-Regularity If OA + 0~ < n, then the closure of A + B has domain D(A) N D(B) [[Axll ÷ 11/3zll < Cl[Ax + Bx[[ for x e D(A + B).

(4)

REMARK i) The Dore-Venni theorem in [17] assumes that X has the UMDproperty and A and B have bounded imaginary powers (BIP), i.e.

IIA~Sll~
with a + b < ~r.

The last condition obviously corresponds to OA+ 0~ < ~r. Our assumption on B is stronger than bounded imaginary powers, but our assumption on A is weaker than BIP (for a direct proof that BIP implies R-sectorial in UMD-spaces,see [11]). Since in manyapplications B is an operator such as ~ or -A, for which the existence of an H~o-calculus is known, our theorem often allows to obtain more general results. ii) Recently Clement and Priiss ([11]) showed that one can derive some important special cases of this theorem from the multiplier Theorem2.b, by an extension of the Coifman-Weiss transference method. iii) It is useful to note that if A is R-sectorial of type w, then ~, ~~ (0, 1) , is R-sectorial of type c~w(this follows directly from the representation (3) of R(A, ~) in [23]). c) Sketch of proof for X = Lq(f~,#) and 0 ~ o(A) V~ ~(B). It is shown in that A + B has a closure A + B (with possibly a larger domain than D(A)~ D(B)), 0 ~ ~0(A + B), and that the resolvent has the representation R(0, ,4 + B) = ~ R(-z,

A)n(z,

where "~ is the boundary of E(~) for a ~ with Ou < ~ < ~r - OA. To prove that D(A + B) C D(A) and that (4) holds we will show that AR(O, A + B) is bounded on X. Then it follows that BR(O,A ÷ B) is bounded too. By an argument with contour integrals we have for x ~ D(A)

A1/4

(ei~

M(z) = AR(-z, andI is a simil ar integ ral over

208

Weis

With the same techniques one can show (see [43]) that Theorem 3.b remains true for 0 ¯ g(A) if one assumes (11) instead of the commutativity of A and A similar approach exists for products of sectorial operators ([51], [42]’). f) H~c-calculus for operator-valued functions Let B be a sectorial operator on a Banach space X with a H~(E(0o))-functional calculus. Wecan extend Theorem 3.b and its proof to construct a functional calculus for certain operator-valued functions. Fix 0 > 0o. Let R?-l~o(E(O)) be the space of all bounded analytic function F : E(O) ~ B(X) such that i) for all A ¯ ~(0), # ¯ ~0(B): F(A)R(p, B) = R(/z, ii)

{F(A) :A ¯ ~(0)} is R-hounded.

Then R(F) = R({F(A) : A E Z(0)}) defines a norm R’H~(Z(O)). Denote by R7~,o(~(0)), all elements in RTi~(E(0)), for which there is an s > 0 so that g-~f belongs to RT~(~(0)), where 9(A) = A(1 -2 . Fix a ~ with 0o < ~ < 0. Then, for F ~ RTi~,o(r,(O)) we define

F(B):=

1 f~ F(z)R(z,B)dz

(13)

Nowwe can estimate (12) in the same way we estimated (5) with F(A) AR(-A, A) and get [IF(B)[]B(x) < CR(F). Approximating the functions in R74.~ by functions in R~,o gives THEOREM ([25]) Let B have an Hcc(~(t?o))-calculus on a Bauach space X. for all ~ > ~o, (12) defines a continuous algebra homornorphism F ¯ RT{~(E(0))

F(B) ¯

B(

REMARK a) Theorem 3.b can be obtained from this functional calculus with F(A) = AR(-A, b) Special cases of this theorem appear in [28] and [11] (e.g. for generators bounded groups on UMD-spacesand of positive contraction semigroups on Lp). c) As noted before, the above proof works for complemented subspaces X of r-concave Banachlattices E with r < ~x~ (e. g. E = Lp(Lq) with p, q < cx3).

4

MAXIMAL REGULARITY

IN

Lq-SPACES

In this section X is always a space Lq(~],#) with 1 < q < ~x). As we ~nentioned in Section 1.f), R-boundedness of v C B(X) is then equivalent to ll( _CIl ~. (~/ ,T~,~)’~llx< ,X~,~)~l’X

(1)

where the finite constant C is independent of Ti ¯ r, z~ ¯ X. First we consider a simple continuous version of (1), then we introduce Rv-boundedness in order deal with maximal estimates and Gaussian bounds.

209

MaximalL~-Regularity a) R-boundedness for functions of operators a strongly continuous function on an interval I.

Let t E I -~ N(t) ~ B(X) be

LEMMA The set {N(t) : t ~ I} is R-bounded if and only if N(t) is a (pointwise) multiplier in Lq(L2), i. e. for f ~ Lq(~,L~(I)) we l/~llx CI[(~ [ f(t )l~dt)1/21Ix II (~ IN(t)f(t)’edt) (2) Proof. "===~" Let f : I ~ L~(fl) ~ X be continuous in X with compact support. (Such f are dense in Lq(L2).) For a partition {B~}of I into finite intervals of equal length and tj ~ Bj, consider M(t) = ~ N(tj)x,~(t), J

g(t)

= ~ f(t~)x,~(t). J

Then with C = R({N(t) : t I})

~ I C,(B~)’/e~[(~lf(t~)le)x/zl’=

C~ (f Ig(t)ledt)~/a

If we refine the partition {Bj } then (3) approximates (2). "~" By [12], Lemma3.3, it is enough to consider ti ~ I with t~ < t~ < ... < t~. Put B~ = {t e I : It - t~l < ~/2} with 5 > 0 so small that the balls Ba,..., B~ do not overlap, and consider the function gs(t) = ~=~ ~-~/~Xn~ (t)x~ By (2) we have n

and the claim follows for J Nowwe can restate

Theorem 1.e ~ follows:

COROLLARY If A generates a bounded analytic semigroup on X = L~(fl,#), 1 < q < ~, then each of the following conditions is equivalent to m~imal Lpregularity. i) There is a C < ~ such that for all f ~ L~(L2) ll(f:~ [tR(it, A)f(t)12dt)W2~{x ~ C~[(fS~ [f(t)12dt)I/2[[x ii)

There is C < ~ and a ~ > 0 such that for all ]~] ~ 0 and f ~ L~(L2) l~(f~ lTto,,(f(t))12dt)i/2]l

~ Cll(f~ lf(t)12dt)i/2ll

Similarly we could restate Corollary 4.4. of [48].

210

Weis

b) Rp-boundedness Ill order to apply interpolation we have to extend this notion.

theory to R-boundedness

DEFINITIONWecall a family of operators r C B(X) Rp-bounded for 1 < p < oo, if there is a constant C such that for all T~,..., Tn E r and xl,..., xn E X

By Rp(r) we denote the smallest constant for which this inequality holds. REMARK i) For p = q all bounded sets r in B(Lq) are Rp-bounded(by Fubini’s theorem). ii) For 1 < p, q < oo a set. T C B(Lq) is R,-boundedif and only if r’ = {T’ : T ~ 7-} ~ 1 1, ~ ~ = 1 (since Lq(lp)’ = Lq,(lp,)). C B(Lq,) is R~,-bounded with ~ + p-7 = ~ + ~ iii) If v C B(Lq) is R~,-bounded,then so is the closure of absco(7-) in t, he strong operator topology (same proof as [12], Lemma3.2.) Put 7,(00,01) = {A ~ C \ {0}, < argA <_ 0~}. As a v ar iant of Stein’s int erpolation theorern we obtain PROPOSITION Let N($), ~ E(Oo,O1), bea f amily of lin ear maps N($ ) : L 1 L~ ~ L1 + L~, such that for all x ~ LI f~L~ the function ~ -~ N(~)x ~ L~ + is continuous and bounded on E(0o,01) and analytic inside E(Oo, 01). Assume that for po,p~,qo,ql ~ [1,c~] and j = 0,1, the maps s ~ R+ -~ N(sei°¢)x ~ Lq¢ are continuous for x ~ L1 CI L~ and {N(te i°~) : t > O} is Rp~-bounded in B(Lq~).

(4)

Thenfor every ~:~ ~ (0, 1) and 1 1 - = (l-e)-+e, q qo

1 1 1 - = (1 - c~)-- +a--, P Po P~

0 = (1 - c~)00 + c~0t,

the set {N(teie) : t > 0} is a Rp-boundedsubset of

]~(Lq).

Proof. Put U = {,~ E C : 00 _< Re ~ < 01} and let C be larger than Rpo({N(ei°°t) t > 0}) in B(Lqo) and Rm({N(eiO’t) : t > 0}) in B(Lq~). For fixed points h,...,t, ~ ~ and A ~ U consider the maps M(,~) (L~ rqL~)(l£) --+ (L~ + Loo) ) M()~)(x~,...,

x,) = (N(t~eiX)x~

, N(t,~e~X)x,,).

By our assumption (4) we have that M(O i +is): Lq~(l~,~) ~ Lq~(l’p’~)

211

MaximalL~-Regularity

are bounded forj = 0, 1 and S E l~ with HM(Oj+is)[[<_ C. By the abstract Stein interpolation theoremin [46] it follows that M(O)is boundedin [Lq0 (lpno), Lql (I;~)]a Lq(l~) with IIM(0)ll <_ C, i.e.

I V(t.ei°)x.lP) k=l

<_c ( IIL~ II

Lq

,--

" []

c) Domination and maximal functions It is clear from (1) and (2) R-boundedness and inaximal Lp-regularity are inherited by dominated operators, i.e. if

IT, xl s,(Ixl) forall

(5)

x e X, i e N,

then (T~} is R-boundedif the set (Si} is. This observation is particularly useful connection with maximal estimates. LEMMA Let A ~ E(O) -~ N(X) ~ B(L~) be a bounded analytic q e [1, 2]. /f N(t) satisfies a maximalestimate 1 sup t>0

ft J0


IN(s)lxds

function

for

e Lq,

(6)

L~

then {N(A) : A e E(~)} is R-bounded for some 0 < ~ < 0. Proof. M(A) = A-~ f: N(#)d# is also a bounded analytic filnction assumption we have for all xi ~ Lq, ti ~ 1~ and I x = supi [xi sup IM(t~)xil i

II~~ II

on E(0).

sup IM(t)[(x)[1~ cIl~ll~,, t>O

i.e. {M(t) : t > O} is R~-bounded. Also {M(ei~°t) : t > O} is Rg-bounded for all 1 I~[ < 0. Choose 00 < 0 and a so that ~ it follows from 4.b that {M(e~’t) : t > 0} is R-bounded. The same is true for {M(e-~vt) : t > 0} and, again by 4.b, we obtain that {M(A) : A ~ E(~)} R-bounded. By the proof of [48], 4.4. iii) it follows that {N(A),A E E(~)} bounded. [] d) Contraction semigroups Now we can improve a result of Lamberton [26], whoconsidered semigroupscontractive in Lq for all 1 _< q _< cx). COROLLARY If A generates an analytic positive and contractive Lq for some 1 < q < c~, then A has maximal Lp-regularity.

semigroup on

Proof. By [18], Sect. 5.4, N(t) = Tt satisfies a maximal esti~nate as in (6). q ~ (1, 2], it follows from 4.c that (Tz : z ~ E(~)) is R-bounded for some ~o and we may apply Theorem 1.e. If q ~ (2, cw) we apply a duality argument with Remark 4.b.ii). []

212

Weis

e) Gaussian estimates As announced in [48], we also obtain a short proof of the following result from [20]. COROLLARY Assume that ,4 generates an analytic semigroup with a Gaussian estimate as in 1.c.ii). Then A has maximal Lp-regularity. Proof. It is well known(see e.g. [42]), that the Gaussian semigroup satisfies maximalestimate as in (6), so that we can apply 4.c., 1.e and duality. Alternatively, one can extend the Gaussian estimate to a sector E(~o) (by an argument of Davies, see e.g. [50]) and apply observation (5). Recently it was observed in [11] that maximal estimates can be used to show directly that Poisson estimates (as in 1.c.ii) imply R-boundedness.

REFERENCES 1. 2. 3. 4. 5.

6.

7. 8.

9. 10.

11. 12. 13.

Amann,H., Linear and Quasilinear Parabolic Problems, Birkh/iuser, Basel, 1995. Berkson, E. and Gillespie, T.A., Spectral decompositions and harmonic analysis on UMDspaces, Studia Math. 112, (1994), no. 1, 13-49. Bergh, J. and LSfstrffl, J., Interpolation Spaces, Springer, 1976. Bourgain, J., Some remarks on Banach spaces in which martingale differences are unconditional, Arkiv Math. 21 (1983), 163-168. Bourgain, J., Vector valued singular integrals and the H-BMO duality, Probability Theory and Harmonic Analysis (New York), Dekker, NewYork, (1985), 1-19. Burkholder, D., A g’eo~netric condition that implies the existence of certai~ singular integrals of Banach-space-valued functions, Proc. of Conf. on Harmonic Analysis in Honour of A. Zygmund(Chicago, 1981), Wadsworth Publishers, (1983), p. 270-286. Coulhon, T., Duong, X. T., Maximal regularity and kernel bounds: observations on a theorem of Hieber and PrSss, 1997, preprint Cowling, M., Doust, I., McIntosh, A. and Yagi, A., Banach space operators with a bounded H~ functional calculus, J. Austral. Math. Soc. Ser. A 60 (1996), 51--89. Cl6ment, Ph. and Guerre-Delabribre, S., On the regularity of abstract Cauchy problems of order one and boundary value problems of order two, preprint Cl6ment, Ph. and Li, S., Abstract parabolic quasilinear equations and application to a groundwater flow problem, Advances in Mathematical Sciences and Applications Gakkotosho, Tokyo, Vol. 3, (1993/94), p. 17-32. C16ment, Ph. and Prfiss, J., An Operator-Valued Transference Principle and Maximal Regularity on Vector-Valued L,-Spaces, this volmne Clement, Ph., Pagter, B., de Sukochev, F. A., Witvliet, H., Sd~auder Decomposition and Multiplier Theorems, to appear Da Prato, G. and Grisvard, P., Sommesd’opdrateurs lindaires et ~quations diffdrentielles op&ationelles, 3. Math. Pures Appl. 54 (1975), 305. 387.

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213

14. Dore, G., LP-regularitj , for abstract differential equations, (in: Functional Analysis and related topics, editor: H. Komatsu), Lecture Notes in Math. 1540, Springer Verlag, 1993. 15. Dote, G., Maximal Regularity in Lp-Spaces for an Abstract Cauchy Problem, to appear in: Advances in Differential Equations 16. de Simon, L., Un applicazione della teoria degli integrali singolari allo studio della equazioni differenziali lineari astratto del primo ordine, Rend. Sere. Mat. Univ. Padova 34 (1964), 547-558. 17. Dote, G., Venni, A., On the closedness of the sum of two dosed operators, Math. Z. 196, (1987), 189-201. 18. Fendler, G., On dilations and transference for continuous one-parameter semigroups of positive contractions on Lp-spaces, Annales Universitatis Saraviensis, Series Mathematicae Vol. 9. 1., 1998. 19. Hieber, M. and Monniaux, S., Noyans de la chaleur et estimations mixtes Lp - Lq optimales: Les cas non autonome, C. R. Acad. Sci., Paris, Ser. I, Math. 328, 233--238 (1999). 20. Hieber, M., Priiss, J., Heat Kernels and maximal LP-Lq-Estimates for Parabolic Evolution Equations, Commun.Partial Differ. Equations 22, 1647-1669 (1997). 21. Hieber, M., Priiss, 3, Functiona/calculi for linear operators in vector-valued Lp-spaces via the transference principle, Diff. Equations 3,847-872 (1998) 22. Kalton, N., Lancien, G., A solution to the problem of the LP-maximMregularity, 1999, preprint. 23. Kato, T., Note on Fractional Powers of Linear Operators, Proc. Japan Acad. 36, 94-96 (1960) 24. Kunstmann,P., MaximalLp-regularity for second order elliptic operators with uniformly continuous coefficients on domains, preprint 25. Kalton, N. and Weis, L., in preparation 26. Lamberton, D., Equations d’~volution lindaires assocides ~ des semigroupes de contractions dans les espaces Lp, J. Funct. Anal. 72, 1987, 252--262. 27.. Lancien, G., Counterexamples concerning sectorial operators, Archiv Math. 71, (1998), 388-398 28. Lancien, G. and Le Merdy, C., ,4 Generalized H°~-Functional Calculus For Operators on Subspaces of L~ and Application to MaximalRegularity, Illinois Journal of Mathematics 42,470-480 (1998) 29. Lancien, F., Lancien, G. and Le Merdy, Ch., A joint functional calculus for sectorial operators with commuting resolvents, Proc. Lon. Soc. 3 77 (1998), 387-414. 30. Layzenskaja, O. A., Solonnikov, V. A., Uralceva, N. N., Linear and Quasilinear Equations of Parabolic Type, AMS,Providence, 1968 31. Lindenstrauss, J. and Tzafriri, L., Classical Banachspaces II, Springer 1979. 32. Le Merdy, C., Counterexamples on L~-maximaJ-regularity Math. Z. 250, (1999), 47-62 33. Monniaux, S., Uniqueness of mild solutions of the Navier-Stokes equation and maximal LV-regularity, C. R. Acad. Sci., Paris, Ser. I, Math. 328, 663-668 (1999). 34. Monniaux, S. and Priiss, J., A theorem of Dore-Venni-type for noncommuting operators, Trans AMS349 (1997), 4787-4813

214 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49..

50. 51. 52. 53.

Weis Pazy, A., Semigroups of Linear Operators a~d Applications to Partial Differential Equations, Springer, 1983. Pisier, G., Les in~galit~s de Khintchine-Kahaned’apr~s C. Borel, Seminaire s~r la g~ometrie des espaces de Banach 7 (1977-78), lgcole Polytechnique, Paris Priiss, 3., Evolutionary Equations and Applications, Birkh~user Verlag, 1993. Priiss, 3. and $chnaubei~, R., Solvability and maximal regularity of parabolic evolution equations with coefficients continuous in time, preprin~ Schweiker, S., MaximalLp-Regularity of Second Order Differential Equat, ions on the Line, preprint S~ein, E., Singular Integrals and Differentiability, Properties of Functions, Princeton University Press, Princeton 1979 Stein, lg., Topics in HarmonicAnalysis related to the Littlewood-Paley Theory, Annals of Mathematical Studies 63, Princeton 1970 ~trkal~, ~., A Theorem on Products of Non-CommutingSectorial Operators, this volume ~trkal3, ~., Sums of Non-Commu~ing Sectorial Operators, in preparation ~rkal3, ~,., Pseudodifferential Operators and MaximalLp-Regularity, in preparation ~trkalj, ~,. and Weis, L., On Operator-valued Fom’ier Multiplier Theorems, submitted. Voigt, J., Abstract Stein Interpolation, Math. Nachrichten 157 (1992), 197- 199 yon Wahl, W., The equation u’(t) = Au(t) + f in a Hilbert space and L~estimates for parabolic equations, J. LondonMath. Soc. 25 (1982), 483-.497 Weis, L., Operator-valued Fourier Multiplier theorems and Maximal L~regularity, submitted. Weis, L., Gaussian estimates and analytic semigroups, in: Partial Differential Operators and Mathematical Physics, M. Demuth, B.-W. Schulze, ed., Birkhiiser 1995, p. 397-404 Weis, L., Perturbation theorems for maximal L~-regularity, in preparation Weber, F., On products of Non-Commutingsectorial operators, to appear in Ann. Scoula Norm. Sup. Pisa C1. Sci 4. Widder, D. V., The Laplace Transform, Princeton University Press, Princeton, 1941. Zimmermann,F., On vector-valued Fourier multiplier theorem, Studia Mathematica, XCLIII, 1989, 201-~222.

The Instantaneous Limit of a Reaction-Diffusion System DIETER BOTHE Germany

1

Fachbereich 17, Universit~it

Paderborn, 33095 Paderborn,

INTRODUCTION

Consider a single irreversible reaction A+B -~ P between mobile species that takes place inside a bounded region ~t C R3. In practice, chemical reactions are often performed inside catalytic pellets of high porosity, hence nonlinear diffusion of the reactands has to be taken into account. If the pellet is isolated this leads to the following model problem where u~ and u2 denote the concentrations of A and B, respectively. in (0, c~) x

in (0, oc)x

(1)

on (0, oc) x OFt in Ft with rate constant k > 0 and a continuous rate function r : 2R+ -~ R such that r(., .) is increasing in both variables with r(a, b) = if f ab= 0; thelatt er is a reali stic assumptions for any rate function. Concerning 99~ and ~2 we assume that both are continuous and strictly increasing with ~Ok(0) = 0; notice that the porous mediumequation (see e.g. [11]) is included as a special case. In this paper we are interested in the case when the rate constant is so large that the influence of reaction is very fast comparedto diffusion. Let us note that this case is encountered often in practice, in particular for ionic or radical reactions. In this situation we consider the limiting process as k tends to infinity, i.e. the passage 215

216

Bothe

to an instantaneous reaction. show that u~(t) -2>\w-(t) J in

Given a uonnegative initial

value u0 E LC~(~)2, we

L~(~)~ uniformly on com pact subsets of (0, c~),

where w+, w- denotes the positive, respectively negative part of the (mild) solution w of wt = A~(w) in (0, c~) O~(w) _ (2) on (0, ~) × 0f~, Ou w(0, .) = in ~ with Wo= u0,~ - u0,~ and ~ : R ~ R given by f ¢2~(r) if r_>0 -~2~(-r) if r <

(3)

To obtain this convergence result we consider (1) as an abstract evolution equation in L~ (fl)~ and use nonlinear semigroup theory to study the singular limit k -~

2

PRELIMINARIES

Weshall use the following notations and facts converning rn-accretive operators. For definitions not explained in the sequel we refer to [1], [3] or [10]. If A is m-accretive in a real Banach space X, then -A generates a semigroup T(t) of nonexpansive mappings T(t) : D(A) -~ D(A), given by the exponential formula, i.e. T(t)x=nl~m~JS,x fort_>0, xeD(A); here J~ = (I + ~A)-~ denotes the resolvent of A. In this situation with u(t) T(t)uo is theunique mildsolut ion of u’+Au~0

oaR+,

u ~ C(R+; X)

u(0)=uo

(4)

for every uo ~ D(A). The facts mentioned so far are still true if A is accretive and satisfies the range condition R(I + AA) D D(A) for all small A > 0. Given f : D(f) C X ~ and uo ~ D(A), we saythat u is a mild solut ion of u’+Augf(u)

on J=[0,

T],

u(0)=’u0

(5)

if u ~ C(J;X), :=](u (’)) ~ L ~(and u is the mild solu tion of the qu asi autonomous problem

u’ + Au~ w(t) onJ, u(0) = u0.

(6)

Recall that (6) with m-accretive A has n unique mild solution u for ever), L~(J; X); this solution will be denoted by u(. ;u0, w). The next result follows fl’om Theore~n 6.8 in [10]. Rememberthat lim inf A~ for a

InstantaneousLimitof a Reaction-DiffusionSystem

217

family of operators (Ak)k>0 is defined by (x, Ak for k > 0 such that x~ --~ x and yk LEMMA 1 Let (Ak)k>0 be a fmnily of accretive operators in a real Banach space X that satisfy the range condition and let Tk(t) denote the semigroup generated by -Ak. Let A C liminfA~, satisfy the range condition and T(t) be the semigroup generated

by -,4.

Then u~ E D(A~.) with lim u~ = Uo ~ D(A) implies

lim Tk(t)u~ = T(t)uo on R+, where the convergence is uniform on bounded interk--~ c~

vals. To be able to apply this perturbation result to the problem under consideration, we also need some information concerning the abstract formulation of nonlinear diffusion equations of the type (2). Define the operator A in Ll(f~) be means Av = -Acp(v)

on D(A)

O (v)

{v e n1 (fl): ~(v) W1’1 (~’~), At/~(V) e ~ (fl ), 0--

= on(7)

where A~o(v) is understood in the sense of distributions. Then (2) corresponds to autonomous problem (4). Before we state some basic properties of this particular operator, we need to recall the following facts: X = L~ (fl) becomesa Banachlattice if equipped with the usual partial ordering v _< ~ iff v(x) <_ ~(x) a.e. in ~. Then F : X -~ X is order-preserving if v < g implies F(v) <_ F(g). An operator A in X is T-accretive if

max( f (w- )H(v- V) dx) for

all v,g ~ D(A),w ~ Av ,i ’O ~ Ag

where H(r) denotes the Heaviside function with H(0) --- [0, 1] and H(v) for v ~ X is short for {w ~ X : w(x) ~ H(v(x)) a.e. in ~}. If this holds then the resolvents of A are order-preserving. LEMMA 2 Let 9t C Rn be open bounded with CU-boundary and ~ : R -~ R be continuous and strictly increasing with ~o(0) = 0. Let X = L1(12) and define operator A in X by means of (7). Then the following holds. (a) .4 is m-accretive and T-accretive with D(A) (b) I(J~v)±lp <_ Iv±l~ for all A > 0, p ~ [1,~] and v ~ Lv(~2). (c) Given v0 E X, the set {u(. ;v0, w) : w ~ W}of mild solutions of (6) is relatively compact in C(J; X) whenever W C L~(J; X) is weakly relatively compact. Parts (a) and (b) follow from Th~or~me 2.1 and Corollaire 2.2. in [2]. Assertion is essentially Theorem 1 in [6], where Av = -A~o(v) with homogeneousDirichlet boundary condition is considered; the arguments given there remain valid in the situation considered here. The following results concerning weakinvariance is a special case of Corollary 1 in

LEMMA 3 Let ,4 be m-accretive in a real Banach space X and K C X be closed with KA := KV~D(A) ~ su ch th at J~ K C K fo r al l ~ > 0. Suppose tha t,

218

Bothe

given Uo E D(A), the set {u(. ;uo, w) : w E W}of mild solutions of (6) is relatively compact in C(J; X) whenever W C Ll(J; X) is weakly relatively compact. Let f : KA--~ X be continuous and bounded satisfying

(8)

lim h-lp(u + by(u), K) = 0 for all u E KA. h-~O+

In addition, assume that f ~naps bounded sets into weakly relatively compact sets. Then initial value problem (5) has a mild solution for every Uo ~ KA.

3

THE INSTANTANEOUS

LIMIT

Weconsider problem (1) as the abstract evolution equation u’+Au=Fa~(u)

onR+,

u(O)=u0

(9)

in X = L~ (~)2, where A and Fk are defined by Au = (,4~u~,A2u2) with Ai given by (7) with 99i instead ofw (i = 1,2),

(-k (.),

Fk(U) = -kr(u~(.),u2(.)),}

’ = {u e X:r(u l( .), u2(.)) e L~(a)}

The subsequent result establishes the convergence of the mild solutions u~ of (1) u~ = (w+, w-) with w being the mild solution of (2). THEOREM 1 Let 9t C R’~ be open bounded with C:-boundary. Let X = L~ ~ (12) with lu[ = [u~[~ + tu2[~ and A : D(A) C X -4 X as well as F~. : D(F~,) -4 X defined as above, where q~l,~.~ : It -+ R are continuous, strictly increasing with ~.(0) = 0, and r : R~_ -4 R is continuous and increasing in both variables such that r(a,b) = 0 iff ab = 0. Given u0 ~ L~(fi;R~_) the initial value problem (9) a unique mild solution u~’(.) for every k > 0, and

u.~(t)J -4 \w-(t) in C([6, rl;X) as k -4 co for all 0 < 6 < r, where w(.) is the mild solution of (2) with ~o from (3) and Wo= -- ~t0,2. If th e initial value satisfies uo,lUo,~ = 0 then ~ = 0 is admissible. PROOF 1. Given uo ~ L~"(f~; R~_) and k > 0, existence of a mild solution of (9) can be obtained as follows. First of all, it suffices to consider initial value problem (9) on J = [0, T], with arbitrary T > 0, instead of R+. Let K = {u ~ X : 0 <_ u,(x) < luo,,G, 0

_~~u(w) lu o,2l~o a. e. in

f~}.

Evidently K is closed with K C D(F~), F~: is continnous and bounded on K, and F~: satisfies (8) as a direct consequenceof the properties of r(., .). Furthermore, F~(K) is a bounded subset of L~(f~)~, hence F~,(K) is weakly relatively compact iu X. By ~neans of Lemma2 it folIows that all assumptions of Lem~na3 concerning the operator A are satisfied. Consequently, Lemma3 applies and yields a mild solution

InstantaneousLimitof a Reaction-DiffusionSystem

219

of (9) on To obtain unique solvability of (9) ootice that the bracket in X is given [l~,~)]

--~

max(~vlSgn~l

d,T4-~v2Sgnu2dx)

for

u,u

E X

where Sgn (r) denotes the sign function with Sgn (0) = [-1, 1] and, e.g., short for (w e LI(~) :w(x) ~ Sgn(u~(x)) a.e. on

=

- (u) -kmin ( fn(r(u~,~)-

Sgnul

r(g~,g~))(Sgn

observe that (r(~, b) - r(g, b))(sgn (a - g) + ~) ~ 0 for a ~ g and every ~ e Therefore F~ is s-dissipative, hence (9) h~ a unique mild solution ~(.) u(t) ~ K on ~+. 2. Let i : N~ ~ N be given by denote the metric projections on [0, is continuous, bounded and increasing in both variables. By the previous step the solutiou of (9) remains the same if we replace -Fk Bk :X ~ X with

Bk

and Bk is continuous, everywhere defined and also accretive. Therefore A~ := A+Bk with D(Ak) = D(A) is m-accretive for every k > 0 by TheoremIII.3.2 in [1]. Hence -Ak generates a nonlinear semigroup Tk(t) and the mild solution of (9) is given

on

u ’(0

Weare going to apply Lemma1 where the first J~ := lira (I + £Aa)-~ exists

step is to show that for all ~ e L~(fl;R~).

k~

(10)

For this purpose let ~ ~ L~(~;R~), A > 0 and k > 0 be given and consider the resolvent equations in ~, O%o~(u~’) _ onOf~

-

+

=

in ~, O
Then ulk ---- (I 4- ~A1)-I(~I yields u~ <_ I~1~ by Lemma2(b) since ~(.,.) >_ 0. To obtain u~ >_ 0 let n~(fl) ~ n~(ft) be given (Rv)(x) = k ÷(v(x),uk2(x)) where k : > 0 i s fix ed. The n A~ + R is T-accretive in L~(fl) since A~is T-accretive and R satisfies

Consequently, the resolvents of A~ + R are order-preserving and therefore

220

Bothe

Integration of the first equation over f~ yields

and multiplication of the first equation by ~ (u~’) and integration over ~ implies

(u~)l~~ I~ ~(u~)l~.NI~l~l~](u~:)l~~ I~l~(l~i~). Together with the analogous inequalities for u~ it follows that (~(u~))~.>o, (~2(u~))~>0 are bounded W~,2(9), hence rel atively com pact in L~(~). Due to the L~-bounds for u~ this also yields relative compactness of (u ~) in L2(9). Therefore, given any sequence k~ ~ ~, there is a subsequence of (u ~’~) which is denoted by (u~) for simplicity, such that U¢"

~ Ui,

V~i(U~’)

~ V~i(Ui),

in L2(fl) for i = 1,2.

It will be shownbelow that the limit u is uniquely determined, which implies that the original sequence (u ~) converges to u for arbitrary kj ~ ~, hence (10) holds. Nowobserve that u~, u~ ~ 0 a.e. in ~ and also u~u~= 0 a.e. in fl; the latter follows from ~(u~,u~) 4 P(u~,u~) in L~(fl) and l¢(u~,u~)l~ 0 which yi elds ~( u~,u~) = a.e. in ~. Hence u~ and u~ are given as u~ = w+ and u~ = w-, respectively, if we let w = u~ -u~. It is consequently sufficient to showthat w is uniquely determined. For this purpose let w k~ and w = u~ - g~. Multiplication of the resolvent =t u~ --~ u~ ~ equations by ¢ e C (~) and integration over ~ yields

hence l ~ ~ implies £wCdx+~£(V~(ut)-V~(u~),V¢}dx=£~¢dx

for

all

¢ ~ C~(").

Nowrecall (see e.g. Chapter II.4 in [9]) that + = Vha.e . in {h > 0}, VI~-= ~’~(~). Due to u~u~ = 0 -Vh a.e. in {h < 0} and Vh = 0 a.e. in {h = 0} for h e W a.e. in ~ we therefore obtain V(~o~(Ul) - ~2(u2)) = V~o(w)a.e. where ~o is given by (3). Hence w satisfies £wCdx+A~{V~o(w),V¢)dx=~¢dx

for

all

¢ E C1(~),

i.e. w is a solution of the resolvent equation w + ABw= N, where the operator B is defined by (7). Consequently w is uniquely determined as w = (I + ~B)-~iii since B is m-accretive by Lemma2(a), and therefore (10) is valid. Actually we obtained somewhat more, namely ;~m~(I + AAk)-~17 = J,x~ exists for all i7 e X+:= L~(12;R~_), (11)

Jag ¯ D := {u ~ Ll(f~) ~ : u~,u~ >_ O,u~u.). = 0} on X÷, LJ~ = (I + AB)-~L on X+ with Lu := u~ - u.~;

/

InstantaneousLimitof a Reaction-DiffusionSystem

221

notice that convergence on X+ follows from (10) since the resolvents of Ak are nonexpansive and L~(ft; ITS_) is dense in X+. 3. By meansof (11) we are able to obtain convergenceof (uk(t)) in case the initial value satisfies uo,lu0,2 = 0. Define the operator Amin X by ~neans of

gr (A~)= { (J~, ~(~ - &~)): ,~ > 0,~ e X+}. Fromthe first line in (11) it follows immediately that A~ C lim inf Ak, hence Am in particular accretive. By definition of A~ it is also clear that Jx are the resolvents of A~ and R(I + ,~A~) D.__~gfor all A > 0. Moreover, the second line in (11) yields D(A~) C D and then D(A~) = D follows D(A~o) = {(w+,w-):

w 6 D(B)} D -) :w 6 L ~(f ~) } = D.

Therefore Ac~ is accretive and satisfies the range condition, hence -A~ generates a nonexpansive semigroup T(t) on D. Application of Lemma1 shows that (u0 k) C X+ with u~ -4 Uo e D implies Tk(t)u~ -4 T(t)uo on It+, (12) where the convergence is uniform on boundedsets. It remains to identify the limit semigroup T(t). For this purpose notice that LI~9 : D -4 L~ (~) is invertible with L~-~ (w) +, w-). Hencethe re pres entation (I + A,4c~)-1 = L~)(I + AB)-~L by (11) yields L(I + ),A~) -n = (I + ),B)-’~L on for all n ~ 1. Let S(t) denote the semigroup generated by -B on L~ (f~). Since S(t) and T(t) are given by the exponential formula, the equation above implies LT(t) = S(t)L on D for t >_ 0, hence \uo,2 = \w-(t)

with w(t,)

= S(t)(uo,l

-- UO,2).

Therefore u~(t)J -4 \w-(t)J

in C([O,r];X) as k -4 oo for all "r > 0

if the initial valne belongs to D, i.e. for u0 ~ L°°(f~; ItS_) with Uo3Uo,~= O. 4. Given an arbitrary initial value Uo ~ L~(~;It~_), let wo = u0,~ - uo,2 and 0 < (~ < r. Wethen have to show

(

u~(t)~ = (uo,~ w+(t)’~ : T( t) (w +°_) uniformly on u~(t)) Tk(t) \Uo,~l -4 \w-(t),] \w o

as k -4 c~, and the latter follows from (12)

T~,(5)\ uo ,-

kWo/

as k -4 o~.

(13)

222

Bothe

To obtain (13) consider ilk(t) integral solution of

uk(t/k).

u’ + [A+kB1]u=O

Exploiting th e fa ct th at uk (.) is the

onlY+,

u(O)=uo

it follows immediatelythat ilk(.) is the integral solution fi’+ To obtain

[¼A+B1]u=0 onR+,

convergence of (fik),

ft(0)

=u0.

wecomputeL := liminf[~A+B,].k_~o

Evidently

Blu E Lu for all u ~ D(A), hence B~u ~ Lu on D(A~ = X since B : X -~ X is continuous and L has closed graph. Therefore gr (B1) C gr (L) and then B~ since B~ is m-accretive, hence especially maximal accretive, and L is accretive. Consequently, Lemma1 yields ~" ~ fi in C([O, ~]; X) as k -~ ec for all a > where ~(.) is the solution fi’+Bl~t

=0 on l~+, fi(O) =uo.

(14)

For the subsequent argumentation we need to knowthe asymptotic behavior of g(t), and here it is helpful to observe that (14) is in fact a fmnily of ordinary differential equations, pa.rameterized by x ~ fL Wetherefore consider a’ = -~(a, b) on R+,

a(O)= ao>_

b’ = -÷(a,b) on R+,

b(0) = bo _>

(15)

Nowrecall that ÷ : 1~2 .4 I{ is continuous and increasing in both variables with ~(a,b) = 0 if a <_ 0 or b _< 0, and 7~(a,b) = 0 with a,b _> 0 implies ab = 0. In particular, the right-hand side in (15) is dissipative in ’~, I"li) . Hence (15) has a unique solution (a(t), b(t)) = (a(t; ao, bo), b(t; ao, bo)) for every ao, bo. Evidently a(t) ~ aoo >_ and b(t) x, ~ boo >_with ~(aoo, boo) = 0, i. e. a~oboo = 0. Furth ermore aoo - boo = ao - bo since (a - b)’ = 0 on R+, hence aoo = (ao - + and b~ = (ao - bo)-. Consequently, a(t; ao, bo) .4 (ao - +, b(t; ao,bo) -+ ( ao - bo)- as t .4

(16)

Let fit (t)(x) a(t; uo ,t(x), Uo ,.2(x)) and fi ~(t)(x) = b(t; uo 3 (x), u0 ,~ (x )) tb r t and x ~ f~. Then fi is the strong solution of (14), hence (16) together with dominated convergence theorem implies (fi,(t)’~ ~(t))

-4 ((uo3-Uo,~)+’~ (uo,~ uo,~)-) =

(Wo_+’~ inXast.4oo.

Nowwe are able to obtain (13) as follows. Given e > 0, there is a > 0 such that

[ \~_(a)

\Wo

InstantaneousLimitof a Reaction-DiffusionSystem

223

Translated to the original time scale the latter means Tk(a/k)

(Uo,,~ \uo,~)

~ (w~l ~ ~ for a11k \w o / -

which implies

ITI>\ (Uo,, uo,2 _

\ Wo ]

<

large k.

Since (Wo+, w~-) belongs to D, exploitation of (12) yields

\Wo

\w0

\ Uo,2

\ Wo )

and therefore

Consequently (13) holds which ends the proof. REMARKS 1. Observe that we cannot expect to obtain convergence of uk to a c¢ limit u in C([O, T]; X), since a jump at t = 0 develops as k -+ cxz; notice that u°~(0) = uo but u~(0+) (( Uo,1 - uo,2) +, (Uo,1 - Uo,2)-). Th is ph enomenon is intuitively clear from the physical background: in the limiting case k = o¢ the concentrations of A and B will be instantaneously reduced at every point by such an amount that one of them vanishes. Thereby a separating interface develops which then starts to move, driven by diffusion of A and B towards this free boundary. Theorem1 contains the main result in [7] where the following special case of system (1) has been considered. ut = d~ Au - kuv vt = d2/kv - kuv O~u = O~v = 0

u(0,.) =uo,v(0,.)

in(0, c~)x in (0, cx3)x on (0, o~) x in fL

In this paper convergence of (uk,v k) to (w+,w-) in L~((O,T) x fl) is obtained in the regular case whenthe initial values satisfy UoVo= O. 2. In [8] the authors study the instantaneous limit for a single irreversible reaction between a mobile and an immobile species. Further assumptions lead to the following reaction-diffusion system with one spatial dimension. ut = uz., - kuv, vt = -kuv fort>O,x>O ~(t,0)=¢(t), ~,(0,.)=0, ~(0,.)=v0>0. Under appropriate conditions on ¢, convergence of uk,v ~ to certain limit concentrations u,v is obtained. Furthermore it is shown that a free boundary (given by a single point p(t) due to the one-dimensional setting) develops as k -~ c~, which separates the two regions where u > 0 and v = 0, respectively u = 0 and v = v0. Here the limit problem for u, p turns out to be a classical one phase Stefan problem.

224

Bothe

Let us note that several processes in Chetnical Engineering lead to related moving boundary problems, called core-shell models in this context. For a realistic model of such a process additional aspects like macroscopic convection and mass transfer resistance have to be taken into account. A specific class of such models which occur for example in semibatch regeneration of exhausted ion exchangers has been studied in [5]. This paper provided a new approach to this type of nonstandard Stefan problems, based on the use of the free boundary as an explicit system variable which then allows application of the theory of accretive operators and nonlinear semigroups. By means of this approach a rather complete analysis of the dynamics of this core-shell process is obtained.

REFERENCES 1. 2. 3. 4. 5. 6.

7. 8. 9. 10. 11.

V. Barbu: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff, Leydeu 1976. Ph. B~nilan: Equations d’fivolution dans un espacc de Banach quelconque et applications, Th~se d’Etat, Orsay (1972). Ph. B~nilan, M.G. Crandall, A. Pazy: Nonlinear Evolution Equatkms in Banach Spaces. (monograph in preparation). D. Bothe: Flow invariance for perturbed nonlinear evolution equations. Abstract and Applied Analysis 1,417-433 (1996). D. Bothe, J. Priiss: Dynamics of a Core-Shell Reaction-Diffusion Sysgem. Comm. PDE (to appear). J.I. Diaz, I.I. Vrabie: Propri~t~s de compacit~ de l’opfirateur de Green gfin~ralis~ pour lYquation des milieux poreux. C. R. Acad. Sci Paris. 309, 221-223 (1989). L.C. Evans: A convergence theorem for a chemical diffusion-reaction system. Houston J. Math. 6, 259-267 (1980). D. Hilhorst, R. van der Hour, L. A. Peletier: The fast reaction limit, for a reaction-diffusion system. J. Math. Anal. Appl. 199, 349-373 (1996). O.A. Lady~enskaja, V.A. Solonnikov, N.N. Ural’ceva: Linear and Quasilinear Equations of Parabolic Type. Amer. Math. Soc. 1968. I. Miyadera: Nonlinear Semigroups. Amer. Math. Soc. 1992. J.L. Vazquez: An introduction to the ~nathematical theory of the porous mediuln equation, pp. 347-389 in Sbape Optimization and Free Boundaries (M.C. Delfour, G. Sabidussi eds.), Kluwer 1992.

A Semigroup Approach To Dispersive

Waves

RADU C. CASCAVALand JEROME A. GOLDSTEIN Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152-6429, USA

1

INTRODUCTION

Of concern are generalizations

of the Korteweg-de Vries equation Ou 03u

Ou

0-7 + ~ +cue= o.

Here u = u(x,t), the time t is real and the spatial variable x ranges over the real line I~ or the circle ’]i" = {x E C : Ixl = 1}, whichwe identify with I~, under addition modulo 2~r. Thus x E "I~ corresponds to periodic boundary conditions. The class of equations we consider is Ou MOU cOt cOx + (F(u)) x ~ G = I~ or ~’,

= 0,

.(1)

u(x,o) = f(x), x hi is defined by the Fourier transform, namely

^ (~)=1~13~,(~) (,~fv)

(2)

for ~ ~ (~ = II~ or Z. Here/~ > 1. Other operators of the form (Mv)^(~) a( ~)~)(~) arise in water wavetheory, but for simplicity we will stick with (2). Special cases include the [generalized) KdVequation (~ = 2) and the fifth order version (~ = 4) studied by J. Bona, P. Rosenau and others. The Benjamin-Ono equation (~ = 1) is barely excluded from our analysis. In that case, M~-~is the composition of the Hilbert transform with the Laplacian o.-~ ¯ The class of nonlinearities we consider includes those of the form

225

226

CascavalandGoldstein

(the odd version), in which case ~°°u

0-~(F(u))

As we wilt assume F(0) = F’(0) = 0, we thus take p > 0. We make the following assumption on F., which we call (HYPF):

F ~ C~+~(~),F(0) = F’(0) lim sup F’(r) r--~:ke¢ ~ < (:X:) for some p < 2ft.

MAINTHEOREM Assume (HYP F) holds. For 13 > ~, the initial value problem ~ (1) has unique global solutions in ( G) ]or both G= ~ and G = ~.For1 < 13<_ the initial value problem has unique global solutions in Hs(G), for s > ~, provided the initial data f are in Hs(G), G = "~ or IR. This theorem naturally requires that the initial data is in H~(G). One can also show that the semigroup leaves HI~+~(G)invariant, if F ~ C~+~(tR). This implies additional regularity; thus if 13 is an even integer, then u is a continuous mapping from IR to H~(G). But this will be discussed elsewhere (see [4]). Webelieve that this result is sharp with respect to the growth of F. Here is why. Weare consideriug semilinear problems of the form dlt

d--[ = Au+ Bu,u(O)= where ,4 is a linear pseudo-differential operator of order fl + 1 and B is a first order nonlinear differential operator; and An + Bu = C(u).~. This formally leads one to expect the solution u to satisfy the (variation of parameters) integral equation t

u(t)

= T(t)f

~0

+ T(t - s)B(u(s))ds,

where {T(t) : t _> 0} is the (Co) group (or semigroup) generated by A. classical theory of ODEin Banach spaces, any local (in time) solution is either global or else blows up in norm in finite time. Multiplying ut = C(u).~ by u and integrating over G implies ~l[u(t)ll~ = 0. Thus the L:(G) norm of u is conserved. Ou Working in H6 with 13 > 1, this blow up should be a consequence of blow up of 07" But for the generalized periodic KdVequation

O-7+-~x3+C U~x=o, with c > 0, Bona et al [3], in nmnerical experiments, got blow up of o,, in finite time if p >_ 4. While these results are not rigorous, they are suggestive, and hence our condition in (HYPF), p < 2~, reduces to p < 4 in this cause. Thus there is possibility that our results are sharp in that the inequality p < P(13) maynot allowable if P(f~) > 2/~.

SemigroupApproachto Dispersive Waves 2

REVIEW

OF SEMIGROUP

227

THEORY

Our proof of this main theorem will be based on semilinear Hille-Yosida theory. We begin with a rapid review of semigroup theory, starting with the Cauchy problem du

(3)

d--~ = Au, u(O) =

in a Banach space X. Suppose that A is linear. Then, according to classical HilleYosida-Phillips-etc. theory, (3) is wellposed(in a certain precise sense) iff it is erned by a (Co) semigroup iff A generates a (Co) semigroup. (See [6], for instance.) By change of norm tricks, one mayconsider only (Co) contraction semigroups. Then A generates a (Co) contraction semigroup iff 7P(A) = X and for all A > 0, A is the resolvent set of A and

(4)

I[(I - A)-II Then the semigroup T and the solution u of (3) are given u(t) = T(t)f

= lim (I-

t-A)-n];

u is a strong solution if f E 7)(A) and a mild solution in general. (See [6].) The Crandall-Liggett-B~nilan theory extends this to nonlinear operators. w E I~ and let A~ = A - wI. Suppose that

Let

II(! - AA~)-tllL~p < 1 for every A > 0; i.e. if A > 0 and ui - AA~ui = hi,i = 1,2, then Ilu also Range(I - A.4~.) D 7)(A~,) for all (small) A >

- u, ll

<_ IIh~ -

hell. Suppose

T(t)f =,~-~oolim(If -tn~)-’~ (where .~ is the closure of A) defines a strongly continuous semigroup T on T~(A) satisfying IIT(t)llLi~ <_~ for a ll t >_0. This is theCrandall-Liggett theo rem [5]. B~nilan [2] defined mild solutions and showed that u(t) = T(t)f is the unique (global) mild solution of (3). The definition of mild solution is complicated need not concern us here. For the semilinear problem du

d--[ = Au + Uu, u(O) =

(5)

where A generates a (linear) (Co) contraction semigroup T or X, a mild solution a continuous function u : [0, oc) -4 X satisfying t

u(t)

= T(t)f

~0

+ T(t - s)B(u(s))ds

for all t > 0. Building upon earlier work of Oharu and Takahashi [9], [10], Goldstein, Oharu and Takahashi [8], [7] developed a semilinear Hille-Yosida theory and applied it to the generalized KdVequation ((1) with fl = 2). For the usual KdVequation,

228

CascavalandGoldstein

S = {S(t) : t E II~} is the group which governs the Cauchyproblem, then on all the Sobolev spaces, IIS(t)IIL~ p = oc for all t :fi 0. Thus, the Crandall-Liggett-B6nilan (C-L-B) theory doesn’t apply. In the following Venndiagram, the lines with slope 1 (see Figure 1) correspond semilinear Hille-Yosida theory, and the lines with slope -1 (see Figure 2) correspond to C-L-Btheory. The inside of the octagon represents the Cauchy problems solvable by linear Hille-Yosida theory. (See Figure 3.)

Figure I

Figure

Figure ~ The region marked R describes Cauchy problems we can solve by semilinear HilleYosida theory but not by C-L-Btheory; this includes all cases covered by our main theorem.

3

SEMILINEAR

HILLE-YOSIDA

THEORY

Consider the Cauchy problem u’(t) = Au(t) Bu(t) It >_ 0], u(0)= f.

(6)

Suppose that ,4 generates a (Co) contraction semigroup T = {T(t) : t _> 0} on By a mild solution of (4) we mean a continuous function u E C(II( + ;X) satisfying the integral equation version of (6) u(t)

= T(t)f

+ T(t - s)B(u(s))ds

for all t _> 0. WhenB is locally Lipschitz continuous, a unique local (in time) solution exists by the usual iteration procedure (cf. e.g. [6]). Then there is

SemigroupApproachto Dispersive Waves

229

T = Tm~×E (0, ~c] such that the solution exists on [0, T) and either ~- = cx) else limt-~r- I lu(t)ll = ~. Thusglobal existence (in this case) follows from avoiding blow up in a finite time, which we can in principle determine by only looking at one solution. This is in contrast to C-L-B theory, where one must always compare two solutions (and have t --+ e-~llu(t ) - v(t)~] be nonincreasing for somereal w). The first semilinear Hille-Yosida theorem was published by Oharu and Takahashi in 1987 [10]. See [11] for an important extension. The context is as follows. Let ¢ : X ~ [0, ~] be proper, convex, and lower semicontinuous. Let E C {f ~ X : ¢(f) < ~}. Let B : E ~ X and suppose that for each r > 0 there is a constant K(r) such that ibr all f, g ~ E~ = {h ~ E : ¢(h) ~ r}. This is a kind of Lipschitz condition involving ¢ and I~-~[. This can be replaced by suitable local quasidissipativity conditions; see

[~q,[sl, [41. A locally Lipschitz semigroup S on E is a strongly continuous semigroup {S(t) t k 0} of operators from E to E satisfying S(t + s) = S(t)S(s), S(O) = I, C([0,~);D) for each f e E, and for all r > 0,~ > 0 there is an w w(r,r) such that

IIS(t)f- S(t)gll~ e~(~")~llf - gll

holds for 0 < ~ < ~ and f, g e E~. {Thus M = 1 s~ill, bu~~ is bo~hspace and ~ime dependenL} The original Oh&ru-T&l~h&shi theorem[10] is as follows. THEOREM Let ~, b ~ 0 be given. The~ ~here is ~ locall# on E satisfying S(t)f

= T(t)f ¢(S(t)f)

+ T(teat (¢(t) +

L{?schi~ s~m~grou ~~

s)B(S(s)f)ds, bt)

for all f ~ E, t >_ 0 (i.e. S gives global mild solutions with an exponential growth rate governedby ¢) if and only if for all r > 0 there is a )~o(r) > 0 such that for

I ~ E~andO < ~ < ~o(r)thereis anI~ ~ V(A)nE~ satislyi,~gI~-~(A+B)f~ and ¢(f~) _< (1 - -t {¢(f) + b A}

Thus, like in the Hille-Yosida theorem, one simply solves the resolvent equation (I - A(A ÷ B))u = for u (f or al l f ~ E and va lues of A t hat dep end on f) and finds a solution of a small enough size. This sufficient condition is necessary. Moreoverthere is no need to establish local existence separately. This is important for equations like the KdVequation where local existence is far from obvious. Weneed an extension of Theorem1 involving several features, the principal one being replacing ¢ by vector valued function ¯ = (¢o,... , Cg-~). The analogue E~for ECXis {h~E:¢i(h)_
fori=0,1,:..,g-1}

The assumption on B is the following local quasidissipativity condition: For every r >_ 0, there exists w,. E II~ such that u, v ~ E~ implies

(B~- By,~ - v) < = ~llu- ~11

230

Cascavaland Goldstein

Weornit a precise statement of the abstract result, except to say that for our dispersive equations we require N = 4. The mild solution u(t) = S(t)f is constructed as u(t) = limn-~oo(I where C = A + B. Then ¢i that we construct are not all convex. Thus it is essential to note that the convexity of ¢ is needed for the necessity of Theorem1 above but not for the sufficiency. But lower semicontinuity of ¢ is needed to pass from the estimate for ¢((/- ¼C)-’~f) to the limit ¢(u(t)) eat{¢(f) + b t} For more details and precise statements, see Goldstein-Oharu-Takahashi [8] and Cascaval [4], both of which are in preparation.

4

THE MAIN RESULT

Wevery briefly outline the proof of the MainTheoremfor the case of G = "I1". (For the special case of the KdVequation and G = N, see [7]). Our problem is u~ - Muz+ (.F(u)).~ = O, x e V, t e

(7)

u(x,O) = f(x), x ~

(8)

Here (we recall (HYPF)) F ~ CO+~(l~;ll{) is a (nonlinear) F(0) = F’(0) = 0 and the growth condition lira sup

function satisfying

< oc

(9)

for some p ~ (0, 2fl), where F’ denotes the derivative of F. Recall the connection between ~ and M: The Sobolev spaces H~(~) are defined Hs(V) = {u e L2(T) : ~ (1 fors>0.

I l Yla( )l < ~ }

Let fI~(’lF) = {u ¯ HS(’l~) : ]vu(s)ds = 0}.

If u satisfies (7), (8) then we have ~7 fv u(x, t)dt 0,whence fvu(x,t)dx

= ~ f(x)dx

=

a constant, for all real t. Thusv(x, t) = u(x, t) I, is H~ (’li ’)-valued if uis an’~ (~) solution. By replacing (7)

v~- Mv~ + (P(v)).~.

SemigroupApproachto Dispersive Waves

231

where/~(r) F(r + Cf), wemaythus assume witho ut loss of ge nerality that c I = 0 and work in the spaces H~(~I’). In order to apply the abstract theorem to our case, the underlying space is taken to be X = H~(~I’). The equation (7) is then of the form (6), by choosing Au = Mu~ and B(u) = -(F(u))~. The main functionals we need are

Co(u) _-- g1 [ u(z).2dx,

¢’2 (~) =- "~1

u e [-I°(qr),

where a(u) = fg’ F(r)dr,

.[.[Mu(x)[~dx ÷

~~

2Z2~ ++21-..L F(u(z))Mu(x)dx ’ u e He(V),

[Mu,~+ F(u)~iedx,

u e/:/~+l(~r).

In the context of the generalized KdVequations, these spaces/:/~(’Ir) have exponents s = 0, 1,2, 3 since fl = 2 (see [7]). One can also introduce (as in [7]) another function ¢4 on /:/~+.2(’11") which makes solutions (corresponding to f ~ HZ+~-(’Ir)) "classical". One uses a fixed point theorem to solve the resolvent equation u - A(Muz - (F(u)):~)

(10)

and gets estimates for Cj(u). The semilinear Hille-Yosida theory gives the corresponding group S, so that u(t) = S(t)f is the mild solution of (7), (8). The estimates on the solution are Cj(u(t)) = Cj(f) for j = O, 1 and all t e(T); if f ~ H Ce(u(t)) e~ltl(¢~(f) + b

(11)

for f e H’~(’Ir) and Cj(f) _< eta for j = 0, 1, where ao,a~ > 0 are given and co, b are functions of a0, ¢3(u(t))

(12)

for f ~ H~+l(~),ltl < ~- and Cj(f) _< aj for j = 0,1,2, where r, ao,a~,a~ > 0 are given and ~ depends on The above statements involve the spaces H~(’Ir). As indicated above, the proofs take place in the/:/s(,r) context. Our proof requires estimates to obtain local (in time) boundedness of u(t) in/:/s(ql’). For this purpose, (9) is needed. However if (9) fails, the methodstill worksand we get global existence for sufficiently small data f. Weindicate briefly the use of (9). For a suitable constant Cx > 0, (9) implies F’(r)

<_Cllrl~ + C,

232

CascavalandGoldstein

holds for all r E IR. Using G(r) = fo r F, F(0) = 0, it follows that G satisfies

a(r) < Cdrlp+=÷ ~" C=r for all r E I~ and some constant C~. Thus,

~

p+2

~¢~-~)+~

2

~

by a Sobolev inequality

by Young’s inequality since p < 2~. This implies +

¢~(u)

since ¢1(~/)

~llD~aull~2- £ G(u)dx,

and the desired lower bound on q51 follows. The proof of the main theorem is much more complicated than the proof in the KdVcase [7], [8]. Anmngother things one needs suitable estimates of expressions involving terms such as DZ(uv). To illustrate the tedmiques used, we show how we get the quasidissipativity condition for the nonlinear term B(u) --- -0F(u), where Let u, v e H~+~-. Then,

I(Bu - Bv, u - v)zl = I(0F(u) OF (v),u - v)~l _ 1, and 7 > }, the following holds:

lID~(uv)-~V~vll~ c(~/, Z){tMI.~IMI~. + IMl~,+~llvlle-~ }. Using the same idea, one can obtain, in our context, the following inequality (u

~+~):

SemigroupApproachto Dispersive Waves

233

This result plays the key role whenapplying the fixed point argumentto solve the resolvent equation (10) and get the estimate (11) for qoi(u). Note that, when (KdVcase) or ~3 = 4 (fifth order KdV), the previous inequality can be checked directly. For noninteger fl > 1, one needs the use of Fourier transform methods. For full details, see [4].

REFERENCES 1.

L Abdelouhab, JL Bona, MFelland, J-C Saut. Nonlocal models for nonlinear dispersive waves. Physica D 1989; 40:360-392. 2. Ph B~nilan. Equations d’evolution dans un espace de Banach quelconque et applications. PhDdissertation, Universit~ de Paris XI, Orsay, 1972. 3. JL Bona, VA Dougalis, OAKarakashian, W McKinney. Conservative, highorder numerical schemes for the generalized KdVequation. Philos Trans Royal Soc London Ser A 1995; 351:107-164. 4. R Cascaval. PhDdissertation, University of Memphis, In preparation. 5. MGCrandall, TMLiggett. Generation of semigroups of nonlinear transformations on general Banach spaces. AmerJ Math 1971; 93:265-298. 6. JA Goldstein. Semigroups of Linear Operators and Applications. Oxfbrd and NewYork: Oxford U. Press, 1985. 7. JA Goldstein. The KdVequation via semigroups. In: A Kartsatos, ed. Theory and Applications of Non-linear Operators of Accretive and Monotone Type. NewYork: Marcel Dekker, 1996, pp 107-114. 8. JA Goldstein, S Oharu, T Takahashi. A class of locally Lipschitzian semigroups and its application to generalized Korteweg-de Vries equations. Unpublished manuscript (1993); expanded revision in preparation. 9. T Kato. On the Cauchy problem for the (generalized) Korteweg-de Vries equation. In: Advances in Mathematics Supplementary Studies, Studies in Applied Math. NewYork: Academic Press, 1983, vol 8, pp 93-128. 10. S Oharu, T Takahashi. Locally Lipschitz continuous perturbations of linear dissipative operators and nonlinear semigroups. Proc Amer Math Soc 1987; 100:187-194. i 1. S Oharu and T Takahashi. Characterization of nonlinear se~nigroups associated with semilinear evolution equations. Trans AmerMath Soc 1989; 311:593-619. 12. J-C Saut and R Temam. Remarks on the Korteweg-de Vries equation. Israel J Math 1976; 24:78-87.

Regularity Properties of Solutions of Fractional Evolution Equations PH. CL]~MENT Faculty of Technical Mathematics and Informatics, Delft University of Technology, P.O. Box 5031, 2600 GADelft, The Netherlands email: [email protected] G. GRIPENBERG Institute of Mathematics, Helsinki University of Technology, P.O. Box 1000, FIN-02015 HUT, Finland, www.math.hut.fi/-ggripenb email: [email protected] S-O. LONDEN Institute of Mathematics, Helsinki University of Technology, P.O. Box 1000, FIN-02015 HUT, Finland email: [email protected]

1

INTRODUCTION

AND MAIN RESULTS

In this paper we contiuue our work on regularity properties of solutions of fractional evolution equations. In particular, we study the equation D~(ut - u~)(t) + Uu(t) = u(0) = u0, t >_

(1)

which is of order 1 + a. The function u is the unknown,taking values in a Banach space X; c~ E (0,1); uo, ul and f are given, with Uo,Ul ~ X and f ~ C([O,T];X) for some T > 0. 235

236

Clement et al.

In (1), D~ denotes the fractional derivative of order a, i.e., def d it = -~Jo gl-a(t-s)v(s)ds, (D~u)(0)

where

de_f ~ ~

gl -a(h - s) v(s) ds,

g~(t) def= 1~-~t ~-1,

t>0,

/3>0,

and where v is (at least) continuous and satisfies v(0) The operator B is taken to be a closed (not necessarily densely defined) linear map of T~(B) C X into X. Thus :D(B) is a Banach space equipped with the usual graph norm. Weassume B to be positive, i.e., that the resolvent set of -B contains IR+ = [0, oo), and that

supll( + 1)(AI B)-llle(x) < For w E [0, rr), we define ~]w de_:f { ,~ e C \ {0} [ ]arg A[ <

w}.

Werecall that, if B is positive, then there exists a number r/ ~ (0,~r) such that p(-B) D E~ and sup__[l(A + 1)(AI + -~ [I n(x) < oo. (2) The spectral angle of B is defined by ¢~ dej inf{w ~ (0, rr)

[ p(-B) D ~-~

and sup

[[(A+I)(AI+B)-II[c(x) <

In a previous paper, [1], we examined the equation D~t(u- Uo) + Bu = f, /3 (0, 1)

(3)

under analogous assumptions on B, Uo and f, and obtained maximal regularity results in certain interpolation spaces. As to the assumptions on uo, they were shown to be both necessary and sufficient. In this paper, we extend these studies to the case/3 ~ (1, 2). Wemainly consider sufficient conditions for the solutions of (1) to be smoothand will return to necessary hypotheses in. later work. Only at the end do we here give some brief commentson the converse analysis of (1). However,in the case where either u0 or u~ vanishes, our results are optimal. Wewrite Dt~ = D’[Dt, with/3 = a + 1, a E (0, 1), and include the initial value of ut in the convolution integral. The case a = 0, i.e., the differential equation ut+Bu=f,

u(0)=u0,

t>_0,

Regularityof Solutions of FractionalEvolutionEquations

237

was considered by Sinestrari [12], and by Da Prato and Sinestrari [6]. Their results provided partial motivation for [1]. HSlder-regularity results for (1) with a ¯ (0, have previously been obtained by Da Prato, Iannelli and Sinestrari [5]. Wecomment briefly on their results below, at the end of this Section. In forthcoming work, we will apply the results given here to fractional partial differential equations and extend them to cases with nonconstant HSlder continuous coefficients. For (3), and with B a spatial derivative of order _< 1, this was done [1] and [2]. Our analysis concerns strict solutions of (1). With f ¯ C ([0, T]; X); u0, ul and a ¯ (0, 1), these are defined as follows: DEFINITION 1 A function u : [0, T] --~ X is said to be a strict solution of (1) on [0, T] ifu, ut ¯ C([O,TI;X),¯ C([0,TI;T~(B)), u( 0) = u0, gl -~ * (ut -u C~([0,T]; X), (, denotes convolution) and (1) holds for all t ¯ [0,T]. Wesummarize our results in Theorem 2 below. In this Theorem, we formulate someexistence, uniqueness and regularity results on strict solutions of (1). Concerning the interpolation spaces determined by an operator B, we use the notation (here 3’ ¯ (0, 1] and p ¯ [1,

7).(3’,p)~°J(x, V(B))~,p, V.(3’)d~J(x, V(B))~. By [7, Thin. 3.1, p. 159] and [8, p. 314] one has the following characterization ~B(3’,~) and ~PB(3’): If r~ is somenumber such that 0 _< ~ < ~r - Cs,

of

The HSlder spaces C7, 0 < 3’ < 1, are defined by C~([0,T]; X) de_~f { f

,,/(t)- z(s),lx}

¯ c(to, rl;x)]

with

Ilfllc, a~2sup IIf(t)llx + sup t~[0,r I

If 3’ ¯ (1, 2), then 7 are defined by

h.~([O,T];X)

de~f

deJ { f I

f’ ¯ CT-I

Ill(t) f( s)llx

O<s
}"

{ f ¯ C~([O,T];X)

"~ It - sl

Thelitt le HSlder spaces h~, 3’ ¯ (0, 1)

limsup

.o

t,s~[O,Tl,O
If 3’ ¯ (1, 2), then ~ de2 {f I f’ ¯ h~-~ }"

Ilf(t)

- f(s)llX

"~ It - sl

}

"

238

Cl~ment et al.

THEOREM 2 Suppose

(i) ~ e (o, (ii)

B is a positive operator in a complex Banach space X with spectral angle

¢, < ~(½- ~); (iii) Uo e/:)(B), Ul ~ D~(~-~); (iv) f ~ C([O, T]; X), where T > O. Then the following statements hold: (a) Let ~/ ~ (0,1) and f E C’~([O,T];X). Then there is a unique strict u of (1) satisfyingBu(t)e C~([O,T];X) if both

solution

(4) and

u, ~ ~, (i-~’ ~)

(5)

hold. Moreover, in this case there is a constant M = M(%a,B,T) such that

<_M~llBuo - f(0)llv,,(,+-~,oo) + llulnlv.(,~+:,oo) +II/ff)(b) Let 7 e (1,1 + ~] and f ~ C~([O,T];X). Then there is a unique strict solution u of (1) satisfying Bu(~) C7([0, T] ; X)if both (4) and

(6)

+ ~),~ hold. Moreover, in this that

case there is a constant

M = M(%a,B,T) such

/ IIBu(t)- f(o) tf’(o)llc,(to,r];x)<Mtl lBuo - Bf (0)llv,,=(~o+ ,oo) + II/(t) - f(0)tf’(O)llc.([o,?r];x),). (c) Let 3’ ~ (1 + a,2) and f ~ C*([O,T];X). Then there exists a unique strict solution u of (1) satisfying Bu(t_) ~ C’~([O,T];X) if both Buo = f(O) (6) hold. Moreover, in this case there is a constant M= M(7, a, B, T) such that

IIBu(_t) - f(O)- t_f’(O)llc,(to,Tl;x) ~÷ Ill(- t) f(O)- -tf’(O)llc~(to,~’l;m ) < M(llul - B-1 ’ 0 f ()11~=(~,~)

Regularityof Solutions of FractionalEvolutionEquations

239

(d) Let ")’ E (0, 1] and f e ~([0, T];:DB(%c~)). Then there is a unique strict solution u of (1) satisfying Bu(t) e Z de__r C([0, T]; X) n B([0, T]; ~B(% ff both Buo - f(0) e ~8(~, and

ul ~ 79(B~-~), B~-~ule ~-(’r,m),

(7)

hold. Moreover, in this case there is a constant M = M(7, c~,B,T ) such that

IIBu(t_) - f(0)ll~¢tO,Tl;V.¢-.oo)) M(llBuo -(0 f)ll~.¢.~,~)

(e) Let 3’ ~ (0, 1) and f ~ h’~([O,T];X). Then there is a unique strict solution

u of(l) satisfying Bu(t)~h’~([O,T];X)ifbothBuo-f(O)~79~(~+~)

ul ~ 79~(~+7~ hold. (~ Let 7 ~ (1, 1 + ~) and f ~ hT([O,T];X). Then there is a unique strict tion u 4 (1) satisfying Bu(t) h~([0, T] ; X)if bot h Buo- f( O)~ Vn( and u, - B-~ f’(O) ~ VB2 k2(1 + a)

solu-

(8)

hold. (g) Let7 ~ [l+a,2) and f ~ h~([0, T];X). Then there is a unique strict solution u of (1) satisfying Bu(t) h~([0, T] ; X)ff both Buo= f( O)and(8) hold. (h) Let 7 ~ (0, 1) and f ~ C([0, T]; ~n(7)). Thenthere is a unique strict solution u of (1) satisfying Bu(t) C([0, T] ; DB(7)) if both Buo- f( O)~ ~B(and ul ~ D(B~-~), B~-~u~ ~ DB(~/)

(9)

hold. The conditions (7) and (9) may be written _ IC~ + 7 + c~’ Ul ~ "U’B2 ~~(,1-"7’"

respectively. For the definitions and properties of fractional powers of densely defined operators we refer to [1, Theorem 10], [14, p. 98-103]. For powers of non-densely defined operators, see [9, p. 54]. Observe that by (ii) of Theorem2, the operator B sectorial in the sense of [9]. Also note that the statements of [1, Theorem10] can be applied to the operator B. To solve (1) we write, in the cases where3’ ~ (0, 1], i.e., in cases (a), (d), (e), ~(t)

= Y(t)

-~- W(t) "-]-

Z’(t)

"~- B-l/(0),

(10)

240

Cl~.ment et al.

where v, w, z solve, respectively, D~vt +/~’v = 0, v(0) = uo -/~-lf(0),

vt(O) :-

D~(wt - ul) + Bw = O, w(O) = O, wt(O) D~z~ + Bz = f(t)

- f(O), z(O) = z~(O)

(11) (12)

(13)

Then u, defined by (10), solves (1). If 3’ 6 (1, 2), i.e., in cases (b), (c), (f), (g), u(t) = v(t) + w(t) + z(t) B-lf(0) tB-l y’ (o),

(14)

where v, w, z solve, respectively, (11) and D~(wt-[Ul-B-lf’(O)])

+ Bw=O, w(0)=0,

D~zt + Bz = f(t)

w~(O)=ul-B-’f’(O),

- f(O) tf ’(0), z( O) = zt (O) =

(15) (16)

Then u, defined by (14), solves (1). First, in Section 2, existence and regularity for solutions of (11) is given. These results are obtained by the resolvent approach. Lemma3 gives the required properties of the resolvent associated with (11). Lemma 4 gives necessary and sufficient conditions on v(0) for By(t) to be either HSlder-continuous, or bounded in an interpolation space determined by B. Next, the resolvent approach is used to analyze solutions of (12) (and (15)). Lemma5 gives the properties of the associated resolvent; in Lemma6 we formulate necessary and sufficient conditions on wt (0) for Bwto be of a specified type. The proofs of Lemma3 and of Lemma5 are based on techniques given in [11, Sec. 2]. See [3] for details. To analyze (13) and (16), we use the method of sums (see Lemma7) Prato and Grisvard [4]. For the application of this method, we need some results (formulated in Lemma8) on fractional derivatives of order between 1 and 2. For derivatives of order between zero and one, the corresponding results were given in [1]. The regularity results on (13) and (16) are given in Lemma Theorem 2is an immediate consequence of Lemmas4, 6, and 9. In the last Section, we commentbriefly on the converse analysis. In particular, a required interpolation statement is formulated. Regularity results for ut+B(k*u)(t)=

f(t),

u(0)=u0,

with (typically) k(t) = -~, /3 E ( 0,1), havebeen obtai ned in [5 results, invert (17) to obtain, D~ut+Bu=hdef

(17) To compare

= -~(td a * f),

where c~ = 1 -/3. To have h HSlder-continuous, take f(t) = p fo r so me p E (c~,l). Then h ~ C’~([0,T];X), with 7 = P-C~; f(O) = h(O) = ut(O) = 0. By [5,

Regularityof Solutions of FractionalEvolutionEquations

241

Theorem4.4], we have that Buo ¯ Du(+z~ oo) implies D~ut ¯ Cx([0, T]; X), hence I+~ ’ Bu ¯ d~([0,T]; X).OurLemma4(a)does,however, givethe stronger result ~ oc) is equivalent to Bu ¯ Cv([O,T];X). in this case Buo ¯ 2

RESULTS

ON HOMOGENEOUS

EQUATIONS

Webegin by considering (11) and employ the resolvent associated to this equation. Thus, we set 1 fr

s(t)vo= ~.i

e~tA~(Al+~I

,,o

B)-avodA,

t O,

>

+

(is)

S(O)vo= vo, where 0¯

-~’

:-=0-"h

(19)

l+a 7’

and

rr,Ode_=f { reit I ltl <_0 } u{ pei° I r _

(b) s(.) c~((O, oo );c(x)). (c) For t > O, the range of S(t) is contained in T)(B). (d) For t > O, BS(k)(t) ¯ £(X); BS(~)(.) Cm((0, ec );£(X)). (e) For any fixed 0 (0 ~+~ - ~), there exists an analytic extension of Bt S(k)(.) to the sector [argz I _< 0. (f) sup~>ol[t~(~+")+kBtS(k)(t)l[e(X) (g) For Vo ¯ X, v(t) de__f S(t)Vo, t >_O, one has that v is a strict solution D~vt+Bv=O,

v(O)

=vo,

v,(O)=O,

t¯[O,T],

(21)

for any T > O, iff vo ¯ D(B) and Bvo ¯ D(B). (h) If v is a strict solution of (21), then v,vt,D~vt,Bv ¯ BC([0, oo);X). For the proof of Lemma3, see [3]. For the solutions of (11) we have the following regularity results. LEMMA 4 Let (i) and (ii) of Theorem 2 hold. Let Vo ~¯ D(B) and v(t_) ae=f S(t)vo. Then the following conclusions hold for each T > O: (a) iet 2/ ¯ (0, l+a], I’ ¢ 1. ThenBv(t_) ¯ C’~([O,T];X) iff Bvo ¯ D,( a-~-5, ). Moreover, in this case there is a constant M = M("/,c~, B), but independent of T, such that

[IBv(_t)[Ic. ([O,TI;X) MllBvollv.(,-~ ,oo).

(22)

242

Ch~ment et al. (b) Let 7 e (1 + a, 2). Then Bv(~_) C’~([O, T]; X)iff v(~_) = v( O)(iff .BVo (c) Let 7 e (0, 1]. Then my(t) C([0, T] ; X) O B([ 0, T]; D~(7~)) if f Bv (~) BVo e C([0, T]; X) ~ B([0, T]; DB(7, iff BVo e DB(7, ~). Moreover, in this case there is a constant M = M(7, a, B), but independent v~ T, such that I[Bv(!)ll~(t0,T];~.(~,~)) ~ MllBvo[].~<~,~). (23) (d) Let T ~ (0,1+~),7#1. Then Bv(t)_ ~ h*([O,T];X) iff Bvo ~ V~(~). ~ (e) Let 7 6 [1 + ~, 2). Then Bv(~) h~([0, T] ; X)iff v(~)= Vo (i ff Bvo = (f) Let 7 ~ (0,1). Then Sv(~) C([0, T] ;~s(7)) if f Bv (~) - Bvo ~ C([0, T];~,(7)) iff Zvo e V,(~).

Observe that in all cases of Lemma4, the hypotheses made and Lemma3 give that v(t) is a strict solution of D~vt + Bv = O, v(O) = Vo, vt(O) Except for some technicalities, the proof of Lemma4 parallels that of [1, Lemma12]. For the analysis of (12) and (15) we need the corresponding resolvent. We fory E X, 1 fr e’XtA-l+~(Al+(~I + B)-lydA’ t > 0, S~(t)y = ~ ,,o (24) $1 (0)y = where the integration path is as in (18). One then has the following result. LEMMA 5 Assume that (i) and (ii) of Theorem 2, and (19) hold. Define S~ by (24), and let k be a nonnegative integer. Then properties (a)-(e) of Lemma3 hold with S replaced by S~. Moreover, for I = O, 1,

sup~f(~+~)+a-~ B~S~) (t)lk(x) ~0

The .function w(t) ~f Sl

(t)Wl

i8

a

strict solution of

D~(w~ - u~) + Bw = iff

w(O) =

w~(O) = w~,

(25)

Wl ~

For the proof of Lemma5, see [3]. Concerningthe solutions w(t) of (25), we then have the following regularity state~nents. def

~ LEMMA 6 Let (i) and (ii) of Theorem Z hold, let ~~ ~ B(~) and w(t) S~(t)w~. Then the following statements hold for each T > ~ l ~+~ ~ ~). (a) Let 7 ~ (0,2), ~ ~ 1. Then Bw(t) Moreover, in this case there exists a constant M = M (% a, B), independent of T, such that

IIBwff)llc,([o,rl;x) < Mllmtllv.~,~+, ~,.

(26)

243

Regularityof Solutionsof FractionalEvolutionEquations Let 7 e (0,1]. Then Bw(t_) e C([O,T];X) nB([O,T];Z)B(%oo)) D(B~--~-a) and BrDawl E DB(7, oo). Moreover, in this case there is a stant M = M(3", ~, B), independent of T, such that IIBw(t_)lt~(t0,Tl;Z).(~;~)) <_ MllB~-~.wlllv.(.~,~).

(27)

(c) Let7 ~ (0,2), 7 ¢ 1. ). Then Bw(t_) e hT([O,T];X) ifJwl e DB2(~(--(iT~ if fwl e :D(Br~z) an (d) Let 7 ~ (0,1). Then Bw(t_) C([O,T];gs(7)) B’-~w~ ~ In all cases of Lemma6, the assumptions made and Lemma5 imply that w(t) is a strict solution of (25). For the proof of Lemma6, see [3].

3

METHOD OF SUMS

AND NONHOMOGENEOUS

RESULTS

To analyze

D~z~+ Bz = h, z~(0) = 40) = o,

(28)

with h ~ 0, we use the method of sums of Da Prato and Grisvard [4]. The following Lemma[1, Theorem8] reformulates [4, Theorem 3.11]. LEMMA 7 Assume that f( is a complex Banach space and that ft, ~ are resolvent commutingpositive operators in f( with spectral angles CAand ¢~, respectively, such that ¢,i + ¢~ < ~r. I]? is one of the spaces T)~i(7,p), T),~(3’), Dh(7,p) or Dh(7), where3’ e (0, 1] and p ~ [1, oc], and if y ~ ?, then there is a unique x such that fix + ~x = y. Moreover, fix and ~x e ~ and there exists a constant C

such thatIlfixll~÷II*)xll~
To apply this Lemma,we write 2 d__e¢ Co([O, TI;X) ae=f {u C([0, T] ;X)lu(0) = 0 } and define/) in 2 by

v([~)= Co([O, TI;V(B)),(,?u)(_t) = B~(t),u e Then/) is a positive operator in ~’~" with spectral angle ¢~) = CB. For the interpolation spaces of/) one has, 3’ ~ (0, 1],

Weconsider the fractional

derivative

as an operator ~+~ in 3~ by

Z)(fi~+~) = { u ~ 21u, ~ 2,g~_~ * u, ~ C~([O,T]i X),(DTu,)(O) = 0}, (2~+,u)(t) de___f D~ut(t), u ~ Z)(ft,+,),

a e (0,1).

The equation (28) can then be written fil+az +/)z = h. In addition, define fi~

244

Cli~ment et al.

and ,z]~ for a e (0, 1) 73(~i.a) dej { u efi [gl-~ *u e CI([O,T];X),(D~u)(O) 0},

Concerning the operators ~1+~, a ¯ [0, 1), one has LEMMA 8 (a) ¢il+a is a positive,

densely defined operator with spectral angle

~(1 + a) and~i1+~= (A~)~+~. (b) For ~/¯ (0, 1) and (1 + c~)~/¯ (0, onehas

v~,÷o(~, ~) = { f l Y¯ c(1+~)’([0, T]; X),y(0) For ~),z~i~_ ~ ~1 ¯ (0, 1) and (1 + c~)~/¯ (1, onehas (~, (X)) ~--* { f [ f ¯ C(1+°~)0([0, T]; X); f(O) = if(0) = (c) For ~? ¯ (0, 1) and (1 + c~)r/¯ (0, 1) U (1, onehas [~ hO+~)"([ 0, T]; X). ~)al+o (O) = ~D.z~1+ o (~’ (~)) The corresponding statements for ~, 0 < a < 1, were given in [1, Lemma11]. Proof of Lemma8. (a) By [1, Theorem 10(b) and Lemma11],

~1+c~

= ~v~l

-----

(.~)~ = (.~l)l+a. To see that ¢A1+o-< ~(l+a) one applies [10, Prop. 4]. that ¢~i~+o < {(1 + a). By [13, Proposition 2.3.2], and as .~1+~ = (.41) ~+~, one then has ¢~i~ < {. By this contradiction, CA~+.= {(1 (b) and (c) follow by [1, Theorem 10(c) and Lemma11] and by the Reiteration Theorem. [] A combination of Lemma7 and Lemma8 immediately implies (b) and (d) Of following Lemma and also (a), (c), if 7 ¯ (0, 1 + a), To prove (a), (c), if ~, ¯ [1 + a, 2), apply ~ to (28), with e ¯ (7 - 1 - a, use [1, Theorem10(c)], and thus reduce the problem to a case already known. LEMMA 9 Let (i) and (ii)

of Theorem 2 hold. Then the following is true:

(a) Let h ¯ C’¢([0, T];X) with 7 ¯ (0,2), 7 ~t 1. /f7 ¯ (0, 1), assume h(O) := If 7 ¯ (1,2), assume h(O) = h~(O) = O. Then there exists a unique strict solution z of (28) such that Bz(t_) CT([0, T] ;X). Moreover, in thi s cas e there exists a constant M = M(7, c~, B, T) such that

(b) Let h ¯ .(" ~I3([O,T];i~B(7,~x~)), 3’ ¯ (0, 1]. Then thereexists a unique strict solution z of (28) such that Bz(t) ¯ .~C~/3([0, T]; ~z(3’, co)). Moreover, in this case there is a constant M= M(7, a, B, T) such that IlBz(t)ll~([o,T];v,(.r,~))

Regularityof Solutionsof FractionalEvolutionEquations

245

(c) Let h ¯ hT([O,T];X) with 7 (0,2), 7 ~ 1. /f7 ¯ (0 ,1 assume h(O) = ff-~ ¯ (1, 2), assume h(O) = At(O) = O. Then there exists a unique solution z of (28) such that Bz(t_) ¯ h~r([O,T];X). (d) Let h ¯ Co([0, T];7:)~(7)), with 7 ¯ (0, 1). Then there exists a unique solution z of (28) such that Bz(t) ¯ Co([O,T];I)B(7)).

4

AN INTERPOLATION

LEMMA

For the analysis of necessary conditions for solutions of (1) to exhibit a specific behavior one needs the following version of earlier interpolation results [2, Lemma 3], [9, Prop. 2.2.12]. Below, CI (I, X) denotes Lipschitz-continuous functions defined on I with values in X and C2 1. denotes functions having the first derivative in C LEMMA 10 Let X and Y be Banach spaces that are continuously injected in a Hausdorff locally convex topological vector space. Let I be a closed, bounded interval and let f ¯ Cc~(I,X) N C$(I,Y), where &,~ (0 ,2]. Then f ¯ C(1-°)5+°~(I, (X, Y)o,~) for each (0, 1) and

fl c,,-o)~,+o~(z.(x,Y)o,~) 2(llfllc~(~;x) + 2allIa-asup~e~llf’(t)llx,i/&> l,~ 1, &g 1. (29) + 2~1II Moreover, i] & ¢ ~, and (1 - 0)& + 0n = 1, then f’ ¯ B(I, (X, Y)~,oo) (29) holds with {If~l{~(1.(x,y)~.oo) on the le~ side o] the inequality. If f ¯ hS(I,X) ~ h~(I,Y), then f ¯ hO-e)~+e~(I,(X,Y)e) ~ ¯ (0, 1 ). Moreover, if ~ is such that (1 - ~)5 + ~1~ = 1, and & ~ ~, then f’ ¯ C(I, (X, Proof of Lemma10. For the relation (29), see [2, Lemma3]. The fact that (29) holds with I]f’llt~(~,(x,y)~,~) on the left side (under the stated assumptionson &, is contained in the proof of [2, Lemma3]. Let f ¯ h~(I,X) ~ h~(I,Y). Then choose f~ ¯ C°°([,X ~ Y) such that f in C~(I,X)~C~(I,Y). By (29), f~ -4 f in CO-°)(~+~(I,(X,Y)o,~), and if & ¢ ~, (1-O)&+O~ = 1, then t , - 4 f ’ i n B(I,(X,Y)e,oo). M oreover, f hO-~)c~+e~(I,(X, Y)e) and f~,~ ¯ C(I, (X, Y)e). But these last spaces are closed in CO-e)~+e$(I, (X,Y)o,oo) and C(I, (X, Y)0,oe), respectively, and so the last part the Lemmafollows. [] Let us briefly indicate an immediate use of the above Lemmafor the first step of the converse analysis. Weonly consider case (a), and take f = By (1), and assuming Bu ¯ C~([0, T];X), 7 ¯ (0,1), one D~(ut - u~ ) C~’([O,T];X). Then u ¯ C~+a([0, T];X). Apply Lemma10 with & = 1 + a, ~ = o- ~+~_~ ~ ¯ (0, 1), r = T)(B) and f = u. This gives = ~’(o) ¯ v.(~,~).

246

Cb~ment et al.

Then, by (a) of Lemma6, Bw E {: 1+o-~. By the relation Bu = Bv + Bwone has Bv E ~; 1+~-~. Thus, by (a) of Lemma4, we arrive at BUoE "DR((I+~_.~)(I+~)

REFERENCES 1. 2.

3.

4. 5.

6. 7. 8. 9. 10. 11. 12. 13. 14.

Ph. Clement, G. Gripenberg and S-O. Londen, Schauder estimates for equations with fractional derivatives, Trans. A.M.S. (to appear). Ph. Clement, G. Gripenberg and S-O. Londen, H6Ider regularity for a linear fractional evolution equation. In Topics in Nonlinear Analysis, The Herbert AmannAnniversary Volume (1998), Birkh~iuser, Basel, 69-82. Ph. Clement, G. Gripenberg and S-O. Londen, Regularity results for some fractional evolution equations, Helsinki University of Technology, Institute of Mathematics, Research Report A 413 (1999). G. Da Prato and P. Grisvard, Sommesd’op~rateurs lindaires et dquations diffdrentielles opdrationnelles, J. Math. Pures Appl. 54 (1975), 305-387. G. Da Prato, M. Iannelli and E. Sinestrari, Regularity of solutions of a class of linear integrodifferential equations in Banachspaces, J. Integral Eqs. 8 (1985), 27-4O. G. Da Prato and E. Sinestrari, Differential operators with non dense domain, Ann. Scuola Norm. Sup. Pisa C1. Sci. (4) 14 (1987), 285-344. P. Grisvard, Commutativit~ de deux foncteurs d’interpolation et applications, J. Math. Pures Appl. 45 (1966), 143-206. P. Grisvard, t~quations diff~rentielles abstraites, Ann. Sci. l~cole Norm.Sup.(4) 2 (1969), 311-395. A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkh~user, Basel, 1995. S. Monniaux and J. Pr/iss, A theorem of the Dore-Venni type for noncommuting operators, Trans. A.M.S. 349 (1997), 4787-4814. J. Priiss, Evolutionary Integral Equations and Applications, Birkh~iuser, Basel, 1993. E. Sinestrari, On the abstract Cauchy problem of parabolic type in spaces of continuous functions, J. Math. Anal. Appl. 107 (1985), 16-66. H. Tanabe, Equations of Evolution, Pitman, London, 1979. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.

Infinite Horizon Riccati in Nonreflexive Spaces

Operators

WOLFGANG DESCH and WILHELMSCHAPPACHERUniversitgt tut fiir Mathematik, Heinrichstrasse 36, A-8010 Graz, Austria

Graz,

Insti-

EVA FA~ANGOV~ and JAROSLAV MILOTA Department of Mathematical Analysis, Charles University, Sokolovska 83, 18675 Praha 8, Czech Republic

ABSTRACT For the least square control problem with an unbounded controller and a bounded observer we prove that the optimal control is of feedback type also for infinite time horizon and a non-reflexive state space. This feedback is given by a Riccati operator which is a unique positive and symmetric solution of the algebraic Riccati equation.

1

INTRODUCTION

We consider the infinite

horizon control problem

Minimize J(u; xo) = []Cx(s)] 2 ~- lu(s)121 ds.

Partially supported by Spezialforschungsbereich F 003 "Optimierung und Kontrolle" Karl-Franzens-Universitgt Graz, grant GAUK 19/1997, and Aktion ~sterreich-Tschechische blik".

247

(1.1) at the Repu-

248

Deschet al.

over all u E L2(0, oc; U), subject ~ = Ax + Bu , z(O) = xo

(1.2)

Here A is the generator of an exponentially stable Co-semigroupin a non-reflexive Banach space X, and B is a bounded linear operator from a Hilbert space U into the extrapolated Favard class F-1. C is a bounded linear operator from the state space X into a Hilbert space Y. Since the range of B is not contained in X, this is a problem of unbounded control. The theory of the least squares regulator with unboundedcontrol in Hilbert spaces is well developed, see e.g. [1], [2] and references therein. It turns out that the formalism of the Riccati equation can be adapted to unbounded control and observation. The value function of the optimal control problem is a quadratic form (Pxo,xo), where P is a bounded, positive definite symmetric linear operator on X. In many cases the optimal feedback is synthesized by a feedback law u*(t) -B*Px(t), but there are intricate situations whenthis expression is not well-defined and the optimal control is not givenby a bounded state feedback. Such problems are likely to occur if the range of the control operator B is relatively large, or if an unbounded input B is combined with an unbounded observation C. Roughly speaking, bounded state feedback is a matter of comparing the range of P with the domain of B*. Our intention is to carry over the Riccati formalism to the case of nonreflexivc Banach spaces with the smoothest kind of unbounded control, namely control taking values in the extrapolated Favard class. This is only a very restricted class of unbounded control (it would be bounded in any reflexive space), but it appears frequently in semigroup settings for delay equations, integrodifferential equations, and age dependent population dynamics ([3]-[7]). Notice that existence and uniquehess of the optimal control is clear by quadratic opti~nizatiou in the Hilbert space L~(0, oo, U). The nonreflexive state space enters the scene only when we consider the relation between the optimal solution x and the optimal feedback. The Riccati equation is originally a Hilbert spaceconcept, but it can be rewritten in terms of the duality between X and X*. In nonreflexive spaces this duality is loaded with technical difficulties since the adjoint semigroupis usually only strongly continuous on a closed subspace of X*, called X~. Nevertheless, in a recent paper [4] we have shownthat the finite horizon control problem 7’

Minimizeg(u; x0, T) =/o [ICx(s)()" + lu(s)12] ds.

(1.3)

admits a Riccati operator P(t) which takes values in Xa), so that B*P(t) is well defined and the optimal control is giveu by bounded state feedback. In this note we derive a similar result for the infinite horizon problem. We~ni~nic an approach well-knownfrom the Hilbert space theory: Given the Riccati operator for the finite horizon problem, we perform a limiting procedure for T ~ oo, which leads to the Riccati operator for infinite horizon and the algebraic Riccati equation. In the next section we give a short overviewover the technical tools required, i.e. the concept of the extrapolated Favard class, and the Riccati equation for the fi~ite horizon control problem. In the subsequent section we state and prove the existence and uniqueness of the Riccati operator for the infinite horizon problem.

Infinite HorizonRiccati Operatorsin NonreflexiveSpaces 2

249

PRELIMINARIES

Let X be a Banach space with norm I I. Let A be the infinitesimal generator of a semigroup S(t) on X and let its growth rate be negative, i.e. S(t) is exponentially stable. Let X-x be the completion of X with respect to the norm Ixl_l :: IA-lx[,x E X. The Banach space X_I is said to be the extrapolated space with respect to the generator A (or the semigroup S(t)), see e.g. [7],[8] and references given there. Evidently, A can be extended from X1 (the domain of with the graph norm) to A_x : D(A_I) = X -~ X_~. Then p(A) = p(A-1) and (A - A_~)-1 is the continuous extension of (A - A)-x. Moreover, the semigroup S(t) has also a continuous extension to the C0-semigroup S_l(t) on X_~ and the generator of S_l(t) is A-1. The growth rate of S_l(t) is the saIne as for S(t), in particular it is negative. Wedefine the Favard class F of the semigroup S(t) by F := {x ~ X;limsupl S(t)x-x I < cx)} t t~o+ (see e.g. [8],[9]) and similarly the Favard class F_~ of S_~. Weintroduce the norm IzlF := supl S(t)x - x t I t>o on F and similarly IXlF_, on F_~. Obviously, A_a maps F isometrically onto F-t. If X is reflexive then E = D(A) ([8],[9]). For a non-reflexive space the adjoint semigroup S*(t) is not strongly continuous in general. As usual, X(v denotes the maximal closed subspace of X* where S*(t) is strongly continuous. By Theorem 3.2.6 in [8], F-1 can be imbedded into X(~*, and F_~ = X~:-)* whenever X is sunreflexive with respect to the semigroup S(t). The following well-known proposition is crucial for our formulation of control problems.See also [3],[41,[7],[8]. PROPOSITION 2.1 map

Let S(t) be a C0-semigroup on a Banach space X. Then the f~S-l*f:=

S_~(t- s)f(s)ds

is continuous from L~(O,T;F_~) into C([O,T];X) for all finite exists a constant c(T) such that

IIS-~*fllc([o,rl;x)

T > 0 and there

_< c(T)

On the basis of this proposition we define the mild solution of the equation (1.2) for B ~ £(U,F_I) and xo ~ X as follows t

x(t)

= S(t)xo

j~0

+ S-a(t

- s)Bu(s)ds

(2.1)

]l)eschet al.

250

Nowwe state the main result of [4] on the existence and properties of the differential Riccati operators. THEOREM 2.2 For each 0 < t < T < c~ there exists a unique operator P(T, t) £(X, °) satisfying t he f ollowing properties : (1) P(T, t) is symmetric in the sense that (P(T, t)x, z) = (P(T, t)z, for all x, zEX. (2) P(T, t) is positive in the sense that (P(T,t)x,x I >_ 0 for all x ~ X. (3) For each x ~ X, P(T, t)x depends continuously on t. (4) P(T, .) satisfies the following version of the Riccati equation: For each x, z ~ X, T

(P(T, t)z,

x) = ~ (CS(s - t)z,

CS(s

T

- f (B*P(T, s)S(s - t)z,

B*P(T, s)S(s -

Moreover, if P(T, t) is given as above, then the optimal control is determined by the feedback law ~t(t) = - B*P(T,t)2(t) and the value function of the optimal control problem is determined by P(T, .): (P(T,t)xo,x0)

= minJ(u;xo,

T-t),

where the minimumis taken over all u ~ L2(0,T - t; U).

3

ALGEBRAIC

RICCATI

EQUATION

The main result is the following theore~n. THEOREM 3.1 Let the following hypotheses be satisfied (HI) A is a generator of an exponentially stable Co-semigroup S(t) on a Banach space X; (H2) B E £(U,F_I) , U is a Hilbert space; (H3) C E £(X, Y), Y is a Hilbert space. Then there exists a unique P ~ £(X, @) satisfying t he f ollowing p roperties : (i) (Px, x)x.×.x" >_ 0 for all x e X ; (ii) (Px, y) = (Py, for allx , yGX; (iii) The following weak version o.f the algebraic Riccati equation holds for any x, y ~ F for which there are u, v ~ U such that A_ ~ x + Bu ~ X, A_ ~ y + Bv ~ X: (A_lx, PY)xe. ×xo + (A_iy, Px)xo. ×xo (Cx, Cy)~. - (B*Pz, B*Py)u

(13.1)

Infinite HorizonRiccati Operatorsin NonreflexiveSpaces

251

Moreover, the optimal control fi for ti~e problem (1.1),(1.2) is given by the feedback law ~t(t) = -B*P~(t) , (3.2) where ~ is the mild solution of ~=Ax-BB*Px

,

x(O)

=

(3.3)

For the value function the equality min J(u;

Xo)

=

(Pxo, Xo)

uEL2(R+;U)

holds for any z0 ¯ X. Proof: The proof is given in three steps. 1) Construction of P, Since the differential Riccati operator P(T, t) is symmetric and non-negative, the Cauchy type inequality [(P(T,t)x,y)l

2 <_ (P(T,t)x,x).(P(T,t)y,y)

holds for any x,y ¯ X. Because of the assumption (H1) we have T-t

(P(T,t)x,x)

= (P(T - t,O)x,x)

~, ICS(s)xl2ds<_cllxl j~O

where c~ is independent on t,T. Moreover, if T1 > T2 and u ¯ L2(O, Tr;U), then J(u; x, T2) ~_ g(u; x, TI). Consequently, (P(T2, O)x, x) <_ (P(TI, O)x, Therefore there exists lira (P(T, t)x, =: T--~oo

for all x ¯ X independently on t ¯ I~+. By symmetry we have 4(P(T, t)x, y) = (P(T, t)(x + y), x + y) - (P(T, y), ( x - y) so that we infer the existence of lira

(P(T,t)x,y)

=: a(x,y)

for any x, y ¯ X. The bilinear form a(x, y) is symmetric, non-negative and bounded. This means that there exists P ¯ £(X,X*) such that (Px,y) = a(x,y). We have shown that P(T,O) -~ in the weakopera tor topol ogy. By sy mmetry and monotonicity we get again a Cauchy inequality ](y,Px-

P(T,O)x)] ~ <_ (y, Py- P(T,O)y).(x, Px- P(T,O)x) 2. _< Cllyl (x, Px- P(T,O)x).

From this we deduce that P(T, O)x converges to Px in the strong topology, and as a consequence Px ¯ X~. Evidently the operator P satisfies (i),(ii).

252

Deschet al.

2) P solves (3.1). In (4, ibrmula (3.8)] we proved the following formula

(P(T, t)4, 4) = [ICx(s)l2 ÷ lu(s)l 2 -[~4,~) ÷ B*P(T, s)x(s)l’2]ds,

(13.4)

where x(.) is the mild solution of the problem &(s) = Ax(s) + Bu(s) , x(t) on the interval It,T]. Choose u E L~(I~+;U). Then the corresponding mild solution x E L2 Iq C(]~+;X), and P(T,s)x(s) -+ Px(s) pointwise in the norm of X® for T -+ oe. As B ~ £(U, F_~) and F_~ is continuously imbedded into °*, B*P(T, s)x(s) -~ B*Px(s). Because of the uniform boundedness of B*P(T, s), the dominated convergence theorem yields [ICx(s)l 2 + lu(s)l 2 -lu(s) + U*Px(s)12]ds

(Px(t), x(t))

(3.5)

Suppose now that Xo e X is such that there is Uo E U for which A_~xo + Buo ~. X. If u ~ W~’~(ll~+, U),u(O) = Uo, then the corresponding mild solution x is a classical one (Proposition 4.3 in [4]). Therefore, we have (using the symmetry of P (3.5))

~

(Px(t),x(t))

= 2(Px(t),A_~x(t) -= -ICx(t)l ~ - lu(t)l ~ + lu(t) + B*Px(t)]~ ¯

Especially, for t = O, 2(Pxo, A-~xo + Buo) = -ICxo] ~ - luol 2 + ]uo + B*Pxo]2 .

(3.6)

Evaluating (3.6) for x + y instead of x0 and subtracting (3.6) for x and y, we

(Px, A_~y + By) + (Py, A_~x + Bu) -(Cx, Cy) + (B*Px, v) (B*Py, u) (B*Px, B*Py) for any x,y ~ F,u,v ~ U such that A_~x + Bu ~ X,A_~y+ Bv ~ X. Further, (Px, A_~y + Bv)xex.~: = (A_~y, Px).,~.x.~

+ (Bv, Px).~.xX~

and (By, Px)xe. x.~e = (B*Px, v)u and (iii) follows. 3) Uniqueness. The equation (3.3) has a mild solution as the linear operator F : C([v, v + T]; C([T, "r + T]; X) defined by Fx(t)=

S_~(t-s)BB*Px(s)ds,

te[r,r+T],

(~.7)

253

Infinite HorizonRiccati Operatorsin NonreflexiveSpaces

has the norm [IFII < 1 for sufficiently small T independently on 7 E ll~ +. From (3.5) we infer immediatelythat IPxo, xo) is the value function of the infinite horizon problem, and that the optimal control is obtained by the feedback ~t(t) = -B* P~(t). Nowlet Q ¯ £(X, (~) s atisfy ( i)-(iii). I f x is a c lassical sol ution to (1. 2) for u E W1’2, then we obtain after a simple computation

: 2(Qx(t),A_x(t) + Bu(t)) : -tCx(t)l- lu(t)l + lu(t) + B*Qx(t)l Taking into account the exponential stability of S_1, we get limt-~c~ (Qx(t), x(t)) 0, and by integration,
+ B*Qx(t)12]dt

(3.8)

The validity of (3.8) can be easily extended to any x0 ¯ X and u E L2(I~+;U). This shows again that (Qx, ~:) is the value function, so that (Qx, x) = (Px, for all x E X. Since P and Q are symmetric, we obtain Q = P. COROLLARY 3.2 Let P be as in Theorem 3.1.

Then

(i) there exists a continuous extension P_~ of P, P-1 : F_I ~ X* (ii) P maps:D(A) into :D(A*) (iii) for x ~ T)(A) the algebraic Riccati equation PAx + A*Px + C*Cx - P_~BB*Px = 0

(3.9)

is satisfied. Proof : (i) Since F_~ is continuously is a linear continuous form on X and that (x, Py}xe.×x~ = {P-~x,y}x.×x £(F-1, X*) and (P-~Bu,y)x. xx (ii) and (iii)

imbedded into °*, t he map y~ (Py, x) therefore there exists P_~x ~ X* such for y ¯ X,x ~ F_~. Moreover, P_~ ~ = (u,B*Py}[~

x,y ¯ ~) (A),u = v = 0.From(3.7)and

(3.10)

(a.10)we have

(Px, Ay) = (P_IBB*Px, y) - (C*Cx, y) - (PAx,y) i.e. Px e 7?(A*) and (3.9) follows. ACKNOWLEDGMENT The authors thank to L.Maniar for pointing in the previous version of this contribution.

out the error

REFERENCES 1.

B.van Keulen. H~-Control for Distributed Systems : A State Approach, Systems & Control : Foundations & Applications, Birkh/iuser, 1993.

254 2.

3.

4. 5. 6. 7.

8. 9.

Deschet al. I.Lasiecka, R.Triggiani. Differential and Algebraic Riccati Equations with Applications to Boundary/Point Control Problems: Continuous Theory and Approximation Theory, Lecture Notes in Control and Information Sciences 164, Springer Verlag, 1991. W.Desch, W.Schappacher. Somegeneration results for perturbed semigroups. In: P.Clement, S.Invernizzi, E.Mitidieri, I.Vrabie, eds. SemigroupTheory and Applications. Lecture Notes in Pure and Applied Mathematics 116, M.Dekker, 1989, pp. 125-152. W.Desch, J.Milota, W.Schappacher. Least square control problems in nonreflexive spaces, to appear in Semigroup Forum. O.Diekmann, S.A.van Gils, S.M. Verduyn Lunel, H.-O. Walther. Delay Equations. Functional-, Complex-, and Nonlinear Analysis. Springer Verlag, 1995. G.Greiner. Perturbing the boundary conditions of a generator. Houston J. Math. 15: 527-552, 1987. R.Nagel, E.Sinestrari. InhomogenousVolterra integrodifferential equations for Hille-Yosida operators. In:Lecture Notes in Pure and Applied Mathematics 150, M.Dekker, 1994, pp. 51-70. J.van Neerven. The Adjoint of a Semigroup of Linear Operators. Lecture Notes in Mathematics 1529, Springer Verlag, 1992. P.L.Butzer, H.Berens. Semi-Groups of Operators and Approximation. Springer Verlag, 1967.

A Hyperbolic Variant Simon’s Convergence

of Theorem

Alain HarauxCNRS Laboratoire d’Analyse Num6rique,Universit6 P. et M. Curie, Paris, France

The main object of this report is convergence to an equilibrium of weak global energy-bounded solutions of the wave equation utt + cut - Au + f(x,u) where ~2 is a bounded smooth domain of ~v, c > 0 and

is a function analytic in s satisfying some relevant growth conditions. Such a result has been obtained recently in a joint work with M.Jendoubi [8] as a culminating point of our attempts to understand in depth the fundamental theorem of L. Simon [17] on convergence of bounded trajectories of semilinear parabolic systems with analytic nonlinearities. The convergence problem for gradient or gradient-like systems has already a rather long history starting around 1960 with some investigations of S. Lojasiewicz on the steepest descent problem in differential geo~netry. Even in finite dimensions the question is more complicated that it maylook at first sight. S. Lojasiewicz [12, 13] was led to the idea that analyticity of the potential is the key to convergence and, under this condition, established a topological property of stationary points which allows to derive convergence via a simple differential inequality. Muchlater, in 1982, J. Palls and W. de Melo [15] somehowconfirmed the necessity 6f the Lojasiewicz condition which may fail even for C~ gradient systems. Very soon after, in 1983, L. Simon achieved his remarkable work on parabolic problems. In tile mean time, Tile convergence property had been established for various gradient and quasi-gradient flows by B.Aulbach [1], H. Matuno[14] and T.J. Zelenyak [18]. The dimension condition used in the proofs of Aulbach and Matano has been later exploited in a quite general context by Hale-Raugel [3] and allows convergence proofs under standard regularity assumptions, in particular analyticity is not needed. However, apart from the case of thin domains, in higher space dimensions 255

256

Haraux

this hypothesis is not very natural. Also the counterexalnple of [15] suggests that we have to be careful even when dealing with the heat equation. And maybe there is something really deep hidden behind this strange-looking Lojasiewicz Lemma. The hyperbolic case studied below as well as a recent result of E. Feireisl and F. Simondon [2] for degenerate parabolic equations seem to confirm once more the interest of this simple intuitive idea: analyticity of the potential has a tendancy to kill small oscillations, even if the solution itself is not analytic!

1

FINITE

DIMENSIONAL

GRADIENT

SYSTEMS

In this section we consider the finite dimensional Euclidian space X = l~ N with usual norm denoted as I.I and a potential field F : tt: N ~ l~. For simplicity we assume F N) E C201~ and we denote by VFthe gradient of F. Weconsider the first governed by the equation u’+VF(u)=0,

order gradient system

tk0

(1.1)

First we recall a simple well-knownfact on bounded solutions of (1.1) PROPOSITION 1.1 If u is any 91obal bounded solution

of (1.1),

we have

lim dist(u(t),g) with £={aeX,

VF(a)=0}

PROOF First we notice that F is a Liapunov function of (1.1). Moreprecisely for any solution u of (1.1) we have

-lu’(t)l~ <_ ~F(u(t)) =

0

(1.2)

Since u, and therefore F(u) is bounded, (1.2) implies (1.3)

u’ E. L~(O, +e~; X) Then by differentiating

the equation u" = -V~F(u)u ’ ~ L~(O, +oo; X)

(1.4)

and therefore

- lu’(t)l

(1.5)

e

By (1.5) the function p(t) I¢(t)l ~ te nds to a l imit as nonnegative and p ~ L~ (0, +oc; X), we conclude that lim (u’(t))

t--~+~x)

t

Since p is

HyperbolicVariant of Simon’sConvergenceTheorem

257

This last property implies the result sihce then by the equation, any limiting point of u(t) as t -~ +e~ belongs to g = (a E X, ~7F(a) 0}. REMARK 1.2. It is natural

to wander whether in fact we have

lim u(t) = a £ The situation is as follows 1) If N > 1, the answer is negative in general. More precisely, if N = 2 the omega-limit set of a bounded trajectory can be a full circle even for F ~ C°°, of. Palls-De Melo [15 ]. 2) If N = 1 we have convergence since any non constant trajectory is monotone. If N > 1 we have convergence assuming that F is (real) analytic. THEOREM 1.3 (Losjasiewicz) Let F : l~ N -~ ]~ be real analytic and let u be any boundedglobal solution o[ (1. I). There exists a ~ ~ such that lim lu(t) - aI = 0

(1.6)

The proof of theorem 1.3 relies on the following deep lemmadue to Lojasiewicz [12, 13 1. LEMMA 1.4 (Losjasiewicz) Let F : NN _.+ ~ be real analytic and let a ~ g. There are two real numbers (depending on a in general)

oe(o,1/2); such that

Vue X,lu - al _IF(u)~-° F(a)l

(1.7)

The proof of Lemma1.4 is quite involved and is based on the theory of real analytic manifolds. On the other hand, it is very easy to understand why (1.7) not satisfied in Palls-De Melo’s counterexample, and the proof of Theorem1.3 is a simple consequence of Lemma1.4. Part of the depth of Lemma1.4 is contained in the quite non-trivial and essentially global information that any critical point of F close enough to a has to be at the same energy level as a. Indeed when VF(u) = and lu - a t _< r], (1.7) gives F(u) = F(a). On the other hand there is apparently no way of globalizing (1.7) completely. It would be very useful, especially in view infinite dimensional generalizations, to find either a non-constructive elementary proof by contradiction or (even if it is very complicated) a constructive method allowing to determine 0 in terms of F and a. PROOFof Theorem 1.4 (admitting Lemma1.3) Let a be any limiting point u(t) as t -~ +oo. As a consequence of Proposition 1.1, we have a ~ £. Introducing v = u - a and C(v) = F(u + a) -

258

Haraux

we may assume that in fact a=0 After this reduction,

F(0)=0

(VF)(0)=0

since F(u(t)) is nonincreasing and a = lim U(tn) for some

sequence tn tending to infinity,

we have Vt > 0, F(u(0) >_

(L8)

Also by (1.2) we have also , since ~ =-VF(u)

= -I-’(t)llVF(-(t))l In addition, either F(u(T)) for someT fin it e, in wh ichcase F(u(t) ) = 0 for all t k T and as a consequence of (1.2) we conclude that u(t) ~ a. Or F(u(t)) > 0 for all t > 0. In this last case ~(-[F(u(t) °) : -O[F(u(t) )] °-’ ~ F(u(t) ) 0 IVF(u(t ) ~-° )llu’ [F(u(t))] and by (1.7) this implies

°) >Olu’(t)l ~(-[F(u(t))]

(1.9)

on any interval of (0, +oc) on ~vhich the condition

lu(t)l_< is fulfilled. Inequality (1.9), implying an ~ bound on u ~, i s q uite s uggestive o f convergence. To complete the proof we select s > 0 such that

1 sup[F(s)l° < ~

and

~ Isl<~

~

(1.10)

Then let tl > 0 be such that lu(h)l <_

(1.11)

aud T := sup{t~ > t,,

[u(t)l

~, Vt E ( t, ,t~)}

By using (1.8) for t = te we find, integrating (1.9)

’¢t.~z (t,,T),

t~ 9~t

1 r/ lu’(t)ldt < ~[F(t,)]° < ~

If we assume combining(1.10), (1.11) and (1.12) , we obtain by letting t2

N(T)I
(1.12)

259

HyperbolicVariant of Simon’sConvergenceTheorem a contradiction

with the definition

of T. Hence T = +~ and lu’(t)ldt <

In particular a(t u’ ~, E +~) L and ’a(t) is convergent as t

2

A RESULT

OF L.SIMON

L. Simon[17] cousidered in particular the semilinear heat equation ut-Au+f(x,u)--O u=0 where ~ is a bounded smooth domain of

inR+ onl~ + ×Ogt l~ N,

C

×~

(2.1)

> 0 and

f : f~ x IR ---+ I~ is analytic in u, uniformly with respect to x ~ f~ He obtained THEOREM 2.1 (L. Simon) Let u be any bounded global solution on [0, +co) × ~. There exists ~ ~ g such that lim Ilu(t)

- vlI~ = 0

of (2.1) bounded

(2.2)

with

$ = e C( )n HI(a)/ - zs f( x,o) ina} The proof of Theorem2.1 is quite subtle and we shall not indicate its details. It relies essentially on the Lojasiewicz Lemmaand most of the difficulty consists in extending this Lemmaon an infinite dimensional setting. Although Lemma1.4 has an essentially global character since it involves the nonlinear energy functional, Simon has used the linearized equation around an equilibrium ~o ~ g. Fredholm’s alternative for the linearized stationary operator allowed him to apply Lemma1.4 to an auxiliary finite-dimensional nonlinear potential. Then he was able to come back to the concrete energy functional modulo a translation procedure replacing ~ by 0. The end of proof is parallel to the proof of Theorem 1.3, replacing the potential F(u) by the energy functional

= 1/~lVul2 with

dx + /~F(x,u)dx

(2.3)

It

j~0

F(x, u) := f(z,

s)ds

(2.4)

260

Haraux

Here note that even if f does not depend on x in the original equation (2.1), fact it does after the translation procedure since f(u) is replaced by g(x,u) f(u ÷ ~(x)) - f(u) REMARK 2.2 1) The original proof of Theorem 2.1 [17] seemed to use quite strong regularity properties of the solutions. Someprecise remarks on the natural functional setting will be given in Section 4. 2) The same result is no longer true if f is Ck with respect to u, cf. P. Polacik and K.P. Rybakowski[16] for a counterexample with f depending on x and k arbitrarily large. However their technique depends essentially on the x-dependence and they cannot handle the C~ case.

3

SECOND ORDER GRADIENT-LIKE

SYSTEMS

In this section we consider the second order differential system governed by the equation u" + c~’u’ + ~F(u) 0, t _ > 0 (3 .1) For simplicity we assume N) F e Cz(I~ and we denote by VF the gradient of F. First we have a simple general result whoseproof is essentially identical to the proof of Propositon 3.1 PROPOSITION 3.1 If u is any global bounded solution

of (1.1),

we have

lim dist(u(t),£) with ~ = {a ~ X, VF(a) = 0} PROOF First we notice by the equation, boundedness of implies that u’ and u" are also bounded. In addition for any solution u of (1.1) we have d~{ lu’(t)] 2 + F(u(t))} = -lu’(t)l 2 < 0

(3.2)

Since u, and therefore F(u) is bounded, (3.2) implies u’ ~ L~(0, +~c;X) Then by differentiating

the equation

u’" + au" = -V~F(u)u ’ e L~(0, +~c; X) Multipying by u" and integrating on [0, t) we deduce that u"e L~(0, +~x);

(3.3)

HyperbolicVariant of Simon’s ConvergenceTheorem

261

Therefore ~-~]ud’(t)l s = 2(u’, u") E t (0, +ec; X) The rest of the proof is identical to that of Proposition 1.1. REMARK 3.2. It is natural

to wander whether in fact we have lim u(t) = a E

If N = 1 we have convergence (cf. e.g. [4, 5]). If N > 1, the answer is probably negative in general. In [7] the following result w~ obtained THEOREM 3.3 Le¢ F : ~N ~ ~ be real analytic solution of (1.1). There exists a ~ ~ such that

and let u be any bounded global

lim ~u(~) - a~ =

To prove this result,

(3.4)

we had to consider the modified Liapunov function

H(t) = ~lu’(t)l ~ + F(u(t)) + e(VF(u(t)), For e > 0 small enough, using Lemma1.4, it can be shown that for t large ~t {(H(t)) °} _< -O-~(lu’(t)l + IVF(u(t))l)

(3.5)

Then the proof parallels the first order case.

4

THE MAIN RESULT

Wenow consider the second order evolution problem utt+cut-Au+f(x,u) u=0

=0 inlR + x~ on~ + )
(4.1)

where Ft is a bounded smooth domain of ll~ ~v, c > 0 and f:Ft

x~--~

In Ill] M.A. Jendoubi adapted to equation (4.1) the method of L. Simon [17] for infinite dimensional systems of parabolic type (cf. also [10] for an abstract reformulation of Simon’s results under natural regularity conditions). He proved the following result.

262

Haraux

THEOREM 4.1 Assume that f is analytic in s, uniformly with respect to x Of (x,s) and 02f ~ and f(x,s), ~s2 (X,S) are bounded in ~ × (-/~,/~) Vfl > O. Let a solution of (1.1) and assume that there exists p >_ 2 such that t3 {u(t, .),ut(t, t_>l

.)} is precompact in W2’P(Ft) W"P(~)

(4.2)

with p >N-~ if N<_6, and p > N if N>6. Then setting $ = {V E H’~(~) there exists ~ E S such that lim {l[utllw,.,(~) Il u(t, .) ~(. )l lw~.,(n)} = 0. This result is interesting because no growth condition on f is required, on the other hand it is only applicable to strong solutions and the condition of boundedness in W~,~(~) × WLP(~)is restrictive and not always easy to check in practice. hypothesis was motivated by the necessity of using an infinite dimensional extension of the classical Lojasiewicz inequality. In fact when trying to extend the method of 3, in a first approach it is natural to .consider, after replacement of u by u - ~, the functional

=dx ¯ (t)=~ I~1

+ E(~)+~ f~ [-/~u+ f(x, ~)].~

with E(u) = l ~¢ lVulU dx + L F(x, u)dx as a possible Liapunov functional. ¯ ’(t)

~( -c~+~f’(x,u))[ut[~-dx+~

= ~(-~+~]’(x,u)),ut,~dx-~

Howeverthe computation gives jf ~[-Au + f( x,u)]uttdx+~/~ ,V

ut,~dx

,-Au+f(x,u),2dx-a~[-Au+f(x,u)]utdx

The three firs~ terms fi~ just OK,bu~ the las~ one is of higher order wi~h ~he bad sign.In order to overcome~his di~culty, Jendoubi considered ~he modified functional H(t) = ~(t) + 6 ~ IVutl2 dx + ~ ~1- Au + f(x,u),:

dx + ~ ~ f’(x,u),ut,

By using (4.2), he proved that for ~ small enough and t > H’(t)

~ -¢

f (iw,,i +1-+ f(x,u)l2} dx

~ dx.

263

HyperbolicVariant of Simon’s ConvergenceTheorem To conclude the proof of Theorem4.1, Jendoubi used the following result

LEMMA 4.2 Under the hypotheses of Theorem 4.1, let ~ E S, then there exist 9 ~ (0, ½) and a > 0 such that Vu ~ W2’P(12) n H~(~), []u- ~l]2,p < a implies

II - ~xu+ f(x,u)lb:(m> 1-a 6lE(u)- E(~o)l

(4.3)

for some ~ > O. Actually (4.3) does not look too natural since when f = 0, the energy is comparable to the square of the norm of -Au in H-l(f~) rather than L2(f2). As a matter of fact, in [8] we established LEMMA 4.3 Assume that f is analytic in s, uniformly with respect to x ~ 12 and either N = i and f(x,s), and ~-~/(x,s) are bounded in ~ × (-~,~3) for all l~ > or N >_ 2, f(x,O) ~ L~(~t)

I-~(~,s)l < c(1 + Isl~) a.~.on~ × (-~,~)

(4.4)

for some C >_ 0 and c~ >_ 0, (N - 2)~ < 2. Given ¢: ~ S, then there exist ~ e (0, ½) and a > 0 such that Vue H~(f~), Ilu - ~llH~(a) < a implies

II- ~Xu+f(x,u)llH-,(~) 61E (u) (~ -~-° E

(4.5)

for some ~ > O. From Lemma4.3, following the strategy of the previous proofs, we obtained the following final result THEOREM 4.4 Assume that f is analytic in s, uniformly with respect to x ~ f~ and either N = 1 and f(x, s), and °o~s(x, s) are boundedin 12 x (-l?, 17) for all/? O, or N >_ 2, f(x,O) ~ L~(~) ]-~(x,s)l

_< C(1 + Isl ~) a.e.on f~ x (-oe,c~)

(4.4)

/or some C >_ 0 and a >_ O, (N - 2)a < 2. Let u be a solution of (1.1) such that ~ {u(t, .),ut(t, t>_o

.)} is boundedin go~ (~t) x L2(12).

Then there is a solution qo of

such that lim {[[Ut[IL:(~)flu(t, .) -- qO(.)[[So~(~)} = O

(4.6)

264

Haraux

REFERENCES 1. 2. 3. 4. 5. 6.

7.

8. 9.

10. 11.

12. 13.

14. 15. 16. 17. 18.

B. Aulba.ch, Approach to hyperbolic manifolds of stationary solutions, Springer-Verlag Lecture Notes in Math. 1017 (1983), 56-66. E. Feireisl ~z F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions, to appear (1999). J. Hale & G. Raugel, Convergence in gradient-like systems with applications to PDE, Z. Angew. Math. Phys. 43 (1992) 63-124. A. Haraux, Syst~.mes DynamiquesDissipatifs et Applications, R.M.A. vol 17 (Masson, Paris, 1991). A. Haraux, Asymptotics for some nonlinear O.D.E. of the second order, Nonlinear Anal. TMA10 (1986) 1347-1355. A. Haraux, "Semilinear hyperbolic problems in bounded domains", Mathematical Reports, vol.3, Part.1 (Edited by J.DIEUDONN15,).Harwood Academic Publishers, NewYork (1987). A. Haraux & M.A. Jendoubi, Convergence of solutions of second-order gradient-like systems with analytic nonlinearities. Journal Diff. Eq 144 (1998), 313-320. A. Haraux & M.A. Jendoubi, Convergence of solutions of the wave equation with analytic nonlinearities. Calculus of variations and PDE,to appear (1999) A. Haraux & P. Polacik, Convergence to a positive equilibrium for some nonlinear evolution equations in a ball, Acta Math. Univ. Comeniane, 2 (1992) 129-141. M.A. Jendoubi, A simple unified approcb to some convergence theorems of L. Simon. Journal Funct. Anal. 153 (1998), 187-202. M.A. Jendoubi, Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity, Journal Diff. Eq. 144 (1998), 302-312. S. Lojasiewicz, "Ensembles semi-analytiques", I.H.E.S. notes (1965). S. Lojasiewicz, Une propri~t6 topologique des sous ensembles analytiques r6els. Colloques internationaux du C.NIR.S #117. Les dquations aux d6riv6es partielles (1963). H.Matano, Convergence of solutions of one-dimensional semilinear heat equation, J. Math. Kyoto Univ. 18 (1978) 221-227. J. Palis, W. de Melo, Geometric theory of dynamical systems: An introduction Springer-Verlag, New-York(1982). P. Polacik, K.P. Rybakowski, Nonconvergent bounded trajectories in semilinear heat equations, J. Diff. Equa. 124 (1996), 472-494. L. Simon, Asymptotics for a class of non-linear evolution equations,with applications to geometric problems, Ann. of Math., 118 (1983), 525-571. T.J. Zelenyak, Stabilization of solutions of boundaryvalue problems for a second order parabolic equation with one space variable, Differentsial’nye Uravneniya 4 (1968) 17-22.

Solution of a Quasilinear Parabolic-Elliptic Boundary Value Problem V. PLUSCHKEDepartment of Mathematics Luther-University, Halle, Germany

1

and Computer Science,

Martin-

INTRODUCTION

In this paper we consider the following parabolic-elliptic interface problem: Let f~ E ll~ N, 1, N _> 2, be a simply connected, bounded domain with boundary 0~ E C and I = [0,T]. For given nonnegative g = g(x,t,u) : ~ × I × R -~ II~ we define f~e = fl \ suppzee g(x, t, u) ,

f~p = fl \ f~e ,

F = Of~e n Of~v

and assume that £te, tip are independent of t and u. Then we look for a weak solution of

in Q/,:= f~ × I, on rT:= r × I, on BT:: 0~ × I, x e f~p,

g(x, t, u) u, + A(t)u = f(x,

u(x,t) = u(z,O)= Vo(~)

(1) (2) (3) (4)

N where up =u[~,u~ utah, and A(t)u =-~,~__, = (~(x,~)~) +~0(~,~)~. Since g = 0 for x ~ ft~ the problem is elliptic on ~t¢ × I and parabolic on tip × I. Parabolic-elliptic interface problemsdescribe eddy currents in applications to electromagnetic field theory (cf. MacCamy and Suri [5]). The linear parabolic-elliptic problem of type (1)-(4) is investigated by a number of authors (cf. l, [21, [3 ]) who derived existence and uniqueness under rather general assumptions for g. A parabolic-elliptic interface problem with special nonlinear monotoneoperator A is solved by Zlmal [10]. Wenow investigate the problem with nonlinear dependence of f and g on u. Unlike in [10] even on f~p × I the coefficient g must not be bounded below by a positive constant. Since supp.~e~ g(x, t, u) maycontain sets of zero measure where g = 0 the parabolic problem on f’tp × I may degenerate. The decrease to 265

266

Pluschke

zero ofg is restricted by a condition on 1/g belonging to L~(flp) for all t E I,u E I~. Moreover, our technique is not restricted to Hilbert spaces. Hence we can prove stronger regularity results, e.g. almost everywhere boundedness with respect to t of ut on fp × I and even continuity of the solution u in QT for nonsmooth data. To construct the solution we use se~nidiscretization in time (Rothe’s method) that approximates the quasilinear degenerate problem by a set of linear nondegenerate uniformly elliptic problems. This method was also used by Ka~ur [3, Chapter 6.2] to solve the linear problem in somewhatlarger spaces. For f¢ = q} our problem also includes degenerate parabolic equations. However, in that case we refer to [6] for stronger results. In the present paper we do not suppose most general assumptions on the data in the parabolic region. Finally, note that our degeneration in equation (1) differs essentially from the degeneration b(u), considered by manyauthors (cf. e.g. [1]). Although for smooth b we can write b(u)~ = b’(u)u,, the set where b’ vanishes depends on u, whereas in our case the set of degeneration is supposed to be fixed for all u.

2

PRELIMINARIES

In the following [l’ lip := I[’ [Ip,~ denotes the normin Lp(f) and (., .) the duality between Lp(gt) and Lp, (~), lip + liP ~ = 1. L~,o(f) is the weighted Lebesgue space with weight 0 = O(x) and finite norm [[ullp, ~ = (f~ o(x)[?t(x)[ p dx)1/p. By [[. [[o and [[. [[o,~ we denote the norm in C(~) and HSlder space C~(~), respectively. wpl(f), l)dp~ (f) are the usual Sobolevspaces where l)dp~ (fl)is normedIlull,, :- IIVul] p. For functions restricted to tip we write l)dp~(flp) if the trace vanishes on 0fl~ ~ Off. For fixed t ~ I the operator A(t) generates a bilinear form I/~(fl) x ~,(~) denoted by A(~)(.,.). Moreover, we use the standard evolution spaces C(I,V), C°’~(I,V), and L,(I, 1/’). By c we denote a generic nonnegative constant being independent of the subdivision. Next we formulate the assumptions which are supposed to hold throughout the paper. Note that for every N there are p and r fulfilling the following restrictions. ASSUMPTIONS Suppose tip being simply connected with boundary °’~ Oi~p ~ C and meas~c_~(0~p Cl Of) > 0. Let g and f be Carath~odory functions defined ~×(I×IR). Let further be r > N, a > 1,p >_ max{2, Nr/(N+r)},p > N(a+l)/2a, and 2r(p- 2)/(r - 2) Np/(N- 2) Then we suppose for arbitrary t, t ~ ~ I and u, u~ ~ C(~) (i) U0e b~/~(llv); (ii) g(., t,u) :I x C(~) -~ Loo(~)is uniformly bounded in Loo(~) and fulfils Lipschitz condition IIg(., ~,~) - g(., ~’, u’)ll~ _<e, (1~ - ~’l ~ - u’ll ~). Furthermore, g(x, t, u) >_ for al l (x , t, u) ~ f × I × I~an 1/g(.,t,u) :I × C(fi---~) -> L~(fip) is boundedin L~(fp). (iii) a~ ~ C°’~(I,C(~)), ao ~ C°’~(I,L~(~)) ellipticity condition

with ao >_ 0, and it,

holds the

QuasilinearParabolic-Elliptic Boundary ValueProblem ~i,k aik (x, t) ~k >_ a

267

f6r all (x, t) E Q--~ and ~ E ~v, a >

: I x C(~) ~ Lp(~) is bounded in L,(~) and fulfils (iv) f(.,t,u) condition Ilf(’, ~,u) f(.,t’ ~’~" (v) Compatibility condition: ¢ ~ L.,~(.,o,~o)(~) such

the Lipschitz

There exist an extension ~0 ~ ~ (~) of Uo and

(g(., 0, Uo)¢, v) + A(0)(~o,v) = (f(.,

(C)

for all v ~ W~,(~) ~ L~(~p) holds. Welook for a weak solution in the following sense: DEFINITION Wecall u a weak solution of problem (1)-(4) u e L~(I , I}V~ (~)) has a derivative ut ~ L~(I, Lp,g(~)) and fulfils the relation ~(g(.,t,u)

ut,v)dt

+ ~ A(o(u,v) dt= f1(f(’,t,u),v)dt

(R)

for all v ~ LI(I,I~V 1 (t)¢~ L~(~p)) and initial condition (4). Wecall u a local weak solution if there is a time ~, 0 < ~ _< T such that u is a a weak solution in ~ := [0, ~]. Due to assumption (ii) a weak solution has a time derivative with ~ = pa/(a + 1) since the estimate

I1 11., <1/g .

~/(~+l)~

in L~(I, L~(~,))

=cllull,

(5)

holds. Moreover,the boundarycondition (3) and the first condition of (2) is fulfilled by the choosen function space for u, while the second condition in (2) is included the relation (R) in the sense that any smooth weak solution by the Gauss formula fulfils the equation (1) as well as transmission condition (2). Wediscuss the compatibility assumption (v). For v e I)V),(~) we see Oo must fulfil the elliptic equation A(0)/)o = f(., 0, ~ro) on f~g. Moreover, ~ro E I)dr ~ (ft), the boundary values of the extension on O~care given by the traces of Uo on F and ~0 = 0 on 0~ \ F. Hence, the extension of Uo into tic is fixed by an elliptic Dirichlet problem on tic. However, relation (C) also includes a transmission condition 0.~ Do,~ = 0~, Uo,v on F in weak sense. There is no freedom to fulfil it. Thus condition (v) is overdetermined: One cannot prescribe an arbitrary even very smooth initial function (4), but it is possible to ’bend’ it somewhatin neighbourhoodof F in order to fulfil (v). This compatibility condition is not supposed e.g. in [2], [3], [7], [10]. But these authors do not prove L~-boundedness with respect to time of the derivatives of u. In our case condition (v) is necessary: LEMMA 1 Let u be a weak solution of problem (1)-(~) with data fulfilling tions (i)-(iv). Thencompatibility condition (v) is satisfied.

assump-

268

Pluschke

PROOF Weprove the assertion for the more general case that ut ¯ Leo(I, L1 (fiTs)), p > 1, and no assumption is madeon 1/g. Of course, this is fulfilled by (i)-(iv) to the estimate (5). Owingto the choosen function spaces there is a sequence {t.n}, t,~ -~ 0, and a constant C such that II.a(.,tn)Nl,r <_.C, Nut(.,tn)Np,~(.,t.,,4.,t.)) < c and (g(.,t,~,u(.,t,d)ut(.,t,

d, v) + A(to)(u(.,t,d,

v) = (f(’,t,.,U(’,tn)),

(6)

for every v ¯ I~, (fl) r-i Loo(fly) holds. Thenthere is a subsequence{t,~. } such u(.,t,~.)~w

inl~l(fl),

w~,:=g(.,t,~,u)~/’ut(.,t,~)~w

inLp(flv).

By our assumptions we have u ¯ C°’l(I, Ll(flp)), hence winv = Uo and thus ’W -~’~ 00. The compact embedding 1~ (fl) C C(fi) even yields uniform convergence lz(’,tn~) --+ ~o as t,~ -~ 0. Moreover, let g~ := g(.,t,~,u(.,t,~)), := g(. ,O, Uo) we have g~ut(.,t.~)

gllp’ 11: w~ ~ _lip’ = ~ w~(^lip’ = y~, lolP’ -g ) w~ +go Yo ~ .

in L~(~tp)

since gk ~ go in Lc~(flT~). Wenowdefine ¢ := g~l/Pw in lip, ¢ = 0 in fl\ flT~, then ¢ ¯ Lpmo(fl) and g~ut(.,tn~) -~ g0¢ in Lp(flT~). Finally, a limit process t.,~ -~ 0 in (6) yields the compatibility conditon (C). In order to solve the problem (1)-(4) by semidiscretization in tilne (Rothe’s method) we subdivide the time interval I by points tj = jh (h > 0,j = 0,... ,n) and look for a solution uj ¯ 1~:,! (fl) of the discretized problem (gjSui,v)

+ Ay(uj,v) = (fj,v)

Yv¯ l~l,(fl)

(Ri)

uo = 0o,

(40)

j = 1,... ,n, where 5uj := (uj - uj_~)/h, gy := g(x, tj,uj_~), ]~ := f(x,t~,u~_~), and Ay(., .) := A(t~)(., .). This is a set of linear non-degenerated elliptic boundary value problems to determine the approximation uj if uj_~ is already known. Since fj, g~uj_~ ~ Lp(~) C (W~, (~)) holds for p ~ Nr/( N + r) and g j ~ 0, ao~ 0 the existence and uniqueness of a solution follows from [8, Theorems7.3 and 5.4]. Observe that the assumption r > N implies uj ~ C(~). Weconclude this section with some inequalities which we need for the a prioriestimates. Anessential tool in our investigations is the interpolation inequality (see [4, pp. 62-681)

Ilwll _
where g < ~ and ~,i~+i/N

< 0 ~ 1 , (7)

which is applied to power type functions w := iui(P-~U~’u defined on ~. Since w = 0 on a part of 09~ (cf. assumptions) IIwlN,~,a~ is an equivalent norm in ~)~ (~). Using Youngs inequality and ~he estimate (5) we obtain ~

c,

u n,

Vue W~(av)nL~(a~),

(8)

with p/2 < s < Np/(N - 2) where ~ = 1 for a = 1 and fl = 3(a,0) > 1 ~v-~(N-~) 1 < a < 1 + s(~s-v) . The last condition on a arises from the fact that we need

269

QuasilinearParabolic-Elliptic Boundary ValueProblem

a0 < 1 in order to derive (8) from (7). If 0 = 1 we obtain from (7) the Friedrichs inequality Vu e 1)¢~1(a) c~ L~(a),

(9)

where kF = kF(s, f~).

3

A PRIORI

ESTIMATES

FOR THE APPROXIMATIONS

In this section we derive estimates for the derivatives of uj with respect to x and for the discrete time derivative. The first estimate follows from the second one since uj can be regarded as a weak solution of the elliptic Dirichlet problem Ajuj = fj-gjSuj in [2, u~lon = O. Then from [8, Theorem6.3] and the positivity of Aj follows

Ilujlll,~
(10)

for all j = 1,... , n and all subdivisions since fj and gj are uniformly bounded in L~(ft) and Lc~(f~), respectively. Hencethe following estimate of (fuj is the crucial result of this section. However, due to the dependence of g on u we only obtain a local estimate. LEMMA 2 For small Lipschitz constant l~ fulfilling interval/~ = [0, ~], :~ < T, and ho > 0 such that

115uyllp,~_< M~Vtj

condition (12) there is a time < ~ 7

holds independent of the subdivision if h < ho. PROOFWetake the difference (Rj) - (Rj-1), j --- 2,... ,n, and test it v = ISujlP-’~6ui. By the use of the notation wj := I(~ujl(P-e)/:5uj and assumption (iii) we estimate (gjSuj, lSujl~-25uj } - (gj_~5uj_~, lSujl~-:Suj ) + klh [lwjll~,~ + kneh llSujllp~n ~

~ h ~(~, ~-~ )~ + l(&_~ - &)(~, where k~ = 4(p-1)/p ~ > 0 and ka e := infnextao(x,t) ~ 0. Note that Vv (p- 1)I~uj[p--2VUj = 2(p- 1)/plSujl (~-~)/~ Vwj. Then we continue by replacing the weight gj-1 by gj on the left side and estimating the right side by means of HSlders inequality, Youngsinequality, and Lipschitz conditions (iv) and (v), I}Sujll~,g~ - (gj~uj-1, ~ l:h

I~ujIp-2~Uj}

+

k~h Ilwill~,~ + ~ kn~hIlSuj[l~,n

1lSujll;-1 + lph 1tSuj-~llp,n~ 115u~ll~,~ + leh IlSuj-xllp,n, 1lSuyll~,~l~

-2)/~-a + ~hII~u~ll~ IIw~ll~,~ II~u~ll~ +~hIlu~ll~ll~u~ll~ p-~dx + /_ IgJ-~- gJl ISuj-xlI~u~l

270

Pluschke

where sz = 2r(p - 2)/(r

- 2) Np/(N - 2). Fu rthermore we hav 1

1

and

These estimates and inequality

(10) then imply

1

+hh[J~p I~uj-llelduylV-~dx+llh f~ Iduy_llP+’dx.

(11)

Observe that the total powerof ~u, in the last two integrals is p + 1 while on the left-hand side we have only the power p. This comes from the dependence of g on u. Wewill manage it below by application of a nonlinear Gronwall lemma, however we have to ensure that no item [l~ujll ~p with ~ > 1 appears on the right-hand side. Otherwise one cannot expect to obtain boundedness of [16ujl[ from the above inequality since it will be fulfilled for every fixed h’just if ~l~uj[I will be sufficiently large. This is not happen if ~lJuj_ll[ ap appears instead of [16uj[[ ~. In order to handle it by means of formula (8) however we have to take care that a > 1 and s > p do not exceed the given bounds. This can only be realized by using (5). Thus we apply HSlders inequality Z, : ( I~u~-ll"

’dx I~j-~l" I~1’-’ dx a~d z~ : [ I~u~-,l" I~u~-,I

with Pl = ~ = ~P/(a + 1), P2 = s2, P3 = 82/(2 -- 1) and p~ = ,~, pe = s3/p, respectively. In both cases we have se = s~ = ap~/(ap - a - 1) < Np/(N - 2) by our assumptions on p. Then we obtain with (5) and Youngs inequMity

where ~ > 1 is sufficiently close to 1 and B = ~(p, ~) >i 1. Applyingnow (8) to norms acting on ~p and Friedrichs inequality (9) to II~u~ll~and II~uyllLwe obtain fi’om (11)

- II~uy_l IIp~,g~_, + k,phIIw~ll?,~ + k~ph Ilau~ll~,., II~uyll~,g~ <

Quasilinear Parabolic-Elliptic

BoundaryValue Problem

Summingup these inequalities

for j = 2,...

271

, ~ yields

i

i

j=2

j=2 i j=2

i-1

i j=2

for every i = 2,... ,n. The critical item is the last If le ~ kas it can be canceled by the corresponding Otherwise we apply Friedrichs HSujH~. If the condition (le is fulfilled

inequality

sum on the right-hand sum on the left-hand

side. side.

(9) to the rest of the sum over ]~Suj]l~,ne

- inf ao) kF(p,~) ~e xI

(12)

< 4(p- 1~ p2

there is an e > 0 such that max{0, le - kae }pkv + e
I1~1~,~, + ~0h~ llwjll~,~ j=2 i

i-1

for every i = 2,... ,n. It remains to estimate II~u~ II~,g, +chIIw~ ~1~,2, We use the compatibility condition (v) and test the difference (R~)-(C) with I~UllP-a~u~. This yield s simil ar to our manipulations at the beginning of the proof

ll~u~lI~,~ + k~hHWlH~,~ + ~ h ll~utll~,~ + (Ao - A1)(Uo,

lSu~lP-~Sul)

~ [~¢[[~,~o [[5u1[[~,~ + -1 lachl[SUl[[~ + ch [[OoH~[[w~ [[~,2 []Su~ [[~v-:)/: + ch [[Oo[[v[[Su~[[~-1 where owing to f~ = f(., t~, ~o), fo = f(’, 0, ~o) no Lipschitz condition with respect to u is used. By the same reason we have

Applying again Youngs inequality,

(8),

and (9) we obtain

We fix now e < min{1/2, k~} and choose then h~ > 0 with h~C~ < 1/2. obtain uniform boundedness of ~l~u~H~ + ch Ilw~ll~,~for all h ~ h~.

Then we

272

Pluschke

The result of these comprehensivemanipulations is the discrete Gronwall inequality i

i

j=l

j=l

II ullp,g , (14) j=l

i = 1,... , n, obtained from (13) and the last calculations. It is essential that the second sum on the right side only runs up to i - 1 since fl > 1 while the first sum also mayrun up to i - 1 if h _< h2 is small enough such that the coefficient c2h is less than 1. Wewrite ck again instead of c~/(1 -c~h~), k = 1,2,3. Omitting the sum on the left-hand side a nonlinear discrete Gronwall lemmaderived by Willett and Wong[9, Theorem 4] yields the bound e~ [[~u,’[~, _ < [c: (~-’)-

- 1)c~ ~ he~(~-’)J (~

fori= 1 ....

j=l

where ei = (1 + c~h)-~ and ~ ~ n is choosen sm~ll enough such that the expression in the brackets remains positive. Since ih = t~ we have e~ ~ e -c~ih = e-c~. This implies

Then there is a ~ ~ T independent of the subdivision such that the expression in brackets is positive and bounded for all ti e [0,~]. This proves the lemmawith h0 = min{h~,h~}. From (10), (14); and the embedding inequalities

(5), (9) we immediatly

COROLLARY 1 Let the assumptions of Lemma 2 be fulfilled. i

Ilull ,r ~ M2,

-< Ma,

Then

p

h

M4

j=l with ~ = ap/(a + 1) and s = Np/(N - 2) hold independent of the subdivision

if

At the end of this section we add an example that illustrates the behaviour of 115ujllp,a if the compatibility condition (C) is violated. Since (C) is imposedto for t = 0 the first difference quotient is of special interest. EXAMPLE 1 For 12 = (0, 2) G I~ we consider the equation .q(x)

u~-uz.~

=0 in (0,2)

×I with

g(x)=

01 for x ~ (1,2) (0,1) forx~

(15)

and conditions (2)_--(4). If we prescribe Uo(x) = 2 - x on f~ = (1, 2) we mayex~end this function by Uo(x) = into f~ e = (0, 1) as a s ol ution of equation (15 ) ful filling

273

QuasilinearParabolic-Elliptic Boundary ValueProblem

~’o,e(1- 0) = [~o,v(1 + 0). However,compatibility condition (C) is not fulfilled O~,e(1- 0) #/)’~,~,(1 + It is an easy exercise to compute ux as a solution of the discretized problem X(1,2) Ul(X) - hut’(x ) = X0,2) (2 - x), u~(0) = u~(2) = 0. Thenwe -e (2-z)/v~ -( + 2-’*)/v~ e

2

2 e(l_z)/v~)

as h ~ 0. This shows: although 5ul converges pointwise for every fixed x E flp we see that 116ulll~,9 = ll6ulll~,n~, = (9(1 e-1/f-~)lh (p-~)/2) is unbounded forp > 1. It is interesting to see that [[Suiiip,9 remains boundedif p = 1. This is no contradiction to Lemma 1 since the reflexivity of Lp(l’l?p) for p > 1 is an essential tool in the proof of Lemma1. Indeed, if p = 1 it would be possible to prove Lemma2 under suitable assumptions on the data without compatiblity condition (C) since in this case there is used an regularized characteristic function as a test function in order to obtain the estimates.

4

RESULT

In this section we prove convergence of the Rothe approximations to a weak solution of problem (1)-(4). Therefore we introduce the Rothe approximations obtained from uj by piecewise constant and piecewise linear interpolation w.r.t, time t, respectively, ~t,~(x,t ) = ~uj(x) ift E (tj_,,tj] [~o(~)if < o

’

(tn(x,t)

= ~uj_,(x)

ift

~ (tj-l,tj]

[~o(z) i~, <_

’

and (tn(x, t) - tj h ui_~(x)

t - tj-1 + ----h---ui(x),

t ~ [tj_~,tj]. ., If wewrite f’~ = f(., ~n, g,~), g,~ = g(., ~,~, fi,~), and .) = A([~)(., .) wi th ~n = if t~_~ < t ~ ty, piecewise constant interpolation of (Ry) over ~ yields ~( ~g u~ , v) dt + ~ A’~(~ ’~, v)dt = ~(f~,

v) dt

~) (R

~ L~(~v)). The results of Lemma2 and Corollary 1 for all v ~ L~(], W~,(~) ~~ the following a priori estimates for the Rothe approximations, II t (’,t)ll,,a"

~ M1,

II t (’,t)ll~,".

II~%,t)-~(.,t’)ll~,.~Mzlt-t’l,

< M~,

II t (’,t)ll~dt

II~(.,t)-~"(.,t)ll~,~

~

(16)

5Mzh~, (17)

It ~ - ~"IIL.(Z,L~(.)) ~ MAh~/O,II~ - ~ IL,(ZL.(.)) ~ MA h~/~, II~(’,t)ll~,~ ~ M~,IW~(.,t)ll~,~ ~ Mu,II~(.,t)lla,~

(18) (19)

274

Pluschke

for all t, t’ E ~f, where the estimates (17) and (18) are an easy consequence of construction of fi’~, fin, and ~in, respectively. These estimates are the basis for convergence assertions for the Rothe approximates which yield our existence result by passing to the limit n -~ cc in relation (Rn). In order to obtain more regularity of the weak solution we need another interpolation inequality. For our r > N and 0 < A ~ 1 - N/r we have the continuous embedding ~/](~) C CX(fi) and it holds for allue~)(flp) (l~r)

Ilullo,- < Ilull , II ll -° with 1/,- 1/p1/r+

A/N + 1IN ~ 0 ~ 1.

(20)

Especially we obtain by means of (17) and (19)

~[~"(.,t).- ~"(.,t’)l[o,~~ clt (1-°) - t’[

Vt, t’ e i

for some0 6 (0, 1), hence there is a ~ > 0 such that

Nowwe are ready to prove the final result. THEOREM 1 Let assumptions (i)-(v) and condition (12) be fulfilled. Then there a local weak solution u of problem(1)-(4) with additional regularity u ~ C~(f~v for some c~ > 0 and ut ~ Lp(~,Ls(f~)), s = Np/(N 2) A subsequence of the Rothe approximations converges to the solution u in the following sense,

as nk -~ ~ where q ~ [1,~),

A < 1 - N/r, u = ap/(a + 1).

PROOFThe compact embedding CZ(f~, × ]) C C~’(~, x/~) for 0 < a and the weak compactness of a bounded sequence in L~(~,~V~(f~)) imply from (21) and (19) the existence of a subsequence {~’~} (we write {fi’~} again choose subsubsequences below) with convergence properties (22~) and (23). standard arguments we see that the limits coincide. Convergenceproperty (23) for the sequence {~n~} follows from (19) with same limit u due to (18~). To prove the second part of convergence assertion (22) observe that {fi’~} bounded in W.~(Q$) due to (163) and (19~). By the.Rellich theorem there subsequence with fi~

--~ u in L~(Q~)

as n~ ~ ~. Let now be A < 1 - N/r. Then there is a 0x such that (20) holds with /~ = 2 for ~1 < 0 < 1. For fixed q > 2/(1 - 01) we then obtain ’![ II~,"~ (., t)- u(.,t)l[oq, a dt <_cshpess []~n~- uH~°,~f/lift ’’~- u[[~’-°)q d~,

275

QuasilinearParabolic-Elliptic Boundary ValueProblem

which yields convergence (22) if 0 is chbosen such that (1 O)q = 2. Theremaining convergence properties (24) and (25) follow from (16) under consideration of In order to prove that u is a weak solution we have to investigate the behaviour of g’~ and fro. as nk ~ c~. From (22) and (18) we obtain ~n~ -4 u in Lv(~, Ls(ft)) as nk -~ ~. Since s > p assumption (iv) yields fn~ ~ f(.,

",u)

in L~(Q~).

(26)

By application of the interpolation inequality (20) we obtain frown (172) and (193)

II~(.,t) - ~(’,t)ll0,~,~. ch,, ,

1-0

hence we have convergence of {~n~} to u in L~(], C(~)). Thus assmnption yields g.n~ ._+ g(., , u) in L~(Q~)which owing to (24) implies g’~ft~ ~ --~ g(., .,u)u,

in L~(],L~(~t))

(27)

Moreover, by an analogue argument like in the proof of Lemma1 we see from (16~) that u e Loo(i, Lp,~(.,t,,,)(~)). Finally, passing to the limit n = nk --~ c~ in relation (Rn) we obtain relation (R) due to convergence properties (23), (26), and (27), assuming first a more regular test function v. Initial condition (4) is fulfilled by construction and the uniform convergence on ~p × ~f. The usual density argument for the test functions concludes the proof. [] REMARK 1 Friedman and Schuss [2] prove uniqueness of a weak solution belonging to L~(QT) for the linear parabolic-elliptic problem. For our nonlinear problem (1)-(4) fulfilling assumption (i)-(v) we can prove uniqueness, too. Assume were two solutions u and u~, we define z := u- u~ and obtain from (R) with v = [z[P-~z for every fixed to ~ ~ (g(.,t,u)ut-g(’,t,u’)u’t,

IzlP-~z}dt

A(t)(z, Iz l’-~’z)dt

Then we consider

and insert (28) into the lastitem. The resulting inequality we estimate similarly as in the proof of Lemma2. Note that the nonlinear Gronwall inequMity used in that proof is no suitable tool to prove uniqueness since c~ > 0 in (14) is essential for its application. However,for given solutions ~ and u’ we mayreplace Ilu~ll and

276

Pluschke

[1~ g(’,t,u(’,t))ll.,m,

by constants, hence differently from Lemma2 now we have the total power p of z on the right-hand side and may use the well-known linear Gronwall lemma. Since the weak solution is unique the convergence assertions for the Rothe approximations in Theorem1 hold for the whole sequence. REMARK 2 The global assumptions (ii) for g w.r.t, u may be replaced by local assumptions supposed for all u E C(12p) with Ilu - U011o,n. _< R for some R :> 0. Then a truncation method yields a weak solution which is a solution of the original problem for small t since u is continuous on f~p x [0, 7~]. This generalization is not possible for f on f~e since we have no information whether Ilu(., t) -/)ollo,as witl small for small t. Weonly knowfi’om (23) that Ilu(., t)llo,n e is uniformly bounded, however the bound depends on the choosen R and may be much bigger than R even if the time intervall is small.

REFERENCES 1.

H.W. Alt and S. Luckhaus, Quasilinear Elliptic-Parabolic Differential Equations, Math. Z., 183:311-341 (1983). 2. A. Frie&nan and Z. Schuss, Degenerate evolution equations in Hilbert space, Transact. of the AMS, 161:401-427 (1971). 3. J. Ka~ur, Method of Rothe in Evolution Equations, B.G. Tcubner, Leipzig (1985). 4. O.A. Lady~enskaja, V. A. Solonnikov, and N. N. Ural’ceva, Linear and Quasilinear Equations o] Parabolic Type, AMS,Transl. Math. Monographs, Providence, R. I. (1968). 5. R. MacCamyand M. Suri, A time-dependent interface problem for twodimensional eddy currents, Quart. Appl. Math., ~: 675-690 (1987). 6. V. Pluschke, Rothe’s method for degenerate quasilinear parabolic equations, In: Z. Dogla, J. Kuben, and J. Vosmansk); (eds.), Equadiff 9 CD-ROM,pp. 247-254, Masaryk Univ., Brno (1998). 7. R.E. Showalter, Degenerate E~:olution Equations and Applications, Ind. Univ. Math. J., 23:655-677 (1974). 8. C. G. Simader, On Dirichlet’s Boundary Value Problem, Lecture Notes in Math. 268, Springer, Berlin--Heidelberg-New York (1972). 9. D. Willett and J. Wong,On the discrete analogues of some generalizations of Gronwall’s inequality. Monatsh. fiir Math., 69:362-367 (1965). 10. M. Zlfimal, Finite element solution of quasistationary nonlinear magnetic field, R.A.LR.O. Anal. Num., 16:161-191 (1982).

Singular Cluster Interactions in Few-Body Problems S. ALBEVERIO Institut fiir Angewandte Mathematik, Universit’~t Bonn, D-53115 Bonn; SFB 256; SFB 237; BiBoS; CERFIM(Locarno); Ace.Arch., USI (Mendrisio); Fakult/it fiir Mathematik, Ruhr-UniversitS~t Bochum,D-44780 Bochu~n. P. KURASOV Dept. of Math., Stockholm University, 10691 Stockholm, Sweden; Dept. of Math., Lule£ University, S-97187 Lule£, Sweden;St. Petersburg University, 198904 St. Petersburg, Russia.

1

INTRODUCTION

The present paper is devoted to the study of the few-body quantum mechanical problem with singular finite rank cluster interactions. The corresponding Hamiltonians play" an important role in mathematical physics, since few--body Hamiltonians with more regular interactions present considerable difficulties which make impossible a detailed analytic study [13]. Also a numerical study of such Hamiltonians using Faddeev equations present very hard problems, since the interaction between the particles does not vanish at large distances. On the other hand few-body problems with singular cluster interaction are useful and intensively studied in statistical physics, since the corresponding Hamiltonians can be analyzed in detail even if the number of particles is very large. Models describing one dimensional particles are of particular interest, since the eigenfunctions of the manybody Hamiltonians can often be calculated using Bethe Ansatz [14]. In fact the well-known YangBaxter equation was first written in connection with the study of system of several one-dimensional particles with pairwise delta interactions. Similar methods were used in atomic physics to study collisions of several part!cles [12] (in three dimensions). Few-body systems in applications are usually investigated by combining the classical description of the dynamics of heavy atoms with the quantum description of the dynamics of electrons and other light particles. The first attempt ~ to construct a three body Hamiltonian describing three quantum particles in R interacting through pairwise delta fnnctional potentials is due to G.V.Skorniakov and K.A.Ter-Martirosian [29]. R.A.Minlos and L.D.Faddcev proved that the corresponding Hamiltonian is not bounded from below and therefore cannot be used in ~7

278

Albever|o and Kurasov

the originally intended physical applications [24, 25]. The operator was defined using the method of self-adjoint perturbations used by F.A.Berezin and L.D.Faddeev for investigating Schbdinger operators with delta potential in R3 [10]. Almost three decades later semibounded three-body operators in dimension three with ge:aeralized two-body interactions were constructed by extending the standard Hilbert space of square integrable functions in R9 [17, 18, 19, 27,.30]. The most interesting model considered used the theory of generalized extensions suggested by B.Pavlov [7, 26]. Different aspects of these models were analyzed recently [8, 23, 20]. A general approach to these operators is described in [7, 22]. Let us mention that a realization of a many-bodylower bounded Hamiltonian with point interactions for particles in R3 has been obtained in the original Hilbert space by using the theory of Dirichlet forms [2]. The few-body systems of two dimensional particles with twobody interactions was studied in [11], where it was proven that the corresponding Hamiltonian is semibounded. It is stressed particularly in [7] that few-bodyHamiltonianswith delta interactions can be efficiently studied by using the theory of finite rank perturbations. Hamiltonians describing point interactions created at manycenters, e.g. of importance in solid state physics, have been discussed in [1]. Rank one for~n bounded interactions were analyzed in detail in an abstract setting by B.Simon and F.Gesztesy [15, 28]. Perturbations in terms of quadratic form were studied in [3]. In [16] rank one form unbounded interactions are defined following the paper by F.A.Berezin and L.D.Faddeev [10] and introducing a renormalization of the coupling constant. See also [4, 5], where these interactions are defined without renormalization of the coupling constant using a certain regularization procedure. It is not hard to extend this technique to obtain few-body operators with form bounded cluster interactions (see e.g. [7]), but operators with more singular interactions need a more detailed investigation. The present paper is devoted to the study of the few-body operator with one singular cluster interaction having finite rank. It is proven that this interaction in general is described by unbounded boundary operators. This operator can serve as an elementary brick in the construction of the few-body Hamiltonian with several singular cluster interactions. The paper is organized as follows. In Section 2 the few-body operator witb finite rank cluster interaction is heuristically defined. A precise definition of this operator is given in Section 3 by separating the center of mass motion and using the extension theory of operators with finite deficiency indices. To describe the analytic properties of these operators a study of rational transformations of Stieltjes functions is given in Section 4. The resolvent of the few-body operator is calculated in Section 5. The few-bodyoperator is described in Section 6 without separation of the center of mass motion. Then the extension theory for symmetric operators with infinite deficiency indices is used. The extension is described by certain bom.~daryconditions involving unbounded operators.

2

FEW-BODY OPERATOR WITH FINITE CLUSTER INTERACTION

RANK

The Schr6dinger operator describing several quantum mechanical particles is characterized by the following property: the original Hilbert space 7-/and the operator

279

Singular Cluster Interactions in Few-Body Problems possess several tensor decompositions n = l,2,...,N

7/ = K,~ ® Hn,

(1) A

= Bn ® IH,, + IK,, ® An;

where B,, and .4,, are positive self-adjoint operators acting in the Hilbert spaces Kn and H,~ respectively. The index n parameterizes the cluster decompositions and the number N of such decompositions is in general different from the number of particles. The operator An describes the motion of the particles forlning the corresponding cluster in the coordinate system associated with the center of mass of the cluster. The operator Bn describes the motion of all other particles in the same coordinate system. Each pair of operators (An, B~) appearing in the tensor decomposition determines the unperturbed operator ,4 uniquely, since it is essentially self-adjoint on the algebraic tensor product of the domains of the operators An and B,~. The few-body operator with singular finite rank interactions can heuristically be defined by N

.4°

:

d.

.4 n=l

+

(2)

j=l

where ~,~j are singular vectors defining the cluster interactions. In what follows we suppose that these vectors belong to the Hilbert spaces 7~-2(An) associated with the corresponding operators A,~ ~2~j E 7/-2(A,~), j = 1,2,...,dr,

n = 1,2,...,N.

The numbers dn E N determine the rank of the cluster interaction. The coupling constants anak form dn x dn Hermitian matrices. Then the perturbation term is formally symmetric. If the vectors ~2,,j do not belong to the spaces 7/-1(A,,) then the perturbation term is not form bounded with respect to the unperturbed operator. To determine the few-body operator in this case it is necessary to carry out a special analysis i.ncluding the extension theory for symmetric operators. The aim of the current paper is to describe few-body Hamiltonians with one cluster interaction. This is the first step towards the definition of general few-body Hamiltonian with finite rank cluster interactions. Let us consider the Hilbert space 7/ and the operator ,4 possessing the tensor decomposition 7/ = K ® H,

(3)

‘4 = B ® IH + IK ® A where B and A are positive self-adjoint operators in the Hilbert spaces K and H respectively. Consider d singular vectors from the Hilbert space 7/-2(A). The the few-body operator with single cluster interaction is heuristically described by d

‘4°=

(4) j,k=l

280

Albeverio and Kurasov

where the coupling constants ajk form an Hennitian matrix. If the vectors ~j are from the Hilbert space H, then the perturbation term is a bounded operator. The perturbed operator in this case has the same domain as the original operator A. The problem of defining the operator with cluster interactions is trivial in this case. Therefore we are going to concentrate our attention on the case of so-called H-independent vectors. d C DEFINITION2.1 The set of vectors (~J }j=l if and only if any nontrivial linear combination d

d

j=l

j=l

~-{-2

(,4)

is called H-independent

does not belong to the Hilbert space H. In what follows without loss of generality we suppose that the vectors form an orthonormal sct in 7-/_2(A), i.e.

=

(5)

H

Any set of independent vectors can be orthonormalized and the new set is independent also. In particular

e The Hermitian coupling matrix can be diagonalized using a certain orthogonal transformation. Therefore it is enough to consider only diagonal coupling matrices. Thus the following heuristic operator will be studied in this paper d

(6) j=l

where the vectors ~?j form an orthonormal subset of H_.~ and tim real coupling constants are different from zero,

(7)

% ~R, %¢0, j=l,2,...,d.

3

CLUSTER INTERACTION OF MASS MOTION

VIA

SEPARATION

OF THE CENTER

The operator ~l~ can be defined using the following formed decomposition d

B ® I~ + IK ® A + E ay((pj,

.)E~j

j----1

(8) =

B ® IH + IK ® A -1-

O~j((~Oj,

.)H(~j

Singular Cluster Interactions in Few-Body Problems

281

The operator d

As = ,4 + E aj(~j,

’}n~j

(9)

is a finite rank perturbation of the operator ,4 and it can easily be defined following [6]. To define the operator As we restrict first the operator A to the domain Dom(g °) = {¢ E Dom(A): (¢pj,¢)H = 0,j = 1,2,...,d}.

(10)

The restricted operator A° is symmetric and densely defined. The deficiency elements at point i are given by 1 A - i ~j’j = 1, 2,...

,d,

and form an orthonormal subset of the Hilbert space H. The intersection the deficiency subspace and the domainof the original operator is trivial

between

Ker (A°* :E i) ~ Dom(A) = {0}. Therefore the restricted operator is densely defined and has deficiency indices (d, d). The vectors W:-7~J form a basis in the deficiency subspace Ker (,4 0* - i) denoted by M, i.e. M= Ker (A°* - i). All self-adjoint extensions of the operator A° can be described using von Neumann formulas. But we prefer to define the extension using Krein’s resolvent formula. Essentially all extensions of the operator A° can be parameterized by an Hermitian operators ~/ acting in M in such a way that the resolvent of the corresponding self-adjoint operator A"~ is given by ~ 1 A+i 1 A-i - 1 .__, A’~ - z A - z A - z "~ + q(z) P~4 A where "Krein’s Q-operator" q can be chosen as follows q(z) = PM’-’~-~_ l+zA z M. q is a d x d ~natrix Nevanlinnafunction, i.e. its imaginary part is positive in the upper half plane "~z > 0. Our choice of q is determined by the normalization condition q(i) = JIM.

(12)

If all vectors ~,~- are from the space 7/_~ (A) then the relation betweenthe coupling parameters al, a~, ¯ ¯ ¯, ad and the Hermitian operator ~, is given by ")’

:

O~(-1)

+

PMAIM,

(13)

~The self-adjoint extensions that would not be described by this formula would have the following property: the operator A° is not the largest common symmetric restriction of the perturbed and unperturbed operators. But this implies that at least one of the coupling constants is equal to zero, which is impossible due to our assumption (7).

282

Albeverio and Kurasov

where a is the following coupling operator defined in M d j=l

The second term in (11) is well defined. In fact using the orthonormat basis in $4 we have APM

(14) H

II

since the deficiency subspace/14 is a subset of 7-/-1 (A). (Weremark that the scalar pr°ducts/ l,,Tf~-i~j, A,-4~-i~kl)H =---(~J’A--~°kl Hare welldefined.) In the case where some of the ~oj are not from 74_1(A) the operator 3‘ cannot be calculated from the coupling parameters without using additional assumptions. Only some partial information concerning the operator 3’ can be recovered. The formal expression (9) does not define a unique operator in this case, but a certain family of self-adjoint operators described by several parameters. The number of free real parameters can be different from d~ (the number of real parameters in yon Neumannformulas), since the operator 3‘ should satisfy some admissibility conditions if the vectors ~oj are not 74_l-independent. Supposethat a certain linear combination of the vectors ~oj belongs to the space 7/-1

¢

d

=

e

then the scalar product ¢, ~-V-~¢

aj

~C

is well defined. Therefore the operator H

necessarily satisfies the admissibility condition

aj~j, A~ + 1 j=l

’ k:l

H j,k=l

where 3‘i~¢ are the coefficients of the operator 3’ in the chosen orthonormal basis

The operators satisfying all admissibility conditions are called admissible. The family of admissible operators 3‘ was described in [6]. Ii~ in addition the operator A and all vectors ~j are homogeneouswith respect to a certain group of unitary transformations, then the operator "), could be determined uniquely by requiring additional natural homogeneity properties of the perturbed operator. This approach has been developed in [4, 5] for perturbations of rank one. In what follows we suppose that the extension of the operator A° is determined by a certain admissible Hermitian operator % which is compatible with the heuristic expression (9). This extension will be denoted by ~. In what f ollows t he operator

283

SingularCluster Interactions in Few-Body Problems

A~ will be substituted for the operator A~. The corresponding few-body operator with single cluster interaction is defined by A"~ ~. = B®IH +I~ ®A

(15)

The last formula determines the operator A~ uniquely, since ~ B ® Itl + IK ® A is essentially self-adjoint on the algebraic tensor product of the domains of the operators B and A~. Hence Dom(,4 7) : Dom(B) x Dom(A~).

(16)

In order to simplify our presentation let us restrict ourselves to the case of perturbations of rank one (d = 1). Weare going to drop the lower index of the coupling constant and singular vector

To calculate the resolvent of the operator A~ we need to study the properties of the Nevanlinna functions describing the interaction.

4

RATIONAL

TRANSFORMATIONS

OF STIELTJES

FUNCTIONS

Let us prove first some facts concerning Nevanlinna functions F which belong to the Stieltjes class, i.e. possess the representation F(z) =/; 1A_ + ZAdP(A)’

(17)

where the real measure p is finite f~ dp(A) LEMMA 4.1 Let F be a Stieltjes function. Then for any real y and any positive e > 0 there exists a certain b = b(y, e) > 0 such that the following estimate holds If(z ÷ iy)l < ~lxl

b

(18)

for all x < A. PROOFConsider the real and imaginary parts of the function F(x + iy) = /; l +~-Z-x--_-~y (z + iY))~ apta). .... The imaginary part ~F(x + iy) = y ; (A - A~+I x) 2 + y2 d0(A) is uniformly boundedfor all x < A. The real part is given by the sum of two integrals as follows: ~¢ f~ ~(1 - y~) -, i~)

L

284 The first follows

Albeverio and Kurasov integral

is uniformly bounded. The second integral

can be estimated as

( )2 for all x < A - 1. To estitnate J’~ dp(A) < e/2 and we get

the latter

f? %x-I_lap(a)

integral

f?ap(a)+l=l

<_ \-~:-~_x A~dp(A) C For all x function therefore lemma is

we choose C > ,4 such that

+

’x’.

_< Xo = C(1 - 71 f~ dp(A)) the last expression is estimated e[xI. The F(x + iy) is continuous on the bounded interval z0 < x < A and is uniformly bounded on this interval (which is empty if x0 > A). The proven. []

The following lemmadescribes rational transformations of Stieltjes LEMMA 4.2 Let F be a Stieltjes

functions.

function. Let a, b, c, d be real numbers such that ad - bc = 1.

(19)

Then there exists a real numberAt, such that for any real y and any positive e > 0 there exists b~ = b~(y, e.) such that the function G(z) aF(z) + b cF(z) +

(20)

possesses the representation a(z)=~z+g(z),

~

(21)

where

Ig(x+iY)I

(22)

for all x < A~. PROOFCondition (19) guarantees that the function G is a Nevanlinna fl~nction and possesses the representation ~ l+Az G(z) = (~ + ,Sz + ~ -A----~ dp~(A),

f_

where f-~o~ dpt(A) < ~, ~,/3 ~ R, /3 > 0. The support of the measure p~ coincides with the set of real points z where the boundary values G(z + iO) are not real or do not exist. The function G is real on the interval (-c~, A) outside the points where cF(z) + d = 0. The derivative ~ ~ + 1 dF(z) ~zz - (~---)~ 2 dp’ (A)

Singular Cluster Interactions in Few-Body Problems

285

is positive for all z < A. It follows that there exists at most one point where F(z) = - ~. Therefore the support of the measure p~ is bounded from below. Lemma 4.1 implies then estimate (22). The lemmais proven. Weremark that the constant/~ appearing in (22) is different fi’om zero only if the function F has a finite limit F(~) at infinity and F(oc)

5

KREIN’S

RESOLVENT

FORMULA

To calculate the resolvent of the operator A~ we will need the following corollary of the two previous Lemmas. LEMMA5.1 /unction

Let y be an arbitrary positive real number. Consider the Nevanlinna

=

1

1

-

I/~ E ~t-1 (A), then the/unction satisfies

~]H A-.~

A +1

¯

the estimate

IG(x + iy)[ <_ Cl(y)(1 + Izl)

(23)

/or all negative x < 0 and a certain C~(y) > O. I] ~ e 7t_~(A) \ ~t_~(A), then /unction can be estimated by

IG(x+i )l < .for all negative x < 0 and a certain positive C2(y) > PROOFThe function q is a Stieltjes function, since the operator A is positive. Lemma4.1 implies that the estimate (23) holds for all ~ ~ 7-/_:(A). Consider the case ~ ~ 7-/_~(A) \ 7/_~ (A). Weare going to prove that the real of q(x + iy) tends to minus infinity when x -~ -~c. In fact the function q can be presented by the following integral

where the measure p is finite, i.e. f~¢ alp(#) < oc, but the integral ~ diverges. The real part of q(x + iy) is given by ~q(x + iy) =~ (#--~:~,2x) - #Y

fo~ x#(p__-_ x_)_ d ,p ).

The first integral in the last formula is boundedfor negative values of x : c~ # -- X ~0o~ py2

~0

(25)

286

Albeverio and Kurasov

The second integral in (25) is negative and can be estimated x 2 /o ~ ,x,# dp(#) ~ ¯~o(~ x#(p_ x)2 +x)y’2 dp(#) >_ x 2 + y~ # - x _ > x ~ + y~ 2 ao #dp(#). The last integral tends to infinity when x -~ -~. It follows that lim G(x+iy)

1

li m

,~-~-~

.~-~-~~ - q(~+ i~)

-0.

Therefore the continuous function G(x + iy) is uniformly bounded on the interval x < 0, i.e. the estimate (24) holds. The lemmais proven. Weremark that if y = 0 then the esti~nates (23) and (24) hold for x ~ (-co, where .4~ is a certain real constant. THEOREM 5.1 The resolvent of the operator A~ = B ® IH + I~ ® A~ at a cert~ain point A, ~A ~ O, is given by the formula 1 _ 1 (A-A A~-A where

q(A-

1 A-

~ A +q(A

B) =(~,I+(A-B)@A A-A

The function

1 "l’--q(xTiy)

(26)

1 l~)h" A:+

COMMENT Let us discuss formula (26) first. f ~ 74 the following inclusion holds 1 (’-’~’~v,f)h

_ B)

Let ~ ~

74-1(a).

Then for any

1 V/~+ 1~ = ( ~V, -’~. -~ Jib 741(B).

satisfies the estimate (23) and it follows that the operator 1

-~ _ q(~ maps 741 (B) onto 74_~ (B). This implies

3’ - q(A - B)

¢P’f

®~ ~ 74_2(A).

(27)

This means that formula (26) defines a bounded operator in the Hilbert space

for ~ Consider now the case ~ ~ 74-2(A) \ 74_~(A). For any f ~ 74 the vector 1 1 The function (’-~L~,f)h (" A-TTtp,(A + i) ~_xf)H belongs to the spac e K. 1 "~-q(.~+iu)l is bounded fornegative x(see(24)) andtheoperator "r-q(a-B) is bounded in K. This implies that condition (27) holds. Therefore formula (26) defines boundedoperator acting in the Hilbert space for any q9 fi 74_~(A).

Singular Cluster Interactions in Few-Body Problems

287

PROOFof Theorem 5.1 Let us denote by YB the operator of spectrM transformation for B - the linear operator which maps the operator B into the operator of multiplication by the independent real variable x. Then the resolvent on a dense set can easily be calculated as follows 1

£

~ f = (W-~)¢ ~ (.TBf)(x)

=~¢

-

1 A--~

f

= (x "r - A)(~ ’stb)(x)

1(~ A-A \[7+q()~-B)

"

Wehave supposed that ¢ E £~, where £~ is an algebraic tensor product of Dora (B) and Dora (a~). The operator ~ i s e ssentially s elf-adjoint o n t his d omain and t his completes the proof of the theore~n. []

6

CLUSTER INTERACTION WITHOUT SEPARATION OF THE CENTER OF MASS MOTION

The resolvent of the operator .4 ~ has been calculated using the tensor decomposition. The operator A~ is a self-adjoint extension of the symmetric operator A° = B®I~ +I~:

°®A

with infinite deficiency indices. Consider the annulating set (I’~eg of regular functionals for the operator A° defined as follows: ¯

if

~ E

qr~_I(A) then

v~ = {~: ¯ = ~(~) ®v, ~(~) e ~_~(B)}, ¯

if ~ ~ ?/_~(A) \7/_~(A)

Let us consider the corresponding subspace of regular elements from the domain °~. of the adjoint operator .4

288

Albeverio and Kurasov

* if ~a E 7/_~(,4) then °*) = {’~b: $ ---’~ + A.A--~+~p(~b)®W,~ e Don,(A), p(~b) ~ ~h!_, Domreg(A

The boundary form of the adjoint operator calculated on the regular elements is given by 0.) =~ U, V ~ Domreg(A

(u, A°*v)- (‘4°*u,v) : (p(U)®~, 9) - (0, p(v)

(28)

A symmetric extension of the operator .40 can be defined in ter~ns of any symmetric operator F by restricting the operator .40, to the domain of flmctions from °*) satisfying the boundary condition Domreg(A

-(¢, ~2): rp(U).

(29)

In order to obtain the perturbed operator possessing the tensor decomposition (8) let us consider the symmetric operator .4r determined by the following boundary operator F=3,-B

%(.42+1)(A

~+1)~o

.

(30)

H

The operator (~o,(.A~+I)(A~+I)AA-1 V-’! ~u is a boundedself-adjoint operator in K commuting with the operator B. The norm of this operator is less than or equal to 1. Therefore the operator F is essentially self-adjoint on the domain Dom(B) the operator B. Weare going to keep the same notation F for the corresponding self-adjoint operator. Let us find an expression for the resolvent of the operator .4r. Consider an arbil trary f ~ 7/and suppose that .~b = ~ + ~f+a P0P) ® ~ = ~-v-~f. Then the fimction ’(,, satisfies the fbllowingequation + ~.4 (.4 - ~)V5 ~ A~ + ~ (P(~) ® ~) and the boundary conditions (29). Applying the resolvent of the original operator ‘4 to the previous equation we get ~_

I+AA 1 ‘4-A .42+1

p(¢)

1 ®~°- A-A f

I+AA 1 ~)~ p(¢) f)’" =-(~’)--251 ~ r+(~, 3i--~ ‘4~:~ )

(31)

This equation can be solved and tile function p(¢) can be calculated if the operator F+ (~o, ~ A 1---~+~ ~o)y is invertible. The operator can be simplified as follows taking

SingularCluster Interactions in Few-Body Problems

289

into account equality (30) hI+AA 1 19~)

r + (~, ~1:~ A" =’r-B

~,(A~+I)(A2+I)~

-

~,~--~A2-+~

h

=’r+

~,

t+(~-B)®A

(32) H

1 A2+l ~° H’

A-A

which holds on functions from DomB. Since ~ E ~_~(A) and the operators and B are positive the following inclusion holds (~,A~--~_~f)H ~ K. The comment after Theorem5.1 shows that the operator F + (~o, L+-~1--A--,~\,, is invertible A-A A2+I K. Therefore there exists p(’¢’) Dom (r) C K which sa tisfies eq uation (3 The component ~ of the function ¢ can be calculated using the formula 1

I+AA

A- >, A2 + i p(¢) ® Thus the function ¢ is given by

The resolvent of the operator operator A"r. This implies that been first defined only on the Thus the following theorem

Ar coincides with the resolvent of the self-adjoint the operator Mr is in fact self-adjoint even if it has regular elements. has been proven.

THEOREM 6.1 The operator .A r which is the restriction the set of regular elements

of the operator A°* to

A~ + 1 (P(¢) ® °*) ~o) ~ Dom~eg(A

(33)

satisfying the boundary condition

(~, ~)H= rp(¢)

(34)

is self-adjoint and its resolvent is given by

(35) for any A; 9A ~ O. COMMENT In the course of the proof of the previous theorem we have shown that the density p(¢) is an element from the domain of the operator F. It is possible prove that the restriction of the operator Ar to the domain of functions possessing

290

Albeverio and Kurasov

the representation (33), boundary conditions (34) and having p(¢) E Dom essentially self-adjoint. Suppose that ~o E 7/-1 (,4). Then the boundary conditions (34) can be simplified as follows. Consider the scalar product (~, C/H, where ¢ is any function fi’om the domain of the operator Ar. Then the following equalities hold

= -~/p(~b)+

=

+

~o,\(A~-i-~.~-SUF1

) +A2+---- ~ p®~o ~,

(36)

p(¢) h

=

+ c) p(¢),

where we have used the fact that the function ¢ satisfies boundary condition (34). Taking into account (13) the latter condition can be written

= p(¢). One can define the operator .Ar using this boundary condition, but this condition cannot be generalized to the case of 7-/-2 interactions, since the scalar product ~ does not necessarily define function from K in this case. (fl’ A’--~A+~ P ®(ill REFERENCES 1.

2. 3.

4. 5. 6. 7.

S.Albeverio, F.Gesztesy, R.Hoegh-Krohn, and H.Holden, Solvable pwblems in quantum mechanics, Springer, Berlin, 1988 (Russian edition, Mir, Moscow, 1990). S.Albeverio, R.Hoegh-Krohn: and L.Streit, Energy forms, Hamiltonians, and distorted Brownian paths, J. Math. Phys., 18:907-917, (1977). S.Albeverio and V.Koshmanenko,Singular rank one perturbations of selfadjoint operators and Krein theory of selfadjoint extensions, to appear in Potential Analysis. S.Albeverio and P.Kurasov, Rank one perturbations, approximations, and selfadjoint extensions, J.Funct.Anal., 148:152-169 (1997). S.Albeverio and P.Kurasov, Rank one perturbations of not semibonnded operators, Int. Eq. Oper. Theory, 27:379-400 (1997). S.Albeverio and P.Kurasov, Finite rank perturbations and distribution theory, Proc. Amer. Math. Soc., 127:1151-1161 (1999). S.Albeverio and P.Kurasov, ;~ingular perturbations of differential operators and exactly solvable SchrSdinger type operators, CambridgeUniv. Press, to appear in 1999.

Singular Cluster Interactions in Few-Body Problems 8. 9. 10. 11.

12. 13. 14. 15. 16. 17.

18.

19.

20. 21.

22. 23.

24. 25. 26. 27.

291

S.Albeverio and K.Makarov, Attractors in a model related to the three body quantum problem, C.R.Acad. Sci. Paris Set. I Math., 323:693-698 (1996). S.Albeverio and K.Makarov, Nontrivia| attractors in a model related to the three-body quantum problem, Acta Appl. Math., 48:113--184 (1997). F.A.Berezin and L.D.Faddeev, Remark on the SchrSdinger equation with sin~ular potential, Dokl. Akad. NAUKSSSR, 137:1011-1014 (1961). G.F. Dell’Antonio, R.Figari and A.Teta, Hamiltonians for systems of N particles interacting through point interactions, Ann. Inst. H.Poincard, Phys. Theor., 60:253-290 (1994). Yu.N.Demkovand V.N.Ostrovsky, Zero-range Potentials and their Applications in Atomic Physics, Plenum, NewYork, 1988. L.D.Faddeev and S.P.Merkuriev, QuantumScattering theory for Several Particle Systems, Kluwer, Dordrecht, 1993. M.Gaudin, La Fonction d’onde de Bethe, Masson, 1983. F.Gesztesy and B.Simon, Rankone perturbations at infinite coupling, J.Funct. Anal., 128:245-252 (1995). A.Kiselev and B.Simon, Rank one perturbations with infinitesimal coupling, J. Funct. Anal., 130:345-356 (1995). Yu.A.Kuperin, K.A.Makarov, S.P.Merkuriev, A.K.Motovilov, and BoS.Pavlov, The quantumproblem of several particles with internal structure. I. The two-body problem, Theor. Mat. Fiz., 75:431-444, 1988. Yu.A.Kuperin, K.A.Makarov, S.P.Merkuriev, A.K.Motovilov, and B.S.Pavlov, The quantum problem of several particles with internal structure. II. The three-body problem, Theor. Mat. Fiz., 76:242-260, 1988. Yu.A.Kuperin, K.A.Makarov, S.P.Merkuriev, A.K.Motovilov, and B.S.Pavlov, Extended Hilbert space approach to few-body problems, J. Math. Phys., 31:1681-1690, 1990. P.Kurasov, Energy dependent boundary conditions and the few-body scattering problem, Rev. Math. Phys., 9:853-906 (1997). P.Kurasov and J.Boman, Finite rank singular perturbations and distributions with discontinuous test functions, Proc. Amer. Math. Soc., 126:1673-1683 (1998). P.Kurasov and B.Pavlov, Few-body Krein’s formula, to be published in Proc. of Krein’s conference, Birkhauser. K.Makarov, Semiboundedness of the energy operator of a system of three particles with paired interactions of (~-function type, Algebra i Analiz, 4:155171 (1992). R.A.Minlos and L.D.Faddeev, Commenton the problem of three particles with point interactions, Soviet Physics JEPT, 14:1315-1316 (1962). R.A.Minlos and L.D.Faddeev, On the point interaction for a three-particle system in quantum mechanics, Soviet Physics DokL, 6:1072-1074 (1962). B.S.Pavlov, The theory of extensions and explicitly solvable models, Uspekhi Mat. Nauk, 136:99-131 (1987). B.S.Pavlov, Boundary conditions on thin manifolds and the semiboundedness of the three-body SchrSdinger operator with point interactions, Mat. Sb. (N.S.), 136:163--177 (1988).

292 28.

29.

30.

Aibeverio and Kurasov B.Simon, Spectral analysis of rank one perturbations and applic.~tions, in Mathematical Quantum Theory. H. SchrSdinger Operators (Vancouver, BC, 1993), volume 8 of CRMProc. Lecture Notes, Pages 109-149, Amer. Math. Soc., Providence, RI, 1995. G.V.Skorniakov and K.A.Ter-Martirosian, Three body problem for short range forces. I. Scattering of low energy neutrons by deutrons, Soviet Phys. JEPT, 4:648-661 (1957). L.E.Thomas, Multiparticle Schr6dinger Hamiltonians with point interactions, Phys. Rev. D, 30:1233-1237 (1984).

Feynman and Wiener Path Integrals Representations of the Liouville Evolution I. ANTONIOU International Solvay Institutes for Physics and Chemistry, Brussels, Belgium / Department of Mathematics and Mechanics, MoscowState University, Moscow,Russia O. G. SMOLYANOV International Solvay Institutes for Physics and Chemistry, Brussels, Belgium / Department of Mathematics and Mechanics, MoscowState University, Moscow,Russia

Weshow that the Liouville equation for any Hamiltonian system can be identified with the SchrSdinger equation obtained by the SchrSdinger-Dirac quantization of some "extended" Hamiltonian system whose phase space is the product of two copies of the initial phase space and hence whose configuration space coincides with the initial phase space. This identification allows us to represent somesolutions of the Cauchy problem for the Liouville equation by Feynmanpath integrals [1] over trajectories in the phase space of the extended Hamiltonian systems. In turn such a representation gives a frame to investigate some models which allow us -- under suitable assumptions -- to develop the Fokker-Planck equation (= forward Kolmogorov’s equation) and the backward Kolmogorov equation and hence the Langevin’s equation (= stochastic Ito’s equation) describing physical Brownian motion as the Ornstein-Uhlenbeck process (see [2-5]). The Feynmanmeasure over a space of trajectories in the phase space (it is called also Hatniltonian, or symplectic, Feynmanmeasure) is defined to be a distribution having a given Fourier transform; the Feynmanintegral of a function is defined to be the value of the distribution on the function, assumed belonging to the domain of the distribution. In the paper the Feynmanintegrals are calculated as limits of suitable sequences of finite dimensional integrals. Besides we represent the Feynman integrals by integrals over the usual Wiener measure on the same space of trajectories. This new representation is used to obtain the representation of Feynman integrals by limits of sequences of finite dimensional integrals. Besides the representation by Wiener measure allows us to justify some limiting procedures which 293

294

Anton|ou and Smolyanov

arise in the process of investigation of some models of nonequilibrium statistical mechanics. Any vector space is assumed to be a space over the field of real numbers, and any topological spaces are assumed to be Hausdorff ones. The terminology and notations of the theory of topological vector spaces are used without cornments. For any locally convex space (LCS) E the symbol E x denotes the space of all continuous linear functionals on E equipped with the topology for which (Ex) x coincides with E as a vector space. A map ~ ofa LCS Et into a LCS E2 is called smooth if it is continuous and Ghteau differentiable infinitely manytimes and if for any n ~ N and for any compact K C E~ the mapping (x, h~,..., hn) ~ f(n}(x)h~ ... K x E~ x .’. x E~ -~ E2 is continuous.

1

FEYNMAN INTEGRALS

OVER TRAJECTORIES

After some definitions which are used in a rigorous approach to the Feynmanmeasure, we will present a quite new result: a representation of a Feynmanintegral over trajectories in the phase space byan integral over Wiener measure. The assumptions, under which this result is obtained, are rather restrictive; nevertheless, they are satisfied just in the case of the integrals whicharise in the representation of the solutions of the Liouville equations. Let E be a LCSand, for x e E, g ~ E×, ~o~(x) = ig(’~’), If ~ -~ i s a LCS of some complex valued functions on E containing the functions {~ : g ~ Ex} ghen the Fourier transform of ’0 ~ ()c/~)x (the elements of ($c~)x will be (:ailed distributions on E) is defined by (~’~)(g) = r/(~oa). If the span of ~o~ x is dense in 3c~ then any Y/~-distribution can be reconstructed by its Fourier transform. Belowwe need not explicit definition of ~rE; so we usually will not rnention ~’E. If b is a quadratic functional on Ex, a ~ E and a ~ C, a ~ 0 then the Feynman a-measure on E with the correlation functional b and with the mean value a (or, shortly, with parameters (b,a,~)) is the distribution ~,~,~ on E, whose Fourier transform is defined by (reb,,~,o)(9) = exp((a/2)b(g) \ / REMARK 1 If a = -1 and b is positively definite then the Feynman measure gSb,a,a can be identified with the Gaussian (cylindrical) measure. REMARK 2 Let E = C[0,1]; then E× is the space of all Borel measures on [0,1]. Let further a = 0, a = i and b : E× --> 1R~ be defined by: b(~z) f rain(s, t)t/(ds)u(dt). Then Cb,~,~ is the "classical" Feymnanmeasure. Let

nowQbeaLCS,

P=Q× andE=QxP.

DEFINITION 1 [6) Hamiltonian Feynman measure on E, with the mean value a ~ Q, is the Feynmanmeasure on E with parameters (b, (a, 0), i), where b is defined by b((q, p)) = 2p(q). Nowwe start to define the most important for the paper example of the Feynman measure -- a sequential Feynmanmeasure. Let S be a finite-dimensional Euclidean space. A (Bochner) locally integrable function g on S taking values in a Banach

295

PathIntegrals Representations of Liouville Evolution space, is called integrable (over S) if for any ~o E 79(s) there exists f g( s)~o(ss)ds, s s -~ 0; then the limit depends only on ~(0) and, by the definition, f g(s)ds s lira f g(s)~(-cs)ds, where c2 E D(S), ~(0) = 1. If T is a Banachspace then for .---~0 S

t ~ (0, oc) the symbolCp([0, t], T) denotes the vector space of all right continuous functions on [0, t], taking values in T, whose distributional derivatives are Radon vector measures. We will define the sequential Feynman measure on the space 8~ = C~([0, t], E) (t > 0) assuming that E (= Q × P) is finite-dimensional. For any z ~ Q and for any finite family a of elements of [0, t], a = {0 = to < tl < ... < t,~(~)+~ t} , le t ~( z,a) be a s et of a ll functions ~ ~ & suchthat the projection of ~(t) onto Q is equal to z and, if k ~ {0, 1,..., re(a)}, the restriction ~ onto [t~, t~+~) is constant; let also d(a) max{(t~+~ - t~) :k = 0, 1,. .. ,re (a)}. DEFINITION2 For any z E Q the sequential Feynman measure ~ on $~ is the functional on the vector space of functions f on 3t for which the following limit exists: m(o’)

f...fexp (i k=o lim ~ E

d(o-)-~o

m(~) ~ Pk(q~:+~-- q~:)) dpdq f/~ ,.,J’~ exp(i k=0 where f~ is the function on the product of re(a) copies of E which is defined f~((q,,p~),...,(q~(~),p,~(~)))= f(g~), g~ e ~(z,~), 9~(t~) = (%,p~), dql dp~ ...dqm(~)dpm(~). For such functions f the entity ~z(f) is defined to equal to this limit and is denoted by f f(q,p)~(dqdp). PROPOSITION1 For any z ~ Q the sequential Hamiltonian Feymnan measure.

Feynman measure (I)z

Proof: Let us define the duality between £t and itself

=

is the

by

+

here ({q,{~), (r~q,rlp) e ~ = Q, x P~., where Q~ Cp([O,t],q), P~ = C~( [O,t],P) and integrals f{p(r)rfq(T)dr, f ~?p(r)~’q(r)dr are integrals, resp., of integrands and r~p with respect to the measures, which are distributional derivatives, resp., of rlq and {q. The latter integrals are defined in a natural way: the only difference with the usual definition is that the product of number-valued functions is now substituted by the paring of elements of P and Q (one can use also the so called Kolmogorov’s integral). Then the doruain of ~: contaius the set and direct calculations showthat the Fourier transform F ~2: of (I)-" is defined (F qh~)(~q, ~) = (~.b (~q, ~p)+i({p( t) -~(0) which just provesthat (~ is a Hamiltonian Feynman measure. The following theorem describes some connections between the sequential man measure and the Wiener measure.

Feyn-

296

Antoniouand Smolyanov

Let again dime < ec and let w be the standard Wiener measure on Cp([0, t],E x E). Let moreover gC be an analytical function on the complexification of ExE and g be its restriction to ExE; we assume also that Vx E E the function z ~ g(x,z) is linear and hence 9(x,z) = go(z)z [= (go(x))(z)] where, for any x ~ E, 9o(x) is a linear continuous functional on E. THEOREM 1 Let g~ be the analytical extension of g0 on the complexilication of E, let %be the complex-valued function on (£t x £t, w ~ w) which is defined by:

) where ~ is an entire function and f denotes the stochastic integral, and let % be integrable w.r.t, any z-shift (z ~ E) of the Wiener measure w. Then ¢~ belongs the domain of the sequential Feynmanmeasure 0~, for any z ~ E, and

0

P~OOF The la~er dhnen~ion~l ~nte~r~I~; under~ome~ufl,~ble clmn~eof v~r~ble~ ~heyc~ube transformed~ntoin~e~r~l~ which~reu~ed~n ~he de~ni~on of ~he ~eq~en~i~l Feynmm~ inCe~r~l. ~EMA~ 8 A ~infil~rs~emen~ c~n be form~l~edfor hffiui~e-dimen~on~l spaces E.

2

HAMILTONIAN SYSTEMS (A COORDINATE-FREE APPROACH)

Webriefly review soIne elements of infinite-ditnensional Hamiltonian mechanics. Our considerations do not depend on the dimension of the corresponding phase space. DEFINITION3 [7,8] A symplectic LCS is a pail" (E,I) where E is a LCS and I e L(Ex , E), + =-I . A hamiltonian sy stem is a c ol lection (E, I, 7 /) where (E, is a symplectic space and 7/ is a complex valued function on E which is called a Hamiltonian flmction. (7-/is usually assumed to be a smooth function; the space is called a phase space of the system.(E, I, 7/)). DEFINITION 4 If (E, I, 74) is a Hamiltonian system then Hamilton’s equation for

(E,~, 74) :’(t) = I(74’(f(t))) with respect to a function f of a real variable t, taking values in the phase space E. The equation for the observables g is the Liouville equation:

~’(t) = -{7/,~(t)}

(2)

PathIntegrals Representations of Liouville Evolution

297

where g is a fnnction, of the real parameter t, taking values in a space of complex valued functions defined on E and {., .} is the Poisson bracket which is defined for any two (Gfiteau differentiable) functions ~, ¢ onE, by {%¢}(x) ~o’(x)(I(¢’(x))); one can also define Poisson brackets for vector valued functions on E and hence one can define equations for vector valued first integrals. Let now E be a Hilbert space. For some distributions u on E one can define Poisson bracket {~,u} and {u,~2} by: {~,u} = -{u,(p}, {u,(p} being the distribution on E which is the derivative of u along the Hamiltonian vector field he : z ~ I((p’(x)), E -~ Thederi vative of a dis tr ibution u alo ngany vector field k is the distribution denoted by dku which is defined by the assumption that for any test function ~ one has, in natural notations, (u, dk~) = (dku, ~) where dk¢p is the derivative of ~ aloug the vector field k, defined by (dk(p)(x) = ~p’(x)k(x); of course in order to make this definition correct one need to assume that dkT belongs to the space of test functions. If moreoverdku = ~3"k" u, where ~"k is a multiplicator in the space of distributions then/3 ~ is called the logarithmic derivative of u along k (el. [9]). In particular if k is a constant vector field, k(x) = ko for any x E E and if both u and dkou are the usual (a-additive) measures on E then/3~o is r-almost everywhere defined function on E which is called the logarithmic derivative of u along k0. If kl, k2 E E and u has logarithmic derivatives along kl and along k~, then u has also a logarithmic derivative along any linear combination of kl and k~, and moreover the mapping [span{k1, k~} ~ k ~ fl" k e £~(u)] is linear. Let again u be a probability measure on E, let u have logarithmic derivative along any vector from a vector subspace E~ of E, E~ being in turn a Hilbert space w.r.t. its ownnorm and let k be a vector field on E (k : E ~ E) whose range is contained in El. Then, if tr k’(x) = for an y x ~ E,where U(x) is thederi vative of k "al ong E~" and the trace is calculated w.r.t. E~, then ~3~exists iff the trace of the random linear operator k ® .B.~ (in E~) exists (a randomlinear operator in E~ is defined be a linear mappingof E~ into the space of E~- valued randomvariables on (E, u)); ~ in this case ~3~ k = tr k ®/3. (see [9]). If the vector field k is Hamiltonianttien tr k’ = 0 (if it exists). Underall these assumptions {u, ~o} =/37~, ¯ u = tr ((I~’) ® ~3")u. The Liouville equation with respect to distributions, is adjoint of (2): ¯ ’(t) = {7-/, ~(t)}

(3)

where ~ is a function of the real variable, taking values in a space of distributions on E and {., .} is the Poisson bracket of 7-/and ~(t). EXAMPLE 1 The system of equations (2) and (3) is the Ha~nilton equation the Hamiltonian systern (~, I~, 7-/e). Here £ is the product of a LCSE(E) consisting of some complex functions on E, and a LCS S(E) of some distributions on E, Ig is the multiplication on i (=v/-L~) and 7{g(~, u) = -iu({7-/, ~}). The statement this exampleis also a corollary to the followiug observation. THEOREM 2 The equation for first integrals Liouville equation for any Hamiltonian system is the Schrbdinger equation which is obtained by the SchrbdingerDirac quantization of the Hamiltonian system (EL, IL,’tlL) where EL = QL × PL , QL = E, PL = E’, IL(p,q) = (q,--p), 7-tL(q,p) = -’H’(q)Ip.

298

Antoniouand Smolyanov

PROOFThe SchrSdinger equation obtained by the quantization nian system (EL, IL, 7/n), has the following form

of the Hamilto-

.Of z-~-[ = iT/’(.)I(f(t)’(.) = i(~t,f(t)} but this is just the equation for first integrals for the Hamiltonian system.

3

FEYNMAN PATH

INTEGRAL

REPRESENTATIONS

In this section we formulate two theorems -- theorems 3 and 4 --- about such representations. Theorem3 gives the representation by integrals over the Wiener measure; Theorem 4 gives the representation by (Feynman) integrals over the Hamiltonian Fcynman measure. The proof of the Theorem 3 is based on Ito’s formula; the Theorem 4 can be deduced from the Theorem3 taking in account Theorem1. But actually the method to guess these theorems was quite opposite. On euristical level Theorem4 is implied by Theorem2 if one uses an euristical representation of solutions fbr SchrSdinger equations by Feynmanpath integrals over the phase space; after that one need to recognize in this representation just the sequential Feynmanintegral and after that one can use Theorem1 in order to get the (formal) statement of Theorem 3; after all that one need only to prove Theorem3 (using Ito’s formula). THEOREM 3 Let (E, I, 7/) be a Hamiltonian system with the finite dimensional phase space and let 74 and also ~? be restriction to E of some analytical fllnction defined on the complexification of E. Let, for any t ~ [0, to] and any z ~ E the function (Vl ,’b’2)~’--)’

¯

~(~Vl(~) ~ 1

I(d

1

exp {for 7/’ (z 1 l --~-~v,(r) -~v ~v~(r)))}~

~(r)).

~vl(t)(Z~

1

1

~V2(~))

be w-integrable. Then the Cauchy problem for the equation (2) with the initial data (0, ~) has a solution f on the segment [0, to] which is defined by l

.~ z + ~v~(t) PROOF

- ~v~(t)

l

__5~V~(T)).

)

i o(dv

It is quite similar to the proof of Ito-Feynman--Kacformula. Let

299

PathIntegrals Representations of Liouville Evolution ¯ I d(--~-~,iv,(T)

+ g’~v.2(~-))

Theu, due to Ito’s formula,

..)) --~( ~ --~v~ (t) + ~ v~v~( t) ) 1 v~(t) __~2x/~v;(t)).i ¢~(t)

+

47

+i)dt

l_~_d(v~(t ) + v/~v,~(t)).

’

47

similarly, 1

1

1 1

Proceeding similarly to [10] (see also [11], [12]) one can deduce from the latter identity that function f satisfies equation (2) and initial data (0,r/). THEOREM 4 Let assumption of Theorem 3 be satisfied and let f be the solution of the Cauchyproblem for the equation (2) with the initial data (0, ~).

l(t)(z)=f

--

~(q(t) )O"(dq

PROOFThis is the corollary to the theorems 3 and 1. REMARK 4 Some similar results to distributions.

4

ADDITIONAL

are. valid for Liouville

equations with respect

REMARKS

One can expect that the representations of the Liouville evolution by functional integrals can be as useful as well knownrepresentations of solutions for Schr6dinger equations, which now belong to the main tools in the quantum theory. Wewill list here ouly a couple of possible applications. One of them is of a pure methodological character: it allows us to obtain, for the Liouville evolution, the already known diagrmn technique. Somepossible applications to problems of nonequilibfium statistical mechanics are potentially more interesting. One of ideas in this direction can be formulated as follows. Using the functional integral representation for the Liouville evolution of a finite dimensional classical (sub)system and an (may be infinite dimensional) reservoir can integrate over degrees of freedom of the reservoir and pass to a limit under the functional integral. Then one can get just a functional integral representation for a solution of the (forward) Kolmogorov’s equation giving a Markovian approximation

300

Antoniouand Smolyanov

for the evolution of the subsystem. The Liouville equation itself is a nonstochastic forward Kolmogorov’s equation. Moreover, any Schr6dinger equation is the Hamiltonian e:quation for an appropriate classical Hamiltonian system (describing an infinite dimensional harmonic oscillator whosephase space is just the Hilbert space of the pure states of the quanturn system). On the other hand any mixed state of a quantum system ca~ be described by a probability on the Hilbert space of pure states of the system and the evolution of probability is precisely a Liouville evolution. Hence a Markovian approximation of the evolution of an open quantum system can be described by the Kolmogorov’s equations with respect to the functions or measures on the Hilbert space of states and these equations can be obtained from the equations for Liouville evolution by the above described method. Wemay extend this discussion to the quantum Liouville-von Neumannequation. Wethank Professor I. Prigogine for his interest and support. The financial support of the European Commission (DG XIII ESPRIT Project NTGONGS) and the Interuniversity Attraction Poles is gratefully acknowledged.

REFERENCES 1. O.G. Smolyanov, E.T. Shavgulidze. Functional integrals. MoscowUniv. Press, 1990. 2. I. Prigogine. Non-Equilibrium Statistical Mechanics. New-York--London.., Interscience Publishers, a Division of John Wiley and Sons, 1964. 3. R.J. Rubin. Momentum autocorrelation functions and energy transport in :harmoniccrystals with isotopic defects. Phys.Rev., V. 131, N’- 3, 1963, pp. 964---989. 4. H. Nakazawa. A statistical mechanical model of Brownian motion. Supplement of Progr.of Theoretical Physics, ~’-" 36, 1966, pp. 172-192. 5. E.L. Chang, R.M. Mazo and J.T. Hynes On the Fokker-Planck equation for the nonlinear chain. Molecular Physics, V. 28, 5’-~ 4, 1974, pp. 997-1004. 6. O.G. Smolyanov, M.O. Smolyanova. Transformations of Feynmanintegral under non-linear transformations of phase space. Theor.&Math.Phys,v.100, ~’-" 1, 1994, pp.3-12. 7. O.G. Smolyanov. Infinite dimensional pseudo-differential operators and Schroedinger quantization./Doklady Math., v. 263, N-* 3, 1982, pp. 558-562. 8. P. Chernofl’, J. Marsden. Infinite-dimensional Hamiltonian systems. Lect.Notes Math, V. 425, 1974. 9.. O.G. Smolyanov,H.v. Weizsaecker. Differential properties of measures. Infinite Dimensional Analysis, QuantumProbability and Related Topics, v. 2, B’-" 1, 1999. 10. B. Simon. Functional integrals in quantumphysics. Prinston University Press, 1979. 11. O.G. Smolyanov. Stochastic Schroedinger-Belavkin equation and corresponding Kobnogorovand Lindblad equations. Vestnik h’IGU, Moscow,.~’-0 4, set. 1, 1998, pp. 19-24. 12. S. Albeverio, V.N. Kolokol’tsov, O.G. Smolyanov. Continious quantum measurements: local and global approasches. Rev. in Math. Physics, vo].. 9, J~’-" 8, 1997, pp. 907-920.

Spectral Characterization of Mixing Evolutions in Classical Statistical Physics I. ANTONIOU International Solvay Institute for Physics and Chemistry, CP 231, ULBCampus Plaine, Bd. du Triomphe, 1050 Brussels, Belgium / Theoretische Natuurkunde, Free University of Brussels Z. SUCHANECKI International Solvay Institute for Physics and Chemistry, CP 231, ULBCampusPlaine, Bd. du Triomphe, 1050 Brussels, Belgium / Hugo Steinhaus Center and Institute of Mathematics, Wroclaw Technical University

1

INTRODUCTION

The decay of correlation functions, which is equivalent to mixing, qualifies physical systems approaching equilibrium and includes all types of chaotic behaviour. Although all ergodic properties have an equivalent description in terms of the spectra of the evolution operators in Hilbert space there is no necessary and sufficient spectral condition for mixing. The only sufficient condition knownis that of absolutely continuous spectrum which guarantees mixing from the Riemann-Lebesgue Theorem. The purpose of this work is to give the complete spectral characterization of mixing based on a recent characterization of decaying measures [1]. After a description of the ergodic hierarchy (in Section 2), we describe the ergodic properties in terms of the evolution operators (Section 3) and review their spectral manifestation (Section 4). The spectral characterization of mixing is obtained (Section 5) from identification of the decaying singular spectrum resulting from the decaying singular measures [1].

2

THE ERGODIC

HIERARCHY

OF DYNAMICAL

SYSTEMS

A classical dynamical system on the phase space f~ is described by the evolution group {St}, with t real for flows and integer for cascades. The phase space is 301

302

Antoniouand Suchanecki

endowed with a a-algebra ~ of measurable subsets of ~ and a probabilit:¢ p. Usually p is the equilibrium measure, i.e. 5’, preserve the measure #: #(S;-:(A))

measure

= #(A), for all A @.

The evolution of dynamical systems can be classified according to different ergodic properties which correspond to various degrees of irregular behaviom. Let, us recall the most significant ergodic properties [2, 3, 4, 5]: (I) Ergodicity expresses the existence of only one equilibrium measure, which meansthat for any t there is no nontrivial St-invariant subset of l’l, i.e. if for some t and A ~ ~ S[-~(A) = A, #-a.e., then either p(A) = orp(A) = 1 . Ergodicity is equivalent to the condition lim T 1 j~0 T-~+oo~ #(S~-~(A)

t3B)dt=p(A)#(B),

for all

A,B 6 ~.

(II) Weakmixing is a stronger ergodic property and is expressed for cascades by the fact that for each A, B ~ ~ l,i~moo #(A th S-’~(B) ) = p(A)#(B) where J is a set of density zero, which may vary for different choices of A and B. Recall that a set J C IN has density zero if lira card[J~ {1,...,n)]

: O.

Weakmixing is equivalent to absolute Cesaro convergence, for discrete and continuous time t: T-~oolim ~1 ~r (III)

[#(S~~ (A) f~ B) - #(A)p(B)[dt Mixing is stronger and means that lim #(A t3 S~-~(u)) = #(A)#(B) for all A, B ~ ~ It is easy to show that weak mixing implies ergodicity and obviously mixing implies weak mixing. Fully chaotic systems are qualified by the even stronger Kolmogorovproperty. (IV) Kolmogorovsystems are qualified by the existence of a sub-a-algebra ~0 of ~, called K-subalgebr’a, such that for ~. = St(~o) we have (i)

~ C i~i~ , for s < t.

(ii)

o-(U ~) = @.

(iii)

O~ I~t = ~-~, the trivial

a-algebra.

MixingEvolutionsin ClassicalStatistical Physics 3

THE EVOLUTION

OPERATORS

303 AND ERGODIC

PROPERTIES

The ergodic properties of classical dynamical systems can be conveniently studied in the Hilbert space formulation of dynamics due to Koopman[6]. Weconsider the Hilbert space L2 = L2(f~,~,#) of square integrable functions on ft. The transformations St induce the Koopmanevolution operators Vt acting on the functions f ¯ L2 as follows: Vtf(w) = f(Stw), w¯ As St preserve the measure # the operators Vt are unitary. In the case of flows the selfadjoint generator of Vt is knownas the Liouville operator Vt

iL .= te

(1)

In the case of Hamiltonian flows the Liouville operator is given by the Poisson bracket associated with the Hamiltonian function H Lf = i{H,f}. The Liouville operator is the starting point of equilibriuln and non-equilibrium statistical physics [7, 8, 9]. The ergodic properties (I-IV) of classical dynamical systems can be described as properties of the corresponding Koopmanevolution operators Vt or as properties of the selfadjoint Liouville operator L in the case of continuous time. This approach allows us to study and classify in a unified way dynamical systems in terms of operator theory and functional analysis. The description of evolution in terms of the Koopmanoperators Vt over L2 does not imply any loss of information about the underlying dynamics St. This was established by the converse to Koopman’s Lemma[10] which is an extension of the Banach-Lamperti theory of implementation of isometrics. In fact a positivity preserving group of evolution operators Vt on L~, Vtf >_ O, if f >_ 0, is implmnented by a unique measure preserving point transformations St of f~, i.e. Vtf(w) can be written as f(Stw). The ergodic properties (I-IV) can be expressed as properties of the corresponding evolution operators as follows: (I)

{St} is ergodic if and only if {I~f} is Cesaro convergent, i.e. r

lim 1 T--~c~ (II)

j£0 ~ fi4(g)

{St} is weakmixing if and only if {Vtf} is absolute Cesaro convergent, i.e. lim lf0r

(III)

dt = 0

IfVt(g)ldt =

{St} is mixing if and only if {Vtf} converges weakly in L2, i.e. lim (f, Vtg) =

for any two functions f, g ¯ L2 such that fn f d# = Ji~ g d# = 0.

304 (IV)

Antoniouand iSuchanecki The K-property can be described in terms of 1,~ in a similar way as for transformations St. Putting 74t d--fL2(a,~t,#)

O {1} and 7-/a=~L2(f~,~,#)

we have the analog of conditions (i)-(iii) (i’) (ii’)

characterizing K-flows

1/~(7/0) D Vt(7/0), for s lin( U 7/t) = 7-/(--denotes the closure) tEIR

(iii’)

N 7/t = {0}.

tEIR

Can we express the ergodic properties in terms of the spectra of Kooptnan operators?

4

SPECTRAL

MANIFESTATION

OF ERGODIC

PROPERTIES

The spectrum of the evolution group Vt, t ~ IR, corresponding to a flow is the spectrum of the selfadjoint generator L (1). The spectrum of the evolution group 1/’~, n ~ ~, is the spectrum of the unitary Koopmanoperator V = V:~ . The Caley transform relates naturally the spectra of unitary and selfadjoint operators [11]. The spectrum of an arbitrary dynmnical system may contain, like the spectrum of an arbitrary selfadjoint or unitary operator, any of the three components, namely discrete, absolutely continuous and singular spectrum [12]. Weassume below the knownfacts from the spectral theory of dynamical systems in the Hilbert space of square integrable functions [3]. (I) Ergodicity of a dynamical system amounts to imposing additional conditions on the discrete part of the spectrum. Namely a dynamical system with continuous time is ergodic if and only if 0 is a simple eigenvalue of L. In the case of discrete time ergodicity means that 1 is a si~nple eigenvalue of V. Ergodic dynamical systems may have, of course, eigeni~alues other than 1. However, as we shall see soon, the spectral characterization of properties stronger than ergodicity involves only the continuous part of the spectrum of a dynamical system. (II) It is therefore interesting to knowwhat are the dynamical systems which not have, apart fi’om 1, other eigenvalues. In other words, what ergodic property characterizes the dynamical systems which restricted to the Hilbert space 7/= have continuous spectrum. It turns out that the class of dynamical systems with continuous spectra coincides with the weakly mixing systems [2, 3]. (III) Since mixing systems are weakly mixing, a necessary condition for mixing is that the dynamical group {Vt} has purely continuous spectrum on 7/. This condition is, however, not sufficient. There are exa~nples of weakly mixing systems which are not mixing [4]. On the other hand, if the group {V~} has absolutely continuous spectrum then, applying the Riernann-Lebesgue lemma, it is easy to see that the’dynamical system corresponding to {Vt} is mixing. Therefore the class of dynamical syste~ns with absolutely continuous spectrum is a proper subclass of mixing systems. For an examplethat the converse inclusion is not true see [13]. (IV) The class of K-systems is in turn a proper subclass of dynamical systems with absolutely continuous spectrum. This is a consequence of the Weyl - von

MixingEvolutionsin ClassicalStatistical Physics

305

Neumanntheorem on the canonical commutation relations [5] group {V~.} of unitary operators satisfies properties (i’)--(iii’) has a homogeneous Lebesgue spectrum [5]. Weremark here homogeneousLebesgue spectrum are equivalently characterized a time operator canonically conjugate to Vt [14, 15]:

which says that a if and only if that systems with by the existence of

I/~tTV~ T + tI. On the basis of time operators Misra, Prigogine and Courbage [14, 15] elaborated a rigorous theory of irreversibility. In order to complete the spectral characterization of dynamical systems we must identify the necessary and sufficient conditions for mixing.

5

SPECTRAL

CHARACTERIZATION

OF MIXING

A necessary and sufficient condition for mixing is that the autocorrelation functions decay [4, 5].

/

nfV~fd#~O

for

eachfET/=L

2@1,

(2)

It follows from the above considerations that amongthe above mentioned ergodic properties only mixing does not have a satisfactory spectral characterization as we don’t knowthe necessary and sufficient spectral properties of the evolution operators for mixing. On the other hand, the property of mixing is very i~nportant in physics because not only mixing systems approach equilibrium in classical and quantum systems, but also because the decay of unstable quantum systems is a property of mixing systems. As there exist dynamical systems with continuous spectrum which are not mixing and as all dynamical systems with absolutely continuous spectrum are mixing the natural questions arises: Which property of continuous spectrurn characterizes precisely 1nixing systems? What is the spectrum of mixing systems? Consider, an abstract unitary group Vt = e itL acting on a Hilbert space 74, generated by the selfadjoint operator L with spectral family {Ex}: L =

)~dEx.

Denoteby 74p the closed linear hull of all eigenvectors of L. The continuous subspace of L is the orthocomplement of 74p: 74c = 74 ~ 74,. Recall that the singular continuous subspace 74sc of 74c consists of all f E 74c for which there exists a Borel set Bo of Lebesgue measure zero such that f~o dE)~f = f. By 74a¢ = Hc O 74sc we shall denote the absolutely continuous subspace of 74c. Recall also that 74p, 74¢, 74~ and 74~¢ are closed linear subspaces of 74 which reduce the operator L and that 74 = 74, @74a¢ @74sc. The spectra of the corresponding reductions of L will be called respectively point, continuous, singular continuous and absolutely continuous spectrum of L, and will be denoted by a,(L), ae(L), a~¢(L) and a~¢(L) correspondingly [16]. Let # = #t, denotes, for a given h ~ 74, the spectral measure on a(L) determined by the nondecreasing function Fh(A)=(h,

Exh),

for

AeIR.

306

Antoniouand Suchanecki

Let h = hp + hac + hsc be the decomposition of h correspouding to the direct sum 7/p c~ 7/ac ® 7/sc. Putting lip = Php, ~ac : ~th,¢ and #so = tths¢ we obtain the Jordan decomposition of # # = #p + #so + #ac (3) onto the point, singular continuous and absolutely continuous component. Conversely, given any three finite Borel measuresI~p,/-tac and l-tsc, where p, is concentrated on a countable set of points, and the other two measures are respectively singular and absolutely continuous, one can always construct a Hilbert space 7/and a selfadjoint operator L such that these measures are spectral measures associated with some h ~ 7/. Moreover, the point, singular and absolutely continuous spectrum of L coincides with #p, #so and #ac respectively. This can be proved by taking as 7/the direct sum L2(IR, #p) @L2(]R,/Zac) (~ L2(]R,/Zsc) and as L the operator multiplication by A [16]. It has been shownrecently [1] by S. Shkarin and one of us (I.A.) that the Jordan decomposition (3) of a a-additive measure can be further refined. Nmnely the Banach space Mof a-additive finite complex-valued measures on the real line IR is the direct sum of two linear closed subspaces MD and j~ND, where MD is the set ND of measures p ~ 3/I whose Fourier transform converges to 0 at infinity and 3d is the set of measures # ~ M such that t, ¢ MD for any v ~ ~ \ {0} absolutely continuous with respect, to the variation I#l of #. The corresponding decomposition /.t

= /~D -Jr pND

concerns in fact the singular continuous component in the Jordan decomposition since, obviously, each absolutely continuous componentdecays and the discrete does not. This result can be applied directly to obtain analogous refinement of the spectral decomposition of L or, equivalently, the group {Vt}. Let us call, following the property (2), decaying elements those elements h ~ 7/which satisfy (h,t~h)~0,

ast~.

and denote by 7/~ the set of all decaying ele~nents in the singular continuous subspace 7/~. The space 7/~ consists of all vectors tt ~ 7/~ such that the corresponding measure # = #h is singular with respect to the Lebesgue measure and its Fourier transform is 0 in infinity. 7/sDc is a closed linear subspace of 7/s¢.. Indeed, it is obvious that multiplication by scalars does not lead outside 7/~. Let h~,h~ ~ 7/~ and denote by #l and #2 the corresponding spectral measures. It follows from the inequality: Fh(A) = (E~,(h~ h2), hi + h~) ~ = IJE~hl -4-E)~h2[I

2 +~ NIIExh~+ E.xh~ll [IE~h~- E~h.~ll = 2 (llE~,h,II2 e) + liE,,hell = 2(Fh,(~) Fh=(~))

307

MixingEvolutionsin ClassicalStatistical Physics

that the measure #h is absolutely continuous with respect to both pl and tt2. There¯fore applying [1] Lemma 2 (see also [17] XII, Th. 10.9), we see that h e ~sDc The space 7{sDcis also closed. For, if {hn} E 7{s~, h~ ~ h in the n-norm, then for each Borel set A C ~

:

~,~(.4)

Therefore ~ converges weakly to ~ ([11] IV 9.3, Th.5). Since M~¢is a closed subspace of the Banach space ~ [1] it is also weakly closed. Therefore p ~ ~. Weshall show now that L does not lead out of the space H~ [16], i.e.:

where D(L) is the domain of L. Indeed, if h ~ D(L) then 2 = ~Lh]~ < On the other hand, the spectral me~ure ~Lh is determined by the function fLh(l)

= (Lh, ExLh)

A’2d(h, Ex, h).

Therefore eitXdFLh(A)

ei~A~dFh(A)

and applying [1], Lemma2, once again we see that Lh e ~. Following the notation of [1] let ~ be the space of all h e Hsc such that any measure u ¢ 0 which is absolutely continuous with respect to ~h does not decay. The space H~D will be called the space of non decaying singular elements. Weshall show that H~~ is also L invariant. Let h ~ ~c~ be in the domain of L and let Ph be the corresponding spectral measure. If Lh ~ ~):D then there is a component u of the measure ~Lh sucH that its Fourier transform ~(t) tends to 0 if t ~ ~. By [1] Th.1 u is just the restriction of ~ch to some Borel set A = A(h). Thus ~(t)

eitXlA(t)A~dph(t).

Note that the function ~-~ is integrable with respect to the measure u since A-~ du(A) =

A-~IA(5)a ~ dph(a) = ph(A).

Therefore ([1] Lemma2) ’[’~eita~

-~du(~)~0,

as

t~.

This leads to contradiction because the latter integral is the Fourier transform of the measure Ph restricted to the set A and, by the assumption on h, any measure which is absolutely continuous with respect to ~h cannot decay.

308

Antoniouand Suchanecki

It can also be shown[lS] that both spaces 71sDc and 71sNcD reduce the operator L [16] and that we have the direct sum decomposition 7-/sc = 71~ @7/~cD. This leads to the following direct sum deco~nposition of the whole space 71: 71 = 71. ¯ ~ ¯ 71~ ¯ ~.

(4)

Therefore denoting the corresponding spectra of reduced operators by a~,, and tisNc D respectively we obtain a new decomposition of the spectrum a of any selfadjoint operator which is the missing necessary fact to describe mixing and decay: ti = tip U O’ac U ti~.

U ti~O.

Weare therefore led to the following definitions: "HD = 7-/a¢ ~ 71sDcthe space of decaying ele~nents with respect to "]_/ND

~__. ~.~p

~ "~./sNc D the

space of non-decayingelementswith respect to Vt

o’D : tiac U O’sDcthe decay spectrum of L or Vt O"ND : tip

U

ti~D the non-decay spectrum of L or 1~

and to the following spectral characterization of mixing: THEOREM 1 A dynamical system is mixing if and only if the evolution {Vt}tem on 7t = L~ ~ {1} has purely decaying spectrum, i.e. a(L) :

tiD = .O.ac U tisDc

g~vup

(5)

Proof: As we mentioned in the beginning of this section the necessary and sufficient condition for mixing is that for each h ~ 7{ equality (2) holds. This however means that in the decomposition (4) of 71 consist only of the component71D 71ac This is equivalent that the spectrum of the dynamical system is of the form (5).

6

CONCLUDING

REMARKS

1. Weprovide here the complete spectral characterization I

of dynamical systems

Ergodicity: a(L) ap(L) U ac(L), 0 ~ ti p(L), 0 has multiplicity 1,

~.

II

Weakmixing: a(L) = ap(L) U ac(L), ap(L) = {0}, 0 has multiplicity a(L) = ac(L) on 7/.

III

Mixing: ti(L) = ap(n) U aD(L) = ap(L) a~(L) U aa~(L), ti p(L) = {0 has multiplicity 1 or ti(L) ti P(L) on7-/ Absolutely continuous spectrum: a(L) = tip(L)U ate(L), multiplicity 1 or ~r(L) = aac(L) on

IV

ap(L) = {0}, 0 has

K-systems: ti(L) = tip(L)UO’ac(L), tip(L) = {0}, 0 has mnltiplicity 1, tiac(L) has uniform multiplicity or a(L) = tiac(L) with unifbrm multiplicity on "H.

309

MixingEvolutionsin ClassicalStatistical Physics

2. Although the spectra of the Koopmanoperators in Hilbert space characterize the different ergodic properties mathematically, they do not give any information about the decay rates of autocorrelation functions, i.e. how fast a mixing system goes to equilibrium. This information is however essential for physical and engineering problems. The Brussels-Austin groups have shownrecently [19, 20, 21] that the rates of approach to equilibrium appear as resonance eigenvalues in extended spectral decomposition of the Koopmanoperator in rigged Hilbert spaces beyond the Hilbert space of square integrable phase functions. In fact these resonances determine the natural physical equivalence of dynamical systems [22, 23]. 3. The result of Section 4 allows to provide a spectral characterization of quantum mixing states which decay. The wave functions ¢ are elements of a separable Hilbert space 7{ evolving under the unitary solution Ut, t E lR, of the Schroedinger equation Ot¢=-iH¢,

t~=

l,

Vt = e -iHt ¯ If {E~} is the spectral family of H we have H = f wdE~ Ja (H) I"

e-~’~ dE,~. Ut = [ J~ (H) The survival amplitude of the pure state ¢ evolving under the unitary group Ut is : I~,Ut%b) : ~ H e-iWtd{%b,E~%b ( Therefore ~p is decaying if and only if ¢ is in the decaying subspace ~D of the Hmniltonian H

A more genera] discussion involving the spectra of the L]ouvflle-von Neumann erator Lf : [H, f] in the Hilbert-Schmidt space [24, 25] will be presented elsewhere 4. The distribution of the support of the spectral measures of the evolution operators provide another characterization of mixing and decay. This result will be presented elsewhere. ACKNOWLEDGMENTS We thank Professor I. Prigogine for his interest and support and S. Shkarin for discussions. This work enjoyed the financial support of the European Commission through the ESPRIT PROJECT NTCONGSDG XIII and the Belgian Governmentthrough the Interuniversity Attraction Poles.

310

Antoniouand Suchanecki

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

14. 15. 16. 17. 18. 19. 20.

21.

22. 23. 24. 25.

I. Antoniou and S.A. Shkarin, Decaying measures Math. Docl. (accepted). P.R. Hahnos Lectures on Ergodic Theory, Chelsea Publishing Company, New York 1956. V.I. Arnold and A. Avez Ergodic Problems of Classical Mechanics, Benjanfin, NewYork 1968. W. Parry Topics in Ergodic Theory, Cambridge University Press 1981. I. Cornfeld, S. Fomin and Ya. Sinai Ergodic Theory, Springer-Verlag, Berlin 1982. B. Koopman, Hamiltonian systems and transformations in Hilbert spaces, Proc. Nat. Acad. Sci. USA 17 315-318 (1931). R. Balescu Equilibrium and Non-equilibrium Statistical Mechanics, Wiley, New York 1975. I. Prigogine Non-Equilibrium Statistical Mechanics, Wiley, NewYork 1962. I. Prigogine From Being to Becoming, Freeman, NewYork 1980. K. Goodrich, K. Gustafson and B. Misra, On converse to Koopman’s lemma, Physica 102A 379-388 (1980). N. Dunford and J. Schwartz Linear Operators I : General Theory, Wiley, New York 1988. N.I. Akhiezer and I.M. GlazmanTheory of Linear Operators in Hilbert Space, Vol. I and II, Pitman Publishing Limited 1981. N. Wiener and E.J. Akutowicz, The definition and ergodic properties of the stochastic adjoint of a unitary transformation, Re’ud. Circ. Mat. Palermo (2) 6 205-217 (1957). B. Misra, I. Prigogine and M. Courbage, From deterministic dynamics to probabilistic descriptions, Physica 98 A 1-26 (1979). S. Goldstein, B. Misra and M. Courbage, On intrinsic randomness of dynamical systems, J. Star. Phys. 25 111-126 (1981). H. WeidmannLinear Operators in’Hilbert space, Springer Verlag 1980. A. Zygmund,Trigonometric Series Vol. I,II, Cambridge, University Press 1968. I. Antoniou, S. Shkarin and Z. Suchanecki, Decay spectrum and decaying subspaces, (to appear). I. Antoniou and S. Tasaki, Generalized spectral decomposition of mixing dynamical systems, Int. J. QuantumChemistry 46 425-474 (1993). I. Antoniou, L. Dmitrieva, Yu. Kuperin and Yu. Melnikov, Resonances and the extension of dynamics to rigged Hilbert spaces, Computers Math. Applic. 34 399-425 (1997). I. Antoniou and Z. Suchanecki, Extension of the dynamics of unstable systems, Infinite Dimensional Analysis, QuantumProbability and Rel. Topics 1 127-165 (1998). I. Antoniou and Bi Qiao, Spectral decomposition of the tent maps attd the isomorphism of dynamical systems, Physics Letters A 215 280-290 (1996). I. Antoniou, V. Sadovnichii and S. Schkarin, Newextended spectral decomposition of the Renyi map, Phys. Left. A (1999) (submitted). I. Antoniou, M. Gadella and Z. Suchanecki, General properties of the Liouville operator, Int. J. Theor. Phys. 37 1641-1654 (1998). I. Antoniou, S. Shkarin and Z. Suchanecki, The spectrum of the Liouville-von Neumannoperator in the Hilbert space J. Math. Phys. (in press).

On Stochastic SchrSdinger Equation as a Dirac Boundary-Value Problem, and an Inductive Stochastic Limit V. P. BELAVKIN School of Mathematics, Nottingham University, E-mail address: vpb©maths, nott. ac. uk

NG7 2RD, UK,

Dedicated to Sergio Albeverio.

ABSTRACT ~Ve prove that a single-jump quantum stochastic unitary evolution is equivalent to a Dirac boundary value problem on the half line in one extra dimension. It is shownthat this exactly solvable modelcan be obtained from a Schr6dinger boundaryvalue problemfor a positive relativistic Hamiltonianin the half-line as the inductive ultrarelativistic limit, correspondent to the input flow of Dirac particles with asymptotically infinite momenta. Thus the problem of stochastic approximation is reduced to the to the quantum-mechanical boundary value problem in the extra dimension. The question of microscopic time reversibility is also studied for this paper.

1

INTRODUCTION

The stochastic evolution models in a Hilbert space have recently found interesting applications in quantum measurementtheory, see for example the review paper [1]. In this paper we are going to show on a simple example that classical discontinuous stochastics can be derived fi’om a quantum continuous deterministic conscrvative dynamics starting from a pure quantum state. It has been already proved in [2] that the piecewise continuous stochastic unitary evolution driven by a quantum This work was supported by Royal Society research grant GR/M66196.

grant

311

for UK-Japan collaboration

and EPSRC

312

Belavkin

Poisson process is equivalent to a tilne-dependent singular Hamiltonian SchrSdinger problem, and the continuous stochastic unitary evolution driven by a quantum Wiener process can be obtained as the solution of this problem at a central limit. Here we want to start form a non-singular time-independent dynamics. There exits a broad literature on the stochastic limit in quantumphysics in which quantumstochastics is derived from a nonsingular interaction representation of the Schr6dinger initial value problem for a quantum field by rcscaling the time and space as suggested in [3]. Our intention is rather different: instead of rescaling the interaction potentials we treat the singular interactions rigorously as the boundary conditions, and obtain the stochastic limit as an ultrarelativistic limit of the corresponding Schr6dinger boundary value problem in a Hilbert space of infinite number of particles. Weshall prove that the discontinuous and continuous quantum stochastic evolutions can be obtain in this way from a physically meaningful time continuous (in strong sense) unitary evolution by solving a boundary value problem with an initial pure state in the extended Hilbert space. First we shall describe the boundary value problem corresponding to the singlepoint discontinuous stochastic evolution and demonstrate the ultrarelativistic li~nit in this case. Then the piece-wise continuous stochastic evolution and the continuous diffusive and quantumstochastic evolution can be obtained as in [2, 4]. But before to perform this program, let us describe the unitary toy model giving an "unphysical" solution of this problem corresponding to the free hamiltonian h (p) = -p. This toy model in the second quantization framework was suggested for the derivation of quantum time-continuous measurement process in [5]. Recently Chebotarev [6] has shown that the secondary quantized time-continuous toy Hamiltonian model in Fock space with a discontinuity condition is equivalent to the Hudson-Parthasarathy (HP) quantum stochastic evolution model [7] in the case of commuting operatorvalued coefficients of the HP-equation. Our approach is free from the commutativity restriction for the coefficients, and we deal with time-reversible Dirac Hamiltonian and the boundary rather than physically meaningless discontimfity condition and time irreversible -p. Moreover, we shall prove that the stochastic model can be obtained from a positive relativistic Hamiltonian as an inductive ultra relativistic limit on a union of Hardy class Hilbert spaces. Wecall this limit the inductive stochastic approximation.

2

A TOY HAMILTONIAN

MODEL

Here we demonstrate on a toy model howthe time-dependent single-point stochastic Hamiltonian problem can be treated as an interaction representation of a self-adjoint boundary-value Schr6dinger problem for a strongly-continuous unitary group evolution. Let 7/be a Hilbert space, H be a bounded from below selfadjoint operator, and S be a unitary operator in 74, not necessarily commuting with H. The operator H called Hamiltonian, is the generator for the conservative evolution of a quantum system, described by the Schr6dinger equation ihOt~I = H~h and the operator S (:ailed scattering, describes the unitary quantuna .jump r/~-~ S’q of the state vectors r/ff 7/caused by a singular potential interaction in the system, with the continuous unitary evolution r/~-~ e-~;t~’r~ whenthere is no jump. As for an exaluple of such

Stochastic Equationas a Dirac Boundary-Value Problem

313

junlp we carl refer to the vou Neumannsingular Halniltonian model for indirect instantaneous measurement of a quantum particle position x E I~ via tile registration of an apparatus pointer position y E II{. It can be described [2, 4] by the x-pointwise shift S of y as the multiplication ~ by a (x) = ~°, i n t he Hilbert s pace Lz (II~ 2) of square-integrable functions ~ (x,y), and it does not commutewith the free I-Iamiltonian operator H = 7 (Y~" - 0~) say, of the system "quantum particle plus apparatus pointer". It is usually assumed that the quantum jump occurs at a random instant of time + t , = s with a given probability density p (s) > 0 on the positive half of line 11~ f~ p (s) ds = 1. If H and S commute,the single-point discontinuous in t stochastic evolution can formally be described by the time-dependent Schr6dinger equation ihOtx (t) = Ha (t) X withthe s ingular stoch astic Hamil tonian H~(t) = H + itt6~ (t) In S, where 6~(t) = 6(t-s) = dr(s) is the Dirac &function of z = s-t. Indeed, integrating the time-dependent Ha~niltonian H,. (s) over r from 0 to t for a fixed s ~ N, one can obtain V (t,s) = e-~ f~/%(r)d,r = e-~,tHS&~o(S) = e~,{s-t.)nSA~o(s)e-~,sH ’ where A~ (.s) = f~ 6~ (s)dr is identified with the indicator function 1[o,0 of interval [0, t) for at > 0 (at t_< 0it is zero ifs > 0). The right hand side is for~n of the unitary stochastic evolution V (t, s) which should remain valid even the operators H and S do not commute. First the evolution is conservative and continuous, V (t, s) = e-~ ~ for t ~ [0, s), then the quantum jump S is applied at t = s, and at t > s the evolution is again continuous, described by the Hamiltonian H. As it was noted in [2], the rigorous form of the stochastic SchrSdinger equation which gives such solution even for noncommutingH and S in the positive direction of t, is the Ito differential equation dtV(t,s)+~HV(t,s)dt=(S-I)V(t,s)dlt(s),

t>0,

V(0,

Here dtV (t, s) = V (~ + dr, s) - V s) i s t he forward diff erential corr esponding to an infinitesimal increment dt > 0 at t, and dl~ (s) = t (s - t) is theindi cator function Atd~ (s) = l[t,~+dt} (S), the forward increment of the Heaviside function t ~ 1~ (s) = 10 (s - t), where lo = 1{-o~,0/. The equation (2.2) simply means that t ~ V (t) for a fixed s = z satisfies the usual SchrSdinger equation ihO~V (t) HV(t) if t ¢ s as dlt (s) = 0 for a sufficiently small dt (dr < s - t if t < s, and dt > 0 if t > s), while it jumps, dtV -- (S - I) V at t = s as dlt (s) It=s -- 1 >> Integrating d~)~ (z) = d~V (z) ~ on the domain of the operator H first from z = s with an initial condition X (0) = rh and then from s to t with the initial condition X (s+) := limz-,~, (z) = S:~ (s) one can easily obtain the solution form X (t, s) = V(t, s) r~, where V (t,z)

= e-~Hs (z) ll°’°(z)

-~ zH , S(z) e~ZHSe

without the commutativity condition for H and S.

314

Belavkin

Nowwe shall prove that the stochstic single-jump discontinuous evolution V (t) can be treated as the interaction representation V (t)X ° = e~th(:)Xt for a deterministic strongly-continuous Schrbdinger evolution xo ~ Xt in one extra dimension z E IR with the initial conditions X° (z) = ~ (z)~/E H localized at z > ° (z) = 0 at z _< 0. Here h (/~) = -~fi is the free Hamiltonian, where [~ = -ihO: is the. momentum in one extra dimension of z ~ l~. PROPOSITION 1 The described stochastic Hamiltonian problem (2.2) is unitary equivalent to the self-adjoint boundary-value Schriidinger problem ihO, Xt (z) = (H + il~Oz) t ( z) , Xt (O_) =Sxt(O)

(2.4)

in the Hilbert space 7l ® L2 (~) in the sense that the stochastic evolution V (t) t > 0 coincides with the unitary cocycle V (t, z)X ° (z) = ~ ( z- t ) r esolving t boundary value problem (2.~) with respect to the plane propagation t°~ along z as Xt = e tO" V (t) °, YX° ~~-I ® L2(~) Proof. The boundary value problem (2.4) is well defined on the space of smooth square-integrable ~-valued functions X, and is symmetric as H is self-adjoint, and due to the unitary boundary condition 0= (IIX(-0)II~-IIX(0)II

~) =2 Re{~(z) lx’(z)}dz=

In fact, this problemis selfadjoint as it has apparently unitary solution

x~ (z) = e¼~’"~ (z + t), ~ (s) = S~(~)~o

(2.5)

where Xo (s) = e-~Hzx.° (8). Indeed, substituting Xt (z) = e~zUx~ (z) into the equation {2.4} we obtain the transport equation OtXto (z) = OzX~O(z) with the same boundary condition X~ (-0) SXto (0) an d th e in itial co ndition X] = X0 cor responding to a X° ~ N ® Le (IR). This simple initial boundary-value problmn has the obvious solution X~(z) = X*(z + t) with Xt given in (2.5) ~ (s) = Sq°")(t-~)X° (s), t > 0, X* (s) = S-~I-’.°)(~),~ ° (s), t <

(2.6)

(s} in 7{ and of shift eta’ in L~ (II~) i~nplies the unitarity The unitarity of S~X~ t the resolving map V : Xo ~ Xt in N @L~ (~),

I1211 : = : Moreover, the map t ~ Vt has the multiplicative representation property 17r~ ft : 1,’~+t of the group R ~ r, t because the tnap t ~ Saa(*} is a multiplicative shiftcocycle, S~(*)et°’S ~(*} = et°’S a~+’(s), Vr, t E ~ by virtue of the additive cocycle property for the commutingA~(s) = It (s) - lo [A~+ e"°" A~](s) = 1~ (s) - lo (s) + It (s + t) - lo (s et°’A~+" (s).

Stochastic Equationas a Dirac Boundary-Value Problem

315

The subtraction X (t, z) e~thX* (z ) offree evolution wit h the generator ~X ( z) ihO~X(z) obviously gives

x (t,

s) = x - t) = ° ( s),

= e~(S-t)gsa~o[S)e-~SltX° = V (t,s) Thus the single-point discontinuous unitary e-~,th-cocycle V(t,

8)

:e ~t~V ~(=es-t)H~A~o(s

o

)

-~ e sH

,

tEll~

with /X~ (s) = l[0,t ) (s) for a positive t and s E II~+, solves indeed the single-jumpIto equation (2.2). It describes the interaction representation for the strongly continuous unitary group evolution Vt resolving the boundary value problem (2.4) with initially constant functions X° (s) = ~ at s > 0. REMARK 1 The toy SchrSdinger boundary value problem (2.~) is unphysical three aspects. First, the equation (2.~) is not invariant under the reversion time arrow, i.e. under an isometric complex conjugation ~7 ~-~ ~ and the reflection t ~-~ -t, even if ~ = S-~ and ImH = 0 as the Hamiltonian ~ = i~iOz is not real, hn ~ = lion.. Second, a physical wave function et (z) should have a continuous propagation in both directions of z, and at the boundary must have a jump not in the coordinate but in momentumrepresentation. The momentumcan change its direction but not the magnitude (conservation of momentum)in the result of the singular interaction with the boundary. And third, the free Hamiltonian ~ must be bounded from below which is not so in the case of hamiltonian function h (z, p) = corresponding to the equation (2.~). Nowwe showhowto rectify the first two failures of the toy model, but the third, which is a moreserious failure, will be sorted out in the next sections by considering the toy model as a dressed limiting case. Instead of the single wavefunction X~ (z) on I~ let us considering the pair (¢, of input and output wave functions with

¢’(z) = x’(z), > 0,

= x"

<0

on the half of line II~ +, having the scalar product

They satisfy the system of equations i~Ot~ t(z)=(H +ihO~)¢t(z), f12 ° e~®L~(~+) ~ ( II~ ihO~(~~ (z) = (H - ih0~)~ (z), ~0 e ~R ® +) for a quantmnsystem interacting with a massless Dirac particle in 1~+ through the boundary condition ~t (0) = S¢t (0), where ~t (0) ~ (0_). One can showthat this is indeed the diagonal form of the Dirac equation in one dimension in the eigenrepreseutation of the Dirac velocity c = -az along z ~ ll~ +, with the electric and

316

Belavkin

magnetic field componentsu~:, given by the symmetric and atisymmetric parts u + fi of u on I~ in the case In, u = 0. The componentsof (¢, ~) propagate independently at z > 0 as plane waves in the opposite directions with a spin (or polarization) oriented in the direction of z, and in the scalar case H = C are connected by the Dirac type boundary condition (1 + i#) ~t (0) = (1 ) Ct (0) c orre spondent to a point mass h# at z = 0. The input wave function Ct (z)

e-~tg¢ ° (z + t) _-~t(h+H~

(2.7)

is the solution to the equation (2.4) at z ~ ll~+with X°[~>0 = ¢o which does not need the boundary condition at z = 0 when solving the Cauchy problem in t :> 0. The output wave function satisfies the reflected equation at z > 0 and the unitary boundary condition at z = 0: ihOt~t (z) = (H - il~O,) ~b’ (z), ~’ (0) =

(2.s)

It has the solution ~’ (z) = -{’~ [ ~o (z - t) z (z) + S (t - z)¢o(,_ z )1, where 1~ (z) = 1 - l~ (z). This can be written in the similar way as

° (z - t) = ~-~"(~’+~)6 ° (z) 6~(z) =~-~’~V3

(12.9)

with ]~ = -ih8~ if 0o (z) is extended into the domain z < 0

S (Z) Ct (Z) = ~7 (--Z), S (Z) ~.-H,~ -~H

(2.10)

at t = 0. Note that reflection condition (2.10) remains valid for all t > 0 if ¢t extended into tile region z < 0 by (2.7) for all t ~ l~+: ~bt (-z)= e-~tH s (t + z)¢O (t + z) = S (z)e-~’tH¢O (z + t) = S (z)¢t Extending also the output wave ~t by (2.9) into the region z < 0 we obtain the continuous propagation of ¢, ~ through the boundary in the opposite directions, with the unitary reflection holonome connection (2.10) for all z ~ N. If/~ = where/~r/ = H---~ with respect to a complex conjugation in 7/, then the system of Sc}lrgdinger equations for the pair (¢, ~) remains invariant under tile time relicttion with complex conjugation up to exchange ~-t ~_ #)t. Indeed, in this case the complex conjflgated hamiltonian ~ = -ittO, coinsides with the operator [~ corresponding to h (z,p) = p = f~ (z, p). The boundary value problem is invariant under time reversion if 3 = S-1 as the reflection condition (2.10) is extended to the negative t by the exchange due to S (z) -t = o~ (-z). Thus the reversion of time arrow is equivalent to the exchange of the input and output wave functions which is an involute isomorphism due to

=dz = Ilxll== I1¢(~)11

(z) dz.

Stochastic Equationas a Dirac Boundary-Value Problem 3

A UNITARY

REFLECTION

317

MODEL

As we have seen in the end of the previous section, a unitary quantumstate jump at a randominstant of time s _> 0 is a result of solving of the toy SchrSdinger boundary value problem in the interaction representation for a strongly continuous unitary evolution of a Dirac particle with zero mass. The input particle, an "instanton" with the state vectors defining the input probabilities for s = z, has the unboundedfrom below kinetic energy e (p) = -p corresponding to the constant negative velocity v = e’ (p) = -1 aloug the intrinsic time coordinate z which does not coincide with the direction of the momentum if p > 0. One can interpret such strange particle as a trigger for instantaneous measurement in a quantum system at the time z E ~+, and rnight like to consider it as a normal particle, like a "bubble" in a cloud chamber on the boundary of lI~ d x R+ as it w~ assumed in [2], with positive kinetic energy and a non-zero mass. Our aim is to obtain the instanton as an ultrarelativistic limit of a quantum particle with a positive kinetic energy corresponding to a m~s m0 ~ 0. Here we shall treat the kinetic energy separately for input and output instantons as a function of the momentump ~ ~- and p ~ ~+ respectively along a coordinate z ~ ~+ with the same self-adjoint operator values e (p) ~ 0 in a Hilbert space b its spin or other degrees of freedom. For exampleone can take the relativistic mass operator-function e(p) = (pe + h:~) ~/2, ~ ~

= p~- V

(3.1)

in the Hilbert space ~ = L: (~) which defines the velocities v (p) = pie (p) e~ (p) with the same signature as p. At the boundary z = 0 the incoming particle with the negative momentump < 0 is reflected into the outgoing one with the opposite momentmn-p. The singular interaction with the boundary causes also a quantum jump in other degrees of freedom. It is described by the unitary operator a in ~ which is assumed to commute~ith e (p) for each p as it is in the quantum measurement’model [2] when a = e ~oy with V = 0y in (3.1}. Let ~ be a Hilbert space with isometric complex conjugation ~ ~ ~ ~ ~ ~ b, and L~ (~-) = ~ ~ ~ (~-) b e t he s pace of s quare-integrable v ector-functions f (k) ~ on the half-line ~- ~ k. Wedenote by ~- the isomorphic space of Fourier integrals :(z)=

~1 f~eiazf(k)

dk,

f ~L~(~-).

which is the Hardy class of ~-valued functions T ~ L~ (~) having the analytical continuation into the complex domain Im z < O. One can interpret L~ (~-) as the Hilbert space of quantum input states with negative momentap~ = hk, k < 0 along z ~ ~ and spin states ~ 6 ~. The generalized eigen-functions ~ (z) = exp [ikz]

~, k < O, e (hk) ~ = ~

corresponding to spectral values ~ ~ ~+ of ¢ harmonic waves moving from infinity towards ~./[k I along z. The amplitudes ~ are arbitrary to the identity operator 1 in ~, ~ (k) = sal, as

~=~.

(3.2)

(k) h-~e (h k), ar e gi ven as the z = 0 with the phase speed ¢~ = in ~ if all e (p) are proportional it was in the previous section where

318

Belavkin

The singular interaction creates the output states in the sa~ne region z > 0 of observation where the input field is, by the momentum inversiou p = -Pk ~-~/5 = P~,, reflecting the input wave functions ~ ~ ~- isometrically onto 1

~(~) = ~/ff ~ e-ik~](k)dk

= a~(-s) ’ s

by ](k) = af(k), k < 0. The space $+ = {~:~ e ~-} is the conjugated Hardy subspace $+ = {~ : ~ ~ ~- } of analytical functions ~ (z) = ~ (5) in Im z > 0. reflected wave function satisfies the boundary condition ~ (0) = a~ (0) corresponding to the zero probability current j (z) = ~(z)~~ ~ - ~[~(z)~[ at z : 0, and togetber with the input wave function ~ (8), s ~ 0 represents the Hilbert square norms (total probability) in $- and $+ by the sum of the integrals over the halSregion ~:

]l~(8)ll~ + ll~(s)~] 2 ds =

II~(z)ll~dz

~l@(z)~12dz.

The free dynamics of the input and output wave functions can be described ~ the unitary propagation 1

~ (~) = ~ ff

eik(t¢(k)+z)f

1/2

~ (z) = ~ ei~’(~(~’)-~If(k)

(k) dk = [e-i’e~] (z), dk = [e-i’g~]

(3.3)

(z),

of a supe~osition of the ha~onic eigen-~nctions (3.2) in the negative and positive direction of z E N respectively with the same phase speeds Ck > 1 which are the eigen-values of the positive operators g (k) Ikl -~ e (k). The ge nerating se lfadjoint operators g, g are the restrictions g = e (iOn) I’D-, g = e (iOn) of t he kinetic energy operator given by the sym~netric function e (p) ou its symmetric dense domain ~ ~ L~ (N), to the dense domains ~ = ~ ~ g* in the invarim~t subspaces g~ ~ L~ (N). Instead of dealing with the free propagation of the input-output pair (~,~) the region z > 0 with the boundary condition ~ (0) = ~ (0), it is convenient to introduce just one wave Nnction

¢~(z) = ~ (z), Re0,¢~(z)= ~(-z), Re z < 0(~; considering the reflected wave as propagating in the negative direction into ~he region z < 0. Each ¢(-z) is aHardy class function a~ at z > 0, as wellasit is Hardy class function ~ (-z) at z > 0, but the continuity of the analytical wave functions ~ (z) at Re z = 0 corresponds to the left discontinuity ¢ (0_) = ¢¢ (0)

¢ (z)=~o(-z) ~ (z) + ~o(~) ~ (z), ~o(z)

z~o,

where¢ (0_) is defined as the left lower sectorial limit of¢ (z) at Re z ~ 0, Im z fi~ Obviously the Hilbert subspace ai°g - C L~ (N) of such wave functions is isomorphic

Stochastic Equationas a Dirac Boundary-Value Problem

319

to ~- by the unitary operator a1° = I + 5o (a - I), where 5o is the multiplication operator of ~ (z) by 1 if z < 0, and by 0 if z >_ 0. The unitary evolution group v t : ai°e-n~a-i°,t E ~ for ¢~ (z) = ~pt (z) + 10 (z) (a - 1) ~ (z) t (z),

(3.~)

is unitary equivalent but different from the free propagation e -ire of ~t in R. Each harmonic eigen-function (3.2) having the plane wave propagation

~o[.(z) = e-*~"~ (z) = ~ (z + for the negative k E II~-, is nowtruncated, ¢~ (z) ei kzcr~o(z)O~, and pr opagates in the negative direction as

¢[. (z) = ~o(~(z + ~t) = e-~’~t¢~ (z) # ¢~,(z keeping the truncation at z = 0. Therefore the subtraction Ct (z) = eite¢ t (z) of the free propagation of qot fi’om Ct does not return it to the initial ¢0 = ~rioqo0 but to Ct = a~-qa ° =vt¢o, where ~t = eit~ioe-it~, vt = a~a-i°, ~nd ¢0 ¢0. Thus we have proved the following proposition for the particular case >c = 0 of an operator >¢ ~ ~, defined in the Proposition 1 of previous section as Let ~ be a selfajoint operator in b, and ~, (z) --= e-i~z be the correspondent oneparameter unitary group in I}. Belowwe shall denote by ~,~ and ~,~ the operators of pointwise multiplication by the functions e,, : z ~-~ e× (z) and ~ : z ~ e,, (-z) z ~ II~ respectively. Both these operators are unitary in the Hilbert space L~ (ll~). If h~ = ]z is an operator in L~ (tl 0 which is given as a pseudo-differential operator h (z, ~~0 ~) = lit (z,iOz), the operator-function ~/ (z, ~ +~c)=~*~+~ (z) ^~

(z, iOz)~+~(z) =_"~+~

(3.6)

defines the symbol7~ (z, ~) = (z , ~c ÷ ~) of theoperator

It is defined on the exponential functions e~ (z) = -i~-" as t he pseudo-differential operator

PROPOSITION2 Let ~ = ~- be the Hardy class of L~ (~), the Hilbertspaceof functionsqo=~*~owith~oe g~,and~ ~ = ~+. Let the initial boundary-value Schrddinger problem iOt~t (z) = e~ (iO~)~t (z), ~0 ~ $;,z >

~, C L~ (~) =e,go’*"+,where (3.7)

ia~~ (z) = ~ (iaz) ~’ (z) , ~ (o) = be defined by the generators g~, ~ given by the symbols ~ (~) = ~ (~ + ~), ~ (~) ~ (~ - ~) respectively, where ~ (~) is the symmetric.function o] ~, ~ ~, corresponding to the kinetic energy e (p) = ~ (h-~p) > O. Then it is sel]adjoint if the initial output waves ~o are defined in ~ by ~o (-z) = a~ (z) ~o (z), z < O, where a~ =

320

Belavkin

by analytical continuation of each 99~ = ~99o into the domain~-. The solutions to (3.7) can be written

~ (z) = ~ (z),z > 0, ~ (-z) = ~ (z_),z < 0 where ¢~= e-i~¢t, a,~ (z), and

(~.s)

c~t= 99 0 +-a~)7r,~99 (1 ~ ~ , 5~ is pointwise multiplication ^t

eit~z~

~o~-itg,,

by

t (z,iO..)

~_ ff>~

is given by the symbol 7r~ (z, ~) of the orthoprojector #t = en~ioe-ite as in (3.6). Proof. Separating the variable t ~ I~ by 99t = consider the stationary Schrhdinger problem

e-i~.~t99k,

~t =. e-i~.~t~k,

~,, (z, iOz) 99~ (z) = ~k99k(z), ~k (--z) = a~ (z)

let us (3.9)

corresponding to the given initial and boundary conditions in (3.7). Here 99k extended to the domain ~- through the analytical continuation of e~k in Im z < 0, which are the generalized eigen-functions (3.2) of g = (i On) in $~ iff k < 0. Due to the selhadjointness of g in $-, the eigenfunctions ~ = e~+~ of g~ for (3.9) with negative k form an orthocomplete set for the Hilbert sp~e 8~, and the output ~ --1/2 1/2 cigen-functions ~ (z) = ~+~ (z) ~k, where aoOkwith ao = Po apo , form an orthocomplete set for the Hilbert space ~. The solutions to (3.7) can be written in the form (3.3)

~’ (~) = ~

e-~(~’)’~;+,~ (z) y (k)

(z),

~ (~) =

e-i~(~’~t~ (z) ] (~)

(z),

where fo (k) = efo (k) are defined as the Fourier transforms

f (~) =

./i~+~(z) ~o(z) dz, ] (k) =

,~+~(z)

by the initial conditions, analytically extended on the whole line ~. Due to the commutativity of a and g they satisfy the connection ~t (-z) = a~ (z) ~t (z) for t, not only for t = 0. The time invariance of this connection and the unitarity of the time transformation group in the Hilbert space $~ ~ S~, which follows from the unitarity of (3.3) in $~ C L~ (~), means the self-adjointness of the problem (3.7) for the pairs ~ ~ L~ (~, p) in the domain of the generator gz ¯ g~ with connection ~+ (-z) = a~ (z) ~- (z). Introducing

¢~(~)= ~t(z) +~o(z) (~(z) ~)~’ (z)= ~.(z~°(~)~ (~) as in (3.5), and taking into account that

¢~(z_) =a~(~)~o(~-~ ~ (~) =~-~°~-~~ (z) =a~ -~ ( °(-~ ~ (-~), we obtain the representation (3.8) as ~t (z) coincides with ¢~ (z) at z ~ ~t (-z) with St (-z) = ~t (z_) at z ~ 0.

Stochastic Equationas a Dirac Boundary-Value Problem

321

REMARK 2 The SchrSdinger boundary value problem (3.7) is physical in all three aspects. First, the equation (3.7) is invariant under the reversion of time arrow, i.e. under the reflection t ~ -t and an isometric complex conjugation ~ ~-~ ~ together with the input-output exchange ~ ~ ~ if d = a-1 and ~ = ~, where ~ (z) = ~ (-z). Second, the wave functions ~, ~t have continuous propagation in both directions of the momentum along z, and at the boundary z = 0 the momentum changes its direction but not the magnitude (conservation of momentum)as the result of the the boundarycondition ~ (0) ~-~ ~ (0). And third, the kinetic energy operator g,,@g,~ is boundedfrom below as the result of unitary transformation of ~ ~_ ~. (2.4). Indeed, from ~ (~) = e (~) = g (a) it follows that the symbol g,~ (z, a) of the complex conjugated operator ~ is given by g-~ (z) e*~ (z) e (i On) e_~ (z) = e*_~ (z) e (i On) e_~ (z if ,~ = &, as ~ (z) = e-r~ (z) and e_~ (z) = ~ (z). Therefore f~ (z, a) = where g~, (z, a) = ~ (z) is the symbol for the kinetic energy operator g~ = g,, the output wave qS. Thus the time reversion with complex conjugation in (3.7) equivalent to the input-output interchange (~t, ~St) ~ (~St,cpt) which preserves connection between ~ot and ~5t as

=

(z) = (z)

(z) = (z),

where ~ (z) = a,~ (-z) due to ~ -t .

4

THE ULTRARELATIVISTIC

LIMIT

Weshall assume here that the symmetric positive kinetic energy e (p) has the relativistic form IPl, or more generally, e (p) = v/p2 + h2#~" as it was suggested in (3.1). It corresponds to the finite bounds v:~ = ~:1 of the velocity v (p) = e~ (p) p ---~ ~c. Note that the phase speed % = ~(~)/a for the momentap = :t:hn,

= V/1 + ~/~ = v (h~) -~ ,

~ > 0 of the harmonic eigen-waves

e-i~’te~. (z) = e-in(~t+z), " e-ie~t~n (z) = -in(~t-z) has also the limit ~ = 1 at ~ --~ oc. Therefore one should expect that the rapidly oscillating input and output waves ~0t (Z) = e-i~(t+z)~bt (z) , ~t (z) = e-i~clt-z)~ t (z) in the ultrarelativistic

(4.1)

limit p ~ :l:~c will propagate as the plane waves with

¢t(z)=~(z+t)=et°’¢,

~bt(z)=(~(z-t)=e~

(4.2)

if the initial conditions are prepared in this form with slowly changing amplitudes ¢,~ ~ L~ (II~). This propagation will reproduce the boundary-reflection dynamics

322

Belavkin

~t (0) = a¢ (0) on the half line + 9 z = s if theinit ial wave amplitudes are connected by ~b (-z) = a¢ (z) for all z E IR. In particular, the solutions Ct (s)

¢ (s +t), ~ (s) =0, t ~ (s) : ,~-~(o) =,~¢’-~(o) : ,~¢(t - s), to this Hamiltonian boundary vahm problem with the input, wave functions

¢(z) = v~,s > 0, ¢(~) =0,z for the initinl state-vectors ~ ~ ~ will correspond to the single-jump stochastic dynamicsin the positive direction of t with respect to the probability density p > O, f~ p (s) ds = Belowwe give a precise formulation and proof of this conjecture in a more general framework which is needed for the derivation of quantum stochastic evolution as the boundary value problem in second quantization. But first let us introduce the notations and illustrate this limit in this simple case. In the following we shall use the notion of the inductive limit of an increasing family (~)~>0 of Hilbert subspaces ~ ~ $~,,~ < ~’. It is defined as the union ~ = U$~ equipped with the inductive convergence which coinsides with the unitbrm convergence in one of the subspaces $~, and therefore is stronger titan the convergence in the uniform completion ~ = ~. The dual inductive convergence is weaker then the convergence in K, and the iuductive operator convergence in g is defined as the operator convergence on each g~ into one of g~, ~ ~. Let 6- = Ug2, ~+ = Ug~ be the inductive limits for the increasing family + (g~- ,g~)~>o of Hardy classes $2 = ~g- D ~o, ~ = ~.$+ D ~o, ~,o < ~ in the notations of the previous section. Both ~-,~+ are dense in L~ (N), consist of the square-intergable 0-valued functions ¢ ~ 6-, ~ ~ ~+ having zero Fourier amplitudes

g (k)

e-~’~(z) dz, ~ (k)

(z)

for all k R ~ with sufficiently large ~ > 0. If ~ e g2 and ~ e g~, then ~ = e~0 e g-, ~ = ~ ~ g+, and the free propagation (a.3)can be written in the form (.4.1) with ¢~ = ei"ti~.~

t = i;e-i(i-~)ti~

@t = ei~t{~@ t = ~.z*°-i(g-~l)tz~w

"~ ~ ~.

These unitary transformations in g2 and in g+~, written as e~ (z) = ~-"~(~°=)W (z), ~ (z) = ~-,:,,~,~o=)~ are generated by the selfadjoint

(4.a)

operators

w~ (iOn) = ~ (e ( iOn) - ~)-~: = e ( ~ + ion) - ~,

~ (i&)= -~"(e (i&) -~)~"= e (~- i&)-

(4.4)

323

Stochastic Equationas a Dirac Boundary-Value Problem

They leave all subspaces ~7~ and ~+~ invariant respectively, however their generators da~,ga~ are not positive definite for a sufficiently large ~, and are not unitary equivalent for different ~ as ~,~w~e,~ = ~ - ~1 = w~ - ~1, e~w~.e~ = ~ - ~1 = w~ - ~1, where ~ = ~ - ~. Thus we have to prove that the propagation (4.4) has the inductive limit form of plane propagation (4.2) at ~ ~ ~ corresponding to the Dirac form of the limits lira w~. (iOz) = iO:, li~n ~. (iOz) = -iOz for the Schr6dinger generators (4.4). Another thing which we are going to prove for obtaining the single-jump stochastic limit is that the truncated wave

representing the pair (4.4) on the half-line ~+ 9 z as in (3.8), has the discontinuous limit X~(z)=x,(z+t),

X~=¢+(~-~)L¢.

(4.5)

Here ~ = e -it°’~oeit°= is pointwise multiplication by the characteristic function It of the interval -m < z < t which we shall obtain as the inductive limit of the orthoprojector ~t eite(s+io=)ioe-ite(s+io.)

eU~ ioe-i~

(4.6)

at a + ~. This results are formulated in the following proposition in full generality and notation of the proposition PROPOSITION 3 Let ~- be the Hilbert inductive limit of Hardy classes ~[ = ~ 6~ C L~ (R,p) be the Hilbert space of functions 6~, and 6~ = g~6+, where 6 + = U~ ~ 6~. Let the initial boundary-value SchrSdinger problem i0,¢~ (z) = ~,, (z,iO~)¢~ (z),

¢] = ¢ e 6;,z >

io~(z) = ~,~(z,iO=)~i (z) ,~ > o, ~ (o) be defined by the generators

with the symbols w~, &~given in (4.6), (3.1), and the initial as ~ (-z) = a~ (z) ~ (z), z < 0 by analytical continuation of each = ~in to the domain ~-. Then the solutions to (4.7) inductively converge

¢’ (z) ~ (~), z ~ 0, where Xt (z) = e~ (t) X~ (z + t),

(4.7)

324

Belavkin

Proo]. First let us note that the generators in (4.7) have the formal limits lira [~,~¢] (z) : ~:~ (z)i0~ [~¢] (z) = (~ + ~:0:)¢

with ~z = -0~. This follows from (4.6) and iOze~ = ~e~, Oz~ = i&~ as ~ (z) e:~ (z). Thus we have to prove that the solutions to (4.7) have the limits ¢ = lira ~ = lira ,~ in 6~ coinciding with the solutions to the Dirac boundary value problem

~ (z) = (- + ~0~)¢~(z), ¢0= ¢ ~ 6;,z ~0~¢ (~) = (,+ ~) ¢~ (z),z

> 0, ¢~(0)

with the initial ~o analytically deaned as ~o(-z) = ~ (z) ~o(z) in order to keep the solution ~t also in ~Xfor all t. Let us do this using ~he isomorphisms ~ ~ of ~he dense subspaces ~ and ~ C L~ (~). Due this the boundary value problem (~.7) is equivalem

=

~ , o,~ = ¢o ~ 6~,z > 0 ~o,~(~) ,~ > 0, ~o,~. (0) = (0)

with ~ (-k) = ¢ (n- k) - n = ~ (k), and ~,~ (-z) = a~0 (z) as a. for any scalar n. Thus we are to find the ultrarelativistic limit of the solutions 1

e-i(~(-k~-k~ 9 (k) dk,

(4.9)

with ~ (k) = e9 (k) at ~ ~ ~. Here the Fourier amplitudes

g (k)

/7 e-i~¢0(z) dz, ~ (k)

(z)

are defined by analytical continuation of the initial conditions e,.o ~ for a ~o < ~ such that the integration in (4.9) can be restricted to k < no dne to g (k) = 0 = ~ (k) for all k ~ no. Therefore the proof that the unitary evolution (4.9) inductively converges to the plane propagation et°~¢0,et°~o resolving this problem at ~ ~ ~ can be reduced to finding an estimate of the integral

It gives the value ~o ~hemeansquaredistances ¢

~O,n

O,n --

of~,~(~- t) from ~0~ ~.:~C a.do~4~,~(~ +t) from ~o

Stochastic Equationas a Dirac Boundary-Value Problem

325

To this end we shall use the inequality (~f2

~_

1 #2 /A2)l/2 -~<2~’ ---

w > II~1

for the monotonously increasing function k+w~(-k) < n°+w~ (-n °) of k < n°. We shall treat separately the three cases in (3.1): the scalar massless case #0 = 0 when ~ (k) =Ikl, the boundedness case [#1 <- ra when ~ (k) <_ V~ + 2 asin thescal ar case with # = po > 0, and the general vector case when ~ (k) = 2 + p]- V2)~/’2 In the first casek+w~(-k) =k-n+[n-k I=0foraltn>_0andk<~. Thus the plane wave propagation

~o,.(~)=¢o(z + , ¢o,~(z) = ~o(z is extended by ultrarelativistic limit n ---+ ~ from the orthogonal Hardy classes £o:F onto the inductive spaces ~o:F. By continuity they are uniquely defined as the opposite plane propagations on the whole Hilbert space L~ (IR) where they satisfy the connection ~o (-z) = a¢o (z). In the second casek+w~(-k) <m2/2~for all ~<=n-n ° °. > I#l andk < n Using the inequality le z - 1[ < 2Ix[ for any x e C with Ix] _< 1 we obtain the estimate 2rn III(~°,n)ll<_e -i(k+w~(-k))t 1[< 2[t llIk+w~(-k)[[ < Itl-~for the integral I (n°, n) with [[g[[~ = ~ f [[g (k)[[ ~ dk < 1. Hencefor any n° > 0, e > 0 and each t ~ IR there exists an’ < oc such that [[/(n°,n)[[ < e for all n > n’. Namely, one can take ~’ = ~:° +,nax {m, Itt m~/~} such that >c = n-~° > n’-n > m and It I m’~/~<< ~. Thus the plane wave propagation is indeed the ultrarelativistic limit of (4.9) in the inductive uniform sense. In the third case one should replace I) = ~ (I~d) by t he i nductive l imit ~° =Ul~, of Hilbert subspaces I~ of functions in L2 (~d) having the localized Fourier amplitudes h (k) = 0, k ~ (-n, n) for a d. Then #~ - V~ < #o~+ n2 i n each O~, an d []I ( n°, n)[I < It[ (#o2 + n2)/~< if [Ig[I < 1 for the Fourier amplitudesof ~bo e l)~ ®~-o and of~o6 ~ ® £~o, + where ~ are Hardy classes in L~ (I~). Hence for any ° >0, ~z, n EIR ~ > 0 and each t E IR there exists an’ < cx) such that [[I (n°, n)[[ < ~ for all ~ > n’, namely n’ }. = ~° + max{V/~0~+ n2, [t[ (#0~ + ~e)/e Howeverthe estimate It[ (#~ + ~) / (n °) depends now on ndefin ing the c hoice of g (k) in ~}° for each k < °. This p roves t hat t he plane wave propagation i s t he ultrarelativistic limit of (4.9) also in the general vector case, although not in the uniform but in the strong inductive convergence sense. Thus the boundary value problem (4.7) in the ultrarelativistic limit is unitary equivalen_t to the plane propagations (4.2) of opposite waves ’~bo, ’~0 with the connection ¢o (-z) = a¢o (z) for all z ~ R. In the half space z + this obviously can be written as

¢~(z) : 4 (z) ,z >_o, ~ (-~) =Z’o(z_)

326

Belavkin

where X~(z) = X0,t (z + t) is the truucated input wave(4.5) with ¢0 in the capacity ~t v, -t of ~p. Returning back to’~t= e~b ^, o tand=e,~b weshall obtain the representation o (4.8) with

:g’(z)=~;,(z)~o,~,~(~):g~(z)=~,~(t) ~, due to continuity of e,, (t), where:g~ = ~,X0,~is given in (4.8). REMARK 3 The truncated wave Xt = io¢ t + io~b t in the interaction representation X (t) = ei~’txt with respect to the shift group generatedby ~/ = i0,~ satisfies the stochastic single-jump equation d:g (t, z) + i~:g (t, z) dt = (a 1)X (t, z) dlt (z), t > 0

(4.10)

Indeed, the dynamical group e,~ (t) e- i×t is unitary in I}. Tileone-para~netric group e w: is apparently generated by the self-adjoint operator "~(iO,) = iO: in L2 (ll~) which is the symbol of the generator "~ for the shift group evolution e-u#. It is a unitary group in L~ (IR) due to the shift-invariance of the Lebesgue measure on ll~. Hence the truncated wave in the interaction representation is given by

x (t, z)=~(~ -t)= ,~~(t) x~ ( =~;,(z- t) ~,~(z):g~(z)=d(~’-~)’~:g0,t where ?(.o.t = Xo+ (a - 1) (lt - lo) :go with :go = io¢o + ib~o. Taking into account that dtdl, (z) = 0 in the Hilbert space sense as it is zero almost everywhere due dlt (z) = 1 >> dt # 0 only for the single point z = t having zero measure, we obtain d:g (t, z) ei (~-t)’¢ [(a - 1)dlt(z) = [(a - 1) dlt (z) i~dt] = [(a - 1) dlt (z) ixdt]

:go (z) - i.~:go,~(z) dt] ei(Z-O~Xo,t (z)

~(t,z).

Here we used that dlt (z) = dlo (z- t) = 0 if z ¢ t such dlt (z) ei(Z-’)×:go,t (z) = dlt (z) :g0.t (z) = dl,. (z) :go,z due to :go,t (z)It=~ = X0 (z) as It (z) - 10 (z) = 0 for any z > t > 0. Thus we proved that :g (t, z) indeed satisfies the stochastic single jumpequation (4.10) in Hilbert space L~ (II{, p) of the initial conditions :g = ~0¢ + ]0-L~ with respect to the unitary group evolution etO; . Returning to the notations ~ = h-~H, a = S of the Sec. 1 in tile Hilbert space ~ = 7-/ we obtain the stochastic equation (2.2) for the unitary cocycle V (t, s) e-t°" Vt, where Vt = ,~i°et(O~-ih-~ H) s-i°, as a quantum-~nechanical stochastic approximation. Namely, the toy Hamiltonian model for the interpretation of discontinuous stochastic evolution in terms of the strongly continuous unitary group resolving the Dirac boundary value problem in extra dimension, is indeed the ultrarelativistic inductive limit of a SchrSdinger boundary-value problem with bounded from below Hamiltonian g~ (p) = hw~,× (-h-~p).

Stochastic Equationas a Dirac Boundary-Value Problem

327

REFERENCES 1.

2.

3.

4.

5. 6.

7.

S. Albeverio, V. N. Kolokoltsov, O. G. Smolyanov, "Continuous Quantum Measureinent: Local and Global Approaches," Reviews in Mathematical Physics, 9, No. 8, 907-920 (1997) V.P. Belavkin, "A Dynamical Theory of Quantum Measurement and Spontaneous Localization," Russian Journal of Mathematical Physics, 3, No. 1, 3-23 (1995) L., Accardi, R., Alicki, A. Frigerio, and Y. G. Lu, "An invitation to the weak coupling and low density limits," QuantumProbability and Related Topics VI, 3-61 (1991) V.P. Belavkin, "A stochastic Hamiltonian approach for quantum jumps, spontaneous localizations, and continuous trajectories," QuantumSemicalss. Opt. 8, 167-187 (1996) V.P. Belavkin, "Nondemolition Principle of Quantum Measurement Theory," Foundations o] Physics, 24, No. 5, 685-714 (1994) A.M. Chebotarev, "The quantum stochastic equation is equivalent to a symmetric boundary value problem for the SchrSdinger Equation," Mathematical Notes, 61, No. 4, 510-518 (1997) R.L. Hudson, and K. R. Parthasarathy, "QuantumIto’s formula and stochastic evolutions," Comm.Math. Phys. 93, No. 3,301-323 (1984)

A MaximumPrinciple Parabolic Equations

for Fully Nonlinear with Time Degeneracy

JOACHIMVONBELOWLMPAJoseph Liouville, EA 2597, Universit6 du Littoral CSte d’Opale, 50, rue F. Buisson, B.P. 699, F-62228 Calais C6dex (France)

1

INTRODUCTION

AND MAIN

RESULTS

In the present contribution we derive somea priori bounds for classical solutions of fully nonlinear parabolic equations that include possible time degeneracies. Moreover we show that, in general, a comparison principle for such equations cannot hold. The boundary conditions are of mixed type, a Dirichlet boundary condition on one part of the time lateral boundary and a possibly dynamical one on the complementary part. The latter includes as a special case the Neumannboundary condition. Suppose 12 C_ Ii¢ ’~ is a bounded domain whose boundary is decomposed into two disjoint parts 0fl = 01~ e) 02fl, where 0212 is of class C2 and relatively open in 0~. Let v : Of/ -~ It ’~ denote the outer normal unit vector field on 02~ and 0~ the outer normal derivative. For T > 0 we set QT = ~ × [0,T] and introduce the parabolic interior QT and the parabolic boundary qT as QT= (fl tJ c9:~-l) × (0, T] and aT = Q’r \ Q"r. Weconsider fully non linear parabolic equations of the form b(u)t = F (x, t, u, Vu, D2u) =: F[u] and inequalities associated to them, where parabolicity means that F : ~T × It( × ll~ n x I~’’~ ---~ I~ is increasing with respect to q = D~u and b : I~ ---~ t~ is increasing with respect to u outside a compact interval in ll~. 329

(1)

(2)

yon Below

330

Here the order A _< B between symmetric matrices means that the matrix B - A is positive semidefinite. Note that we do not require strict monotonicity. Thus, degeneracies of the principal spatial part as e.g. the porous mediumequation are admitted. Throughout, Condition (2) will be assumed to hold. On 01~ × (0, T] we prescribe an inho~nogeneous DMchlet condition, while on 0~ × (0,T] we consider a possibly dynamical boundary condition/30(u) = 0 with

Go(u) :=~(x,t)O~+ c(x,t)O~u. Throughout we will assume the dissipativity c > 0, a _> 0

condition on 0~ x (0, T].

(3)

Without condition (3) blow up and nonuniqueness phenomena can occur already the case b(u) = u, see [2] and [3]. The investigation of differential equations of the form (1) is motivated by several applications, as e.g. models of hysteresis phenomenain physics, but also of theoretical interest, see [1], [4] and [5], as well as the dynamicalboundaryconditions, see [2] and [3] and the references given there. If b’ vanishes at some value of u, Equation (1) undergoes ti me degeneracy at that val ue. In thi s way, equ ations of the form (1) can display a change of type between parabolic and elliptic problems. The following results present some elements of a qualitative parabolic theory for classical solutions, i.e. solutions in C(-~T ) f’lC 2’1 (QT), as it is well established in the nondegeneratecase b(u) = u, see e.g. [2], [6] and [7]. It is clear that, if b is strictly decreasing in some noneInpty interval, then no classical maximumor comparison principle is possible, since in this case (1) can take locally the form of a backward heat equation. But one can control the solutions outside a compact part of the range, if the r.h.s, in (1) is governed by a generalized Osgoodsign condition. THEOREM 1 Suppose that b ~ C2(I~) and there are constants ~ >_ 0 and Zo i> 1 such that

~ +Z)~’(z)Io,. Izl >zo, ~(., ., z, 0, 0) <(Zz b’(z)

>0 for

Izl>z0,

b(-zo)=

rain

b and b(zo)

(4) max b. (5)

Then any ,solution u ~ C(-~T) ~ 2’~ ( QT) of t he B

~(~)~= F (~, t,,~, W,D~,,)i~

{

on0~x (0, T]

~o(U)=

satisfies m_axlul <_ e2~~r tnax ~maxlul,zo} . QT

[

qT

Note that here b was not assumedto be increasing on the whole real line. If b bears that property, then we can state the following positivity result. Conceivably, the critical values of b play a crucial r61e in separating techniques.

331

Fully NonlinearParabolicEquationswith TimeDegeneracy THEOREM 2

Let u E C(’~T) ~ ~’~ (QT) fulfill t he i nequalities b(u)t >_F(x,t,u,

Vu, D2u)

inQT,

02~× (o, TI

/~o(u) >_

on

u >_0

on qT,

under the conditions

F(.,., z, o, o) > 0 ]orz < o, b E ¢~(~),b(0)= 0, > 0in~, b’(z) =o ~ >o.

(6) (7) (8)

Then u >_ 0 in QT. Can one compare solutions of (1)? The best answer would be a comparison principle with respect to qT: b(u)t- F[u] <_ b(v)t - F[v] in QT, I3o(u) <_ Bo(v) on 0~ × (0, T] and u < v on qT imply that u _< v in QT. It turns out that, in general, this conclusion is false. COUNTEREXAMPLE A comparison principle with respect uniqueness of solutions in C(-~T) C~ ~,~ ( QT) of t he I BVP

to qT would imply

b(u)t = r (x, t, u, Vu, D2u) in QT,

13o(U) =

on c%Q× (0, T],

Ulqr = ~b ~ C(qT). For 1 _< k C N, choose b(u) = ~-~’+1and for a e [0, oc ) se Ya(x,t)

fort>a, for t < a.

(t-a) = 0

Then each y, belongs to C(-~T) ¢~ C2’1

(QT)

and is a solution of

b(u)t Au+ 3(2 k + 1 )u2~+~

in QT,

O,u = 0

on cgf~ × (0, T],

u(., O) =

in

Note that F is of class C~. Thus, even under a one-sided local Lipschitz condition with respect to u on F and even if b is strictly increasing in If{, a comparisonprinciple with respect to the parabolic boundary does not hold for equations (1) with proper time degeneracy. Without degeneracy, i.e. b(u) = u, such a principle holds under canonical conditions, see [2]. But, we can at least carry over a Nagumo-Westphal type conclusion [7,§24] with respect to qT.

332 THEOREM 3 Let

yon Below u,

v E C(-~T)

~l C2’1

(QT)

satisfy

13o(U) = Bo(v) = 0 02f~ x ( 0, T] and the inequality b(u)t - F[u] < b(v)t under the conditions b E C1 (l~) and b’ >_0 in

in

Then u < v on qT implies u < v

in QT.

Finally, we state the following weak maximumprinciple in the case where b has at most one critical value. THEOREM 4 Let u

~ C("~T )

ClC2’I(QT) satisfy

Bo(u) <_

on 02ft x (0, T],

(Bo(U)>_

Suppose that b E Ce(~), b’ > 0 in II~\{0} and F(.,.,.,0,0) Then maXuqT ~ maxu.~T

_< 0 (F(.,.,.,0,0)

_>

\(minUqT "-~ minU.)~T

Wenote in passing the Theorems1-4 apply especially

to equations of the form

(u~k+~), = F (x, t, u, Vu, D~u).

(k ~ N)

Moreover, Theorem1 applies e.g. to cubic l.h.s, b(u) = u(u-a)(u-b) with a < 0 < b that display a backwardparabolic equation in a nontrivial interval containing the origin.

2

PROOFS

Let us recall the basic comparison technique stemming from [7,§24] for Dirichlet boundary conditions and for dynamical ones from [2] or [3]. LEMMA 1 ([2,Lemma

2.1]) Let ~,~ e C(-QT)V~C’2’I(QT) B0(~o) _
satisfy

on 02~ x (0, T],

and the following test point implication: If qo = ¢, V~ = V¢, D2tp <_ D2¢ at some point (x, t) e QT, then Otto < Ore at (x, t). Then qo < ¢ on qT implies ~ < ¢ in QT.

(9)

333

Fully NonlinearParabolicEquationswith TimeDegeneracy PROOFOF THEOREM 1 For ~ > 0, we first

set 99 = b(u) and ¢ = b(’~),

where

"~, qr "7 = (l + ~)e2~tmax~max[ul’z°} Then B0(99) = b’(u)Bo(U) = 0 and, by (3), 2a/3"7 b’(’7) >_ O. Moreo ver , ’7 > Zo _> 1 in ~T and, due to (5), b’(’7) > 0. Thus, as b is strictly increasing Izl _~ zo, 99 < ~b on qT. At a testpoint (x, t) E QTas in (9) we conclude 99=¢ ~ u=7>z0

and b’(u)

>0,

V99 = ~¢ ~ ~99 = b’(u)~u = 0 ~ ~u 0, D299 = b’(u)D~u <_ D~¢ = 0 ~ D~u <_ and using (4), 0~¢ - 0t99 = b’(’7)2/37 F(x,t,u,O, D~ u) _> b’(7)2/37 F(x,t,’7,0,O) >_ b’( 7)/3~f (1 - 7 -~) > 0. Now, Lemma1 yields 99 < ¢ in QT, which implies u < 7 _~ 7(T) in QT, since b is strictly increasing for Izl > zo. In order to show u > -7(T) we apply Lemma1 to ¢ b(u) and 99 = b(- 3’). Usi here that, b’(-’7) > 0 and F(x,t,-’7,0, O) >_ -(/37+/?’7-’)b’(-’7), we proceed similarly and conclude that 99 < ~/~ in QT, which again implies that u > in QT"As ¢ > 0 was arbitrary, the assertion of Theorem1 is shown.

PROOF OF THEOREM 2 For ¢ > 0, we apply Lemma 1 to 99 := 0 and ~p := b(u) ~(1 + t) . Note fi rst th at Bo(¢) >_ 0 by (3) and 99
and b’(u)

>

0 = V99 = ~ = b’(u)Vu =~ Vu = 0, 0 = D~99 <_ D~¢ = b’(u)D~u =~ Deu >_ 0, and using the differential inequality and (6), 0~¢-0t99 = b(u)i + e >_ F(z,t,u,O,O)

+e

Thus b(u) >_ in QT.Thispermits to co nclude that u _> 0 in QTdueto (7).

PROOFOF THEOREM 3 We can apply Lemma 1 di’rectly to 99 = u and ¢ = v: At a testpoint (x, t) ~ QTas in (9) the differential inequality yields due to 0 <_F(x, t, ~, V99,D’~¢)- F(x, t, 99, Vqa, D~’~)< b’(~)Ot¢ - b’(99)Ot99. Thus, necessarily, b’(99) > 0, which showsOre > 0t99 at the point (x, t). This shows Theorem 3. ~

von Below

334

PROOF OF THEOREM4 Fore>0, weapplyLemmalto~=b(u) and¢= b (et + a + maxqr u) in the maximumcase and to ~ = b (~ninq. r u - ~ - et) and ¢ = b(u) in the minimmncase. In both cases, at a testpoint (x,t) E QT as in (9), we can follow the proof of Theorem 1 in order to obtain b~(u) > 0 and cOt¢-cOt~ > br(u)e > 0 at (x,t) and conclude ~o < ¢ in Q~.. As b is strictly increasing, the assertion is shown. ~

ACKNOWLEDGEMENT The author thanks the anonymous referee for helpful comments, as well as M. Cuesta and C. De Coster for stimulating discussions.

REFERENCES 1. 2.

3. 4. 5. 6. 7.

H.W. Alt and S. Luckhaus. Quasi-linear elliptic-parabolic differential equations. Math. Z., lS3: 311-341, (1983). J. yon Below and C. De Coster. A qualitative theory for parabolic problems under dynamical boundary conditions, to appear in J. Inequalities and Applications. J. yon Belowand S. Nicaise. Dynamical interface transition in ramified media with diffusion. Comm.Partial Differential Equations, 21: 255-279, (1996). Ph. B~nilan, J. Carrillo and P. Wittbold. Renormalized entropy solutions of scalar conservation laws, preprint, (1999). Ph. Bfinilan and P. Wittbold. On mild and weak solutions of elliptic-parabolic problems. Advances in Differential Equations, 1: 1053-1073, (1996). O.A. Lady~enskaja, V. A. Solonnikov, and N. N. Ural’ccva. Linear and quasilinear equations of parabolic type. Amer. Math. Soc. Providence RI, (1968). W. Walter. Differential and integral inequalities. Springer Verlag, Berlin, 1970.

Dirac Algebra

and Foldy-Wouthuysen

H. O. CORDES Department of Mathematics, University Calitbrnia

Transform

of California,

Berkeley,

ABSTRACT Weoffer further results and a matured physical interpretation, concerning our invariant algebra P for the Dirac equation discussed in 1983 [1] and 1996 [2]. Especially, P may be fully described either with simple spectral theOry of the Hamiltouian or with a decoupling modulo order -c~ of the positive and negative energy parts of the Dirac equation, similar to the Foldy-Wouthuysentransform. There seems to be evidence indicating that only operators in P qualify as observables--i.e., can be measured with arbitrary precision--a feature comparable to the Heisenberg uncertainty relation.

0

INTRODUCTION

Wewill discuss properties of the Dirac equation O¢/Ot + iH¢ = O, (t,x) with Hamiltonian H = h(x,D),

= (t,x~,x.~,x3)

~

(0.1)

D = (D1,D2,D3), D1= -~0i ~1, where 3

h(x,~)

= Eaj(~j

-Aj(x))

+~+V(x),

with certain constant complexself-adjoint 4 × 4-matrices a j, ]~, and electromagnetic potentials (V(x),Aj(x)). Physical constants li, c,m,e are set = 1. Note, (0.1) represents a first order sym~netric hyperbolic syste~n of 4 PDE’s in 4 unknown functions ~ = (¢~,..., ¢4)T. The differential operator H is self-adjoint with respect to the inner product of H = Le(K~,C~). The symbol h(x,~) is a self-adjoint 4 × 4matrix with two real eigenvalues A~: of constant multiplicity 2 each: £±(x,~) = V(x) :t: V/1 + (~ - :. 335

(0.3)

336

Cordes

With proper assumptions on smoothness and growth of the potentials (0.1) will a semi-strictly hyperbolic system of pseudodifferential equations of type e’ = (1,0), in the sense of [1], [2], since the eigenvalues (0.3) satisfy ~+ - A__> c(1 + I~]), c for large Ix[ + For such systems we discussed the existence of an invariant algebra P of global pseudodifferential operators. Invariance means that the algebra remains unchanged under conjugation A ~ E*AE with the time propagator E = e -itt4 of equation (0.1). Details in Theorem1.1. Weare used to the following Dirac matrices aj, a

= -ia {71-----

0 -i

0

’ /3 = ’ ~ =

0 -1 1 0 ’

(0.4) ~r 3 ~

0 -1

.

In all of this paper we employ global pseudodifferential operators (0do’s) A a(x,D) ~ OpOcm, m = (m~,m~) ~ of "st rictly cla ssical" typ e: A s ymbol a(x, ~) e ~Pcmis O((x) m~(~)’~), (p) = v/~ ’2, its d eriv atives -(~) ~O:~a are O((x),~ {~)m),/~ = mr- ]¢~1,/~ = m~_-1/31, for all multi-indices a,/3. m= (m~, is the "order" of a and A. Assumptions are rigged such that h(x, e~ = (1,0). Wealso use ~ =(0,1), e = (1,1) fo r sp ecial or ders. Op erators in Op~bc= UOp~bCm obey a global "calculus of Cdo’s". Especially there are Leibniz formulas "with integral remainder" and "asymptotic Leibniz formulas" for products and adjoints. Details in [2], I. Our main results are formulated in section 1 and proven in section 2. [Theorem’s 1.1 and 1.2 were discussed in [1] and [3], dating back to 1983.] Here let us attach some physical comments. Equation (0.1), as relativistic wave equation of an electron or positron moving an electromagnetic field with potentials (V, A) was in the center of physical interest during the early 1930’s and later. It is likely that, at that time, the existence of an invariant algebra like P was not known. A Cdo .4 = a(x, D) Op¢c,~ may be regarded as bounded operator H -~ H-m (with the "polynomially weighted" ~Sobolevspace Hs---with normII(x) ’~ (D)s~ ’allL2 ). If mr, m.2 _> 0, then also A defines an unbounded operator of H with domain H.,n C H. Assuming that A* = A, with the formal Hilbert space adjoint ..... of H, one may show easily that every "rodelliptic" such operator is a self-adjoint operator of H. Moreover,every formally selfadjoint such operator of order e or e~ or e 2 admits a unique self-adjoint extension-its closure. [Not true for higher orders!] With this preparation we can state that our algebra P--or at least its formally self-adjoint operators of order e, e1, e2, (or of order 0 or of arbitrary order if rodelliptic) classif~ as physical observables. [In particular, location--the multiplication operators by x~, x.2, xa, momentums-thedifferential operators -ihO~,,..., angular momentum,energy, all are represented by differential operators, belonging to Op$c.] Also, conjugation with e -iHt of mr Observable A amounts to the "Heisenberg transform"--it defines the same observable A at time t, assuming that the physical states (the unit vectors of H) remain fixed. In other words, the self-adjoint elements of our algebra P have the property that they remain ¢do’s for all times.

Dirac Algebra and Foldy-Wouthuysen Transform

337

It is trivial that the energy observable--the operator H, commutingwith e belongs to our algebra P. Tile same is true for the total angular momentum,if the potentials are rotationally symmetric (because that operator com~nutes with H). But uone of the other above standard observables have this property--although "corrections" for them may be found, using Theorem1.1. Now,ibr these "other" observables a feature called "Zitterbewegung" was discovered, already in early Dirac theory--just this fact; their time propagation seems "unphysical". An electron moving with vanishing fields is calculated to execute a rapid oscillatory motion. For the case V = A = 0 several corrected observables were proposed, not exhibiting this oscillation feature, but none of these corrections proved entirely satisfactory ([4], Chapter 1). Our corrections ([1], p. 90) for more general potentials have similar features or are identical. But if we are allowed an opinion, in this context: Wesuggest to allow only the self-adjoint operators of P as true observables. They can be observed since they are stable in time. Measuring another observable--such ~ the space coordinate x~, for example, should amount to finding an element of P which is "close" to x~. Such elements exist, but are not unique. For example, for x = (Xl, x2, x3) we might propose

Xcorr= ¯ - ~c(x, D),

with

the

matrices

~ = e 0 ’ p = 0 e

the physical constants B,m, e, c. Only the first correction was added~there are countably many others, of lower and lower order. Still, an inspection seems to indicate that the correction is of order of magnitude ~the Compton-wavelength of the electron. It is knownfrom experiments [5], [6], [7] that this is the lower limit of accuracy for measuring the position .of an electron. Generally, this see~ns to be a matter similar to the Heisenberg uncertainty relation, which says that position and momentumcannot be measured simultaneously without a definite uncertainty. Here we are suggesting that only observables within P can be measured with arbitrary precision. Location x is not in P, but may be approximated by elements of P with accuracy about h hence it only Can be measured with that accuracy. In [1], [2] we madesimilar reflection with other observables. Here let us point to the following: Assume A(x) ~ 0, and V invariant under rotations: V = V(r), r = ]x~. would like to choose V = ~ = Coulombpotential, with a constant c ~ 0, but then must ’cap’ the singularity at r = 0 to create a C~-function, which then satisfies our general assumptions of section 1. With this V we set up equation (0.1). But also, the potential V should be observable. P does not contain V, but Theorem 1.1 will give V¢orr = V(txl) - E(x). ~c(x, (0.6) with E = grad V = electrical field strength, Ac, of (0.5). Here one finds that the correction symbol E(x)Ac(x,~) tends to get large near x = 0 for not too large [~[ if our cap is applied very close to 0. An interpretation of this: The potential V can only be measured if the electron is not too close to the point 0.

338

Cordes

One then might ask for the physical meaning of Vcorr. Especially: ]is this an expression for the fact that ’other’ forces are acting if the electron gets very close to the nucleus? In particular, since gauge theory suggests a vaguely related theory for a unification of electrical and weakforces ([8], [9]), does the present approach have any meaning, in that context? The author feels very indebted to A. Unterberger and G. Lumer for providing a challenge and encouragementfor continuing the present work (cf. also [10], [11]).

1

INVARIANT ALGEBRA FOLDY-WOUTHUYSEN

Wesummarize some results,

AND TRANSFORM

partly published earlier.

General assumptions: For a(x) = V, Aj require a(a)(x) = O.~a(x) = O((x)-I"l), I~3, for a ll ~ .

(1.1)

Then the (4 x 4-matrix-valued) symbol h(x, ~) and the eigenvalues )~±(x, ~) all belong to ¢c~. Note: No singularities allowed. THEOREM 1.1 Define a class P,~ C Op~PCmof ~Pdo - s A with 4 × 4-symbols a such that (i) At e~Hta(x, D) -ilit ~ Op¢cm, for all t (ii) At ~ C~(I~, Op~bc,n); (iii) O~A~ ~ C(I~, Op¢c,~_ke~), k = O, 1, 2,.... Assertion: (1) For = a(x, D)~ Pm the symbol allo ws a de composition a=q+z, with z Egmm-~, [h,q]

a. :0, x,~ 6 I~

(].2)

(2) Vice versa, if a symbol q ~ ¢c,~ commuteswith h for all x,~ ~ ]~a, there exists z ~ ¢cm-~ with A = a(x,D) E P,~, for a = q + z. Here the "correction symbo|" is an asymptotic stun z = Zz) (rood O(-c~)), with solutions z~ of order commutator equations, recursively, where zj_~ must be adjusted to insure solvability for z~. (3) Suppose A~, A: ~ Pm(both must have a decomposition (1.2)) have the q. Then b = z~ -z~ is symbol of b(x, D) Pm-~; it all ows (1. 2) wit h m- e i nstead of PROOFCf. [1],

Theorem 2.1.

THEOREM 1.2 Under our general assumptions on V,A, there exists operator U of H which is a Cdo : U = u(x, D) Opec0, su ch th at

= 0

H_

a unitary

= r 0 ’ (1.3)

where H± ~ Op~bc~, while F E O(-~x)) is of order -~c. (H± and F are 2 matrices of¢do-s.) The symbols u, h± of U, H±are asymptotic sums, u =

Dirac Algebra and Foldy-Wouthuysen Transform

339

h± = )~+ + Ehj. ; uj and hj:~ are obtained recursively, solving certain cmnmutator equations. Here A+of (0.3) and u0 of (1.4) relate to the diagonalization h(x, ~):

uo=

1+

0

1+1+(.---7 ’’=

u~u0 = 1, x,~ e ]R3, while H±= )~±(x, D)(mod(Op¢c_e~)),

0) {1.4)

with the eigenvalues A+of (0.3). PROOFCf. [3],

Theorem 1.1.

Note: The unitary operator U "decouples" the Dirac equation modulo O(-oo) into a pair of 2 x 2-hyperbolic systems of ~pde’s. An "approximate decoupling" was done in different setting by Foldy and Wouthuysen[12], with similar first few terms. There is work on more precise decoupling under special assumptions([4], p. a09,[13],[14],[15]). For Theorems 1.3, 1.4 below we add the general assumption that the potentials V and A assume a limit as Ix[ + oo. We then may assume (V°,A°) = lim]:rl_~(V(x), A(x)) = 0, since a conjugation of 0t +iH with expi(A°x carries (0.1) into that equation with potentials (V-V°, A-A°) while a Cdo a(x, D) is carried into a(x, A°-D). Moreoverthen, we require that V(a) (x) for all a. Under these assumptions Theorem 1.1 and Theorem 1.2 are related as follows: THEOREM 1.3 The class P,~ of Theorem 1.1 precisely A ~ Op¢cm with

U*AU = D E . ( B C ) whereB,

consists

E~Op¢cm,

of all operators C,D~O(-oo).

Theorem1.3 was indicated in [2], 10. A detailed proof is ibund below. THEOREM 1.4 Let 0 ~ Sp(H), let P± be the spectral projections and (0, oo). Then P,,~ precisely consists of A ~ Op¢cmwith

of H to (-co, 0)

P+AP_,P_AP+ ~ 0(-oo).

(1.5)

THEOREM 1.5 Let the potentials vanish. Then the graded algebra P = i0Pm is invariant (covariant?) under a proper Lorentz transform: Passing from coordinates (t, x) to coordinates (t’, x’) a physical state at t = 0 is turned into a physical state at t’ = 0 by a unitary map R : L2(t = 0) ~ :(t’ =0)--we fo llow th e so lution of (0.1) from t = 0 to t’ = 0 and use a transform of dependent variable, a relativistic contraction. This R also relates P and P’ : P’ = RPR*. PROOF For Theorem 1.5 cf. [16]; for Theorems 1.3, 1.4 see section 2, below.

340 2

Cordes PROOFS AND LINK

TO THE SPECTRAL

RESOLUTION

OF H

Somecomme,~tson spectral theory of Dirac operators with our kind of potentials: H = Ho + Z(x), with matrix-valued Z(x) = V(x) - ~A(x), and H± of (1.4} eI -md-elliptic (cf. [2], II): For large Ixl+l~l we have lh(x,()l > p(~), Isytnb(S±)I > p(~), with p > 0. The same for H - z, H± - z, if Re z it 0, or Izl < 1 [even for sz 2Ax)/ = 0, ((~ +z )+ < (~ 1 -A)) for H±]. We get A+ = (~)+{V+(A where the brackets give a bounded Cdo, while :t:(~) gives H~ = :t:(D), unbounded self-adjoint in He~, diagonalized by the Fourier transform. Hence H±also is selfadjoint in He~ = dora H±. The same argument works for H0 and H : u0 of (1.4) (with A = 0) diagonalizes h0; Ho then is unitarily equivalent to multiplication the diagonal matrix diag((~), (~),-(~),-(~)), with ess.Spectrum = L_ U ess Sp(H°~) = L±, with the half-lines L± ={1 < :t:z < ~c}. For H,H±and Ho,H°~ the difference of resolvents is compact--for example (H0 - i) -~ - (H - i) -4 = (H - i)-~Z(x)(Ho -~ is a product of Z( x)A(D) ~ K(H), A(~) = 1/(~) bounded operators using that a(x)b(D) is compact whenever a(cx3) = b(co) Hence H,H°~ also have essential spectra L+ U L_, and L±, respectively. Their resolvents are meromorphic flmctions in the complement sets. That is also where we found these operators e~ - rod-elliptic. By [2], III, Theorem4.1, H - z, H±- z have a Green-inverse for z ~ C\L+\L_ (z ~ C\L± ), their eigenvectors to eigenvalues in these complementsof their essential spectrum are rapidly decreasing (belong to

S).

Wealso need the resolvent (J- z) -~ of a ¢doJ ~ O(-~). Then J- z, 0 ~ z ~ always is 0 - md-elliptic, hence admits a Green inverse. All eigenvectors of J to z it 0 belong to S. Note, for J = H, H+, or J ~ O(-oc), even the eigenvectors of J : ~ - + S~ belong to S, for eigenvalues z ~ C\ess.Sp. ([2], III, Theorem4.1--for a more general theory of this "spectral invariance" cf. [17], [18]). In particular, the eigenspaces of J as (unbounded) map between any pair of Sobolev spaces Hs, H~ are independent Also, the resolvent (J - z) -~ is a ¢do[~ Op¢c_e~for H, H±~ Op¢co for O(vc)], and even the special Fredholm inverse Iz, defined at an eigenvalue z as inverse of (J - z) [ Ira(J* - ~) : Ira(J* - z) -+ Im(g - z), and 0 in ker(g - z) = (Ira(J* ±, belongs to Op¢cm, m = -e ~ for H, H±, m = 0 for O(-oc). Weneed the following result: LEMMA 2.1 For some s ~ ll~ ~ let A ~ Op¢c,,~ be Fredhohn in L(Hs, H.~_,~). Then there exists a Fredholm inverse B ~ Op¢c-mwhich also is a Green inverse, and A is rod-elliptic. Specifically, if A is invertible then A-t (the only Fredhohninw~rse) belongs to Op¢c_,~. PROOFWeremind of the fact that fl~ = ~r,.(x,D) = (x)"~(D) ’’~, with ~r~(x,~) (x) ~ (~)"~ defines an isomorphismH~ -+ Hs-,., for every s, r, and that II,. ~ Op¢cr has the property that II~-lAIIr

- A ~ Op¢c_~ C K(Hs), for A ~ Opec0, all s,r,

(2.1)

with the compact ideal K(Hs) ([2], III, 5). For Letnma 2.1 we may set m = 0 [nsing II_,~A instead of A], and then s = 0 [using H-~AIIs instead of A].

341

Dirac Algebra and Foldy-Wouthuysen Transform

Thus we now are looking at a Fredholm operator A E Op¢co. Such an operator must be 0 - rod-elliptic, by [2], V, Theorem 10.3. Indeed, A belongs to the C*algebra closure of Op~bco, hence is Fredholm in L(Ho) if and only if its algebra symbol does not vanish. That is, we have Isymb(A)l >_ p > 0, as Ix[ + I~1 large--i.e., A is 0 - md-elliptic. Using [2], III, Theorem4.3 again, conclude that ker.4, ker A* C S. Nowconsider the operator C = A* A + Pker A, with the orthogonal projection Px onto the closed subspace X C H. Clearly PkerA E O(--CK3), and thus C ~ OpCcois invertible in L(H). PROPOSITION2.2 For .4 ~ L(H) consider

the function

Az,~ : e-izDei~Zde-~xe -izD, z,~ ~ ~3.

(2.2)

Then A belongs to Opec0 if and only if Az,; ~ C~(ll~ 6, L(H)), ((x)l#l(o~O~Ao,o)(D)l~l)..,(

C~(IR6,L(H)), for al l c~ ,/3.

(2.3)

PROOFIt was shown in [2], VIII, Theorem 2.1, that the first condition holds if and only if A = a(x, D) with a(x, ~) CB~(]~ 6) = {C~-functions wi th bo unded derivatives}. For such a(x, ~) and A = a(x, D) we get Az,; = a(x + z, D + ~), hence O~O~Az,~= ^(~)’- + z, D + ~).

(2.4)

But we note that

= e

e C"for all

(Z.5)

If in addition to A:,; ~ C~ we have (2.3), then the operator at left (which symbol(x)I~l (f~,,,l~l~,~) a(~) ~, ~)) belongs to Op¢c0--i.e., its symbolsatisfies (2.5), firming the proposition. [For a virtually identical criterion cf. Beals [19].] REMARK 2.3 The condition A~,¢ E C~(II~,L(H)) reduces to "existence 0za 0~A0,0in L(H) for all a,/3", since 0]0~Az,~ = (O]OgAo,o)z,¢, and using that the condition is imposedfor all c~,/3. For the above operator C, it is clear then that C-~ ~ Opec0, and, moreover, that (C - A)-1 ~ Op¢co for every A e C\SpC where SpC C [e,I/e], some e 0. Indeed the condition of Proposition 2.2 applies to A-~ whenever it applies to A, assuming that A-~ ~ L(H). Moreover, we get the positive square root B = C~/2 = ~ ifr(C - A)-~v/~dA E Op¢co: The contour F above will be in Re A > 0, surrounding Sp(C); we get Bz,¢ : ~ 1fr(Cz,; - A)-~ v~dA. Differentiating this for z, ~ we get other well defined complex integrals, converging in L(H): For example O~,(C~,; - A)-~ = -(C~,¢ - A)-~(0~C~,¢)(C~,¢_~) -~, etc., .... Note, we get [(x/, (C- -~] = (C- A) -~[C, (x )](C- A)-~, where IV , (x /] e Op¢c_¢, et c. As a consequence, the operator B will satisfy the conditions of Proposition 2.2, so that B ~ Op¢co. Returning to the proof of Lemma2.1, the polar decomposition A -- QB, where Q = AB-~ ~ Op¢co, then takes place entirely within Op¢co. Note here that

342

Cordes

Q*Q= 1 - PkerA, QQ*= 1 - PkerA*. The first is trivial; for the second, let QQ*u = A(A*A + Pker.~)-lA*u = w. Clearly Q*Qu = 0 if u E kerA*. Let u e (kerA*) ± = hn A. Then u = Av, w = A(A*A + PkerA)-IA*Av = A(1 - Pke~.A)V. Wemay choose v _1_ ker A, then w = Av = u. So, QQ*= 1 in Im A = (Pker ’~)±, but QQ*= 0 in ker A*, confirming that QQ*= 1 - PkerA*. To construct a Green inverse of A, set G = B-1Q* ~ Opec0. Then GA = B-~(1 - Pke-~)B = 1 - Pke~, trivially, while AG = QQ* = 1 - PkerA’, also trivially, q.e.d. In fact we can state COROLLARY 2.4 Under the assumptions of Lemma 2.1 the special (uniquely characterized) Fredholm inverse F with FA = 1 - P~,erA, AF = 1 - Pker A*, with orthogonal projections with respect to the inner products of H.~ and Ht, respectively, are Green inverses. COROLLARY 2.5 For the operators J = H,H+ the resolvent (J - z) -1, z ~ C\Sp(J) as welt as the special Fredholm inverse of J - z, at an eigenvalue z in z = 0, Izl < 1, belong to Op¢c_~. Indeed, for real z the operator V = J- z is self-adjoint; the operator V + Pk,;~ ~. = W: H~ -~ H0 is invertible, hence W-~ ~ Op¢c_~. The desired special Fredholm inverse is given as G = (1 - Pkert,’)W-1; it clearly belongs to Opec_e,. For the resolvent we may directly apply Lemma2.1 to (J - z),k(D) (which is invertible in L(H)). Q.E.D. Next we want to focus on certain spectral projections of H, H+. These operators are self-adjoint, and we knowtheir essential spectra to be L+ U L_, L±, respectively. Consider the integral I,~ = ~ (J - ,~)-ld,k,

J = H,H~=., F = {Re ,~ = ~}, directed upwards, (2.6)

where we assume that ~ is not an eigenvalue of J. For the calculation assuine ~ == 0. The integral is assumed to be a Cauchy principal value at +ee, it then exists in norm-convergence of H. Write =

+

, and set A = ip, for

(2.7) Io = Io(J) = -~r

J(J~ + #’))-~d# = - sgn(J),

where we used the spectral resolution J = f vdE(v), and that .f~’~ v-~dv = ~sgn(v). This is important because we get E(t~) = ½ + I~, while it can be shc,wn that I~ is also meaningful in L(H~), for all s, so that E(n) e Op¢co, due completeness of Opec0 under the Frechet topology induced by all H.,-norms ([2], VIII). Indeed, we know that, for J as map H~ -~ H~, ker(J - ,~) is independent o:[ hence (J - 30-1 exists in L(H~) for all ,k ~ F. Moreover, using the isomorphism l-Is : Hs -~ H of Lemma 2.1, get II~(J- A)II~ -~ = J- A + Z~, where Z~ = II~JH; "1 -J e Op¢c_e~ e L(H). It follows that II~(J-A)-~II; -~ = (J-,~+ Z~)-~ =

343

Dirac Algebra and Foldy-Wouthuysen Transform

((J - A)(1 + (J - A)-1Zs)) -~ = (1 + Ws)~)-l(,] ,~ )-1, where Wsx = (J - A)- ~Zs is small as IA[ gets large. Similarly for (j2 + #2)-1. Thus indeed, the integrals (or the principal value) converge also in L(Hs), for all s. Wehave proven: PROPOSITION2.6 The spectral projections E(A) of each of the operators H, H’x, H+ of Theorem 1.2 all belong to Op~co, as long as A E ll~ belongs to the resolvent set of the corresponding operator. Focusing first on the proof of Theorem1.3 use the Cdo U of (1.3). (0.1) substitute ~b = Uw, and left-multiply by U* for

=

0

H_

’

=

r

0

’

In equation

(2.8)

Wehave decoupled the Dirac equation (modulo O(-~)) into a O¢/Ot + iH+¢ = 0

(2.9)

of2 x 2-systems, using ¢ for w again. Each system (2.9) is symmetric (semistrictly) hyperbolic (of type e~)--with the additional property that H+= A+(x, (mod Op~bc_e~): There is only one eigenvalue of the symbol, modulo order -e 2 = e ~ - e. Theorem5.1 of [1] maybe applied, together with the following corollary: PROPOSITION 2.7 Under the assumptions of [1], Theorem 5.1, if there is only one eigenvalue A(x,~) (of multiplicity n) (where n = 2 here) then the algebra coincides with Opec: It contains all strictly classical Cdo’s. PROOF Wediscuss only our special case with b of [1], (5.1) equal H = ,~(x, D) hi(x, D), h~ 6 ~ee~ - e, ,~ a multiple of 1. Clearly every symbol q E Ce.m commutes with ~, and we thus may construct a z of order m - e such that Op(q + z) ~ Again -z commuteswith ,~, hence an s of order m - 2e exists with Op(-z + s) ~ hence Op(q + s) 6 P. Iterating this we find that a correction w of any finite order may be fouud to bring Op(q + w) into P. It follows that q(x, D) itself belongs to P, as stated. Returning to 4 × 4-matrices we note that both the system (2.8) and the (fully decoupled) system O¢/Ot +iHA¢ = 0 (2.10) are semi-strictly hyperbolic of type e~, and thus must have their invariant algebra P~’ and Q, respectively. Clearly, e",(H" +ra)~’ = U¯ e",HtU, hence also pzX = U*PU. On the other hand, we claim that the algebra pa and Q are identical. look at Zt = ei(g’~+r~)~e-~H~t. Weget OtZt = ei(Ha+ra)tFAe--iHat ~ C°°(N, O(-eo)).

(2.11) Indeed,

(2.12)

344

Cordes

[Note, especially, the families e i Jr, .] : H, HZ~,H/x+ FA are of order 0, theiir derivatives are of order ke~, k = order of derivative, since e i:~ is evolution operator of a symmetric hyperbolic system ([2~, VI, Theorem3.1).] Integrating (2.12) we e i(Ha+ra)~

= (1 + O(t))e ~ = (1 + P(t))e iHa~, O,P e C~(~,O(-~)).

(2.13)

This indeed implies P~ = Q and, of course, P,,~ = Qm, m ~ ~. After Proposition 2.7 we expect Q,~ =

where A,D ~ Op¢c,~, C D B, C ~ O(-~). This will complete the proof of Theorem 1.2: LEMMA 2.8 For A ~ Op¢c~, ifeiH+tAe -i~-t Op¢c,~_je~, for all t, then A ~ O(-~).

= Qt ~ Op¢cm, and if also OjQt

PROOFNote Qt : Rll.~, Rt : eiH+tAe-iH+t ~ Opec,n, ~. : eittt+e-it~-t. Sim~ ilarly, OtQt = cht(A)t’~, with chl(X) = eiH+~(H+X- XH_)e-iH+~ e Op¢c,,.+~, as X ~ Op¢cm. By induction define ch2+~ = H+chJ - chill_, and chj(X) -iH-t. ei~+tchJ(X)e Then ch~(X) e Op¢cm+je~, as X ~ Op~c,~, and by our assu~nption. Nowwe claim that chj ). (A) = 2J Y~ d(mod Op¢Cm+(j_l)e~_e~

(2.15)

Indeed, this may be proven for t = 0 only, using Proposition 2.7. For j =: 1 get ch~(A) ch Y(A) = H+A- AH _ = H+A- A( 9_~ H+) = 2H+A + [A,H+] + ~_~ = 2H+A + ~_~, where ~ generally denotes some operator of O~cr. We used the special assumption on V(x) and (1.4) for H_ = 2V(x)(mod Op~,c_e~), i.e., H_ = ~-e~ -H+, since V ~ ~c-e~. By induction assume that chJ(A) = 2JH~A + ~m+(j-~)~-~. Then ch j+l (A) H+(2:iH~A + ~,n+tj-~)e~-e:) - (2JH~A ~m+(j_~)~_e~)(9_e~ H+). We H~AH+= H?~A + ~m+je’-~,

so, chJ+~(A) = 2J+~H?~A ~, ,+2~,_~:, pr

oving

Nowwe combine (2.14) and (2.15). Note that A = 0 belongs to the complement of ess.sp(H+), so that 0 at most is an eigenvalue of finite multiplicity. By Corollary 2.5 the special Fredholm inverse (for simplicity called H~~) inverting H+ in (ker H+)x =Im H+, and defined 0 in kerH+ belongs to Opy)c_e~. So we get .4~ = 2-JY~Ych~(A) + ¯ .... e = 9,,,-e

+ ~,,~-j~l~ -~, J = 1,2,....

(2.16)

Weclai~n that (2.16) implies that At ~ Op¢cm-~, hence A e Op¢cm_e~. Indeed, the unitary operator Vt-~ is O(0), since the exponentials exp(iH&t) are evolution operators of symmetric hyperbolic systems. Hence Wj = 9m-j~-~ e O(m -- je). For large j the distribution kernel of ~Vj becomes s~noother and its derivatives will decay faster. We may write Wj ~ formal Cdo ~ith symbol w~ wj = O(((x)(~))-l), as j > j(1). Note (2.16) implies equality of symbols symb(At) must satisfy the first few estimates for ¢c,n-~, as j gets large, he:ace symb(At) ~ ¢cm-~, since (2.16) holds for all j. Thus A 60p¢c,,~-~. Iterating A ~ Op~c,,~-ke, k = 2,3,... ~ A e O(-~), q.e.d. With the proof of Lemma2.8 the proof of Theorem1.3 is complete. Then Theorem 1.4 will follow if we look at the spectral projections P = E(0) and p/X = Ez,(0}

Dirac Algebra and Foldy-Wouthuysen Transform

345

H and HA + FA, respectively. (Assume 0 is not an eigenvalue of H and A +FA= U*HUor else take P = E(e) and pa = EA(e) for suitable small e.) We pA = U*PUas well. Let Q = 1 - P, QA = 1 - pA. Then p,Q, pA,QA E OpOco. Let P’, Q’ be the corresponding projections for the operator HA. Then Pa - P’ may be expressed by a difference of resolvent integrals (2.6), with a - z) -~ (HA + F~ - z)- ~ = (Ha - z)- t x a ( a + FA- z )- ~ e O (- ~). I ntegrals conve rge in L(Hs) for all s, as for Proposition 2.6. Hence Pa - P’, Qa _ Q, e O(-~).

Thec°nditi°ns°fThe°rem 00) m°dO(-~)" 1.~ may thus be formulated as P~AQ~, Q~AP~ ~ O(-~). Since P~ U*PU, ~ = U*QU,it is then clear that they equivalently may be fomnulated as (1.5). Q.E.D. als°P~=

( 00 0)i

, Q~= (10

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

11.

12. 13. 14. 15.

HOCordes. A pseudo-algebra of observables for the Dirac equation. Manuscripta Math 45:77-105, 1983. HOCordes. The technique of pseudodifferential operators. London Math Soc Lecture Notes 202; Cambridge University Press, 1995. HOCordes. A pseudodifferential Foldy-Wouthuysen transform. Commin Partial Differential Equations 1983;8(13):1475-1485. B Thaller. The Dirac Equation. Berlin-Heidelberg-New York: Springer Verlag, 1992. WHeisenberg. Gesammelte Werke. Berlin-New York: Springer, 1984. A Sommerfeld. Atombau und Spektrallinien 1,2.5th ed. Braunschweig: Viehweg and Sons, 1931. E Wichmann. Quantenphysik. Braunschweig: Viehweg und Sohn, 1985. D Shirkov, N Bogoliubov. Quantum Fields. Reading, MA:Benjamin, 1982. A Pais, M Jacob, D Olive, M Atiyah. Paul Dirac. Cambridge: Cambridge University Press, 1998. A Unterberger. A calculus of observables on a Dirac particle. Preprint 96.02, Dept Math Univ de Reims, URA1870, 1996. To appear Ann Inst H Poincare (Phys Theor). A Unterberger. Quantization, symmetries and relativity. Preprint 96.09, Dept Math Univ de Reigns, URA1870. Perspectives on Quantization, Contemporary Math, AMS214:169-187, 1998. L Foldy, S Wouthuysen. On the Dirac theory of spin -7 particles. Phys Rev 78:20-36, 1950. E deVries. Foldy-Wouthuysen transformations and related problems. Fortschr d Physik 18:149-182, 1970. DRGrigore, G Nenciu, R Purice. On the nonrelativistic limit of the Dirac Hamiltonian. Ann Inst Henri Poincare - Phys Theor 51:231-263, 1989. B Thaller. Normal forms of an abstract Dirac operator and applications to scattering theory. J Math Phys 29:249-157, 1988.

346 16.

Cordes

HOCordes. Lorentz transform of the invariant Dirac algebra. Integr Equ Oper Theory 34:9-27, 1999. 17. B Gramsch. Relative inversionin dcr Stoerungstheorie von Operatoren und ¢-algebren. Math Ann 269:27-71, 1983. 18. E Schrohe. Spaces of weighted symbols and weighted Sobolev spaces on manifolds. Springer LNMVol 1256:378, 1987. 19. R Beals. Characterization of pseudodifferential operators and applications. Duke Math J 44:45-57, 1977.

On Perturbations for the Continuous Spectra of Semigroup Generators MICHAELDEMUTH Institute of Mathematics, Clausthal- Zeller feld, Germany

1

TU Clausthal,

Erzstr.

1, 38678

INTRODUCTION

In mathematical scattering theory the completeness of wave operators implies the stability of the absolutely continuous and can be used to discuss the behaviour of the singularly continuous spectra. Using the invariance principle the completeness problem can be shifted to the theory of semigroups. The absolutely continuous spectra are invariant if sandwichedsemigroupdifferences are trace class. The singularly continuous spectrum remains empty if semigroup differences decrease sufficiently fast. In stochastic spectral analysis this implies the stability for the spectra of generators of Markovprocesses. In the case of singular perturbations sets of finite capacity have no influence on the absolutely continuous spectrmn. There are sufficient conditions in terms of the equilibrium potential such that the singularly continuous spectrum remains empty.

2

SPECTRAL

INVARIANCE

AND SCATTERING

THEORY

The ~nathematical scattering theory has its origin in quantum mechanics where the dynamics of two quantum systems is compared asymptotically in large time scales. From the mathematical point of view it is a tool in the perturbation theory for the continuous spectra of self-adjoint operators. Let A,B be operators in different Hilbert spaces 7/~, 7-/2, respectively. Assume A, B to be self-adjoint and bounded below. Let J be a bounded identification operator mapping741 to 7/2. Let Pac(A) be the projection operator onto the absolutely continuous subspace of A. Waveoperators are defined by 347

348

Demuth Ft+ (B, J, .4) := s). - lime~tBJe-itApac(A

(Details in two-space scattering theory are given by Kato [8]). Introducing the polar decomposition ~+ = sgn (f~+) the wave operator g/+ is called complete if [sgn (f~+)]* sgn (D+)

(2)

sgn (Ft+)[sgn (f~+)]* Pac(B).

(3)

and For f~_ we have the corresponding definition. The operator sgn (D+) yields a unitary equivalence between A/p,~(A)7~, and B/poc(B)n~ such that the absolutely continuous spectrum remains invariant, i. e., aac(A) = a~(B).

(4)

The completeness problem was combined with the study of the singularly continuous part of the spectrum by results of Enss [6], [7]. The interested reader may find a smnmarizing and extended presentation of the Enss-method in the book by Perry [9]. If the waveoperator f~+(B, J, A) is complete and if its range satisfies ran[~+(B, J, A)] Pc(B)7-t.~,

(5)

it follows that there is no associated singularly continuous subspace, i.e., Psc(B)7-t~ {0}, im plying th e si ngularly co ntinuous sp ectrum tobe empty, i.e .,

(6) Here Pc(B), P~¢(B) are the projection operators onto the continuous and singularly continuous subspace of B.

3

STABILITY

CRITERIA

FOR

The Kato-Rosenblum-Birman-Pearson Lemmacombined with the invariance principle tells us that the wave operator fl+(B, J, A) exists if e-~J -.Je -A is a trace class operator (see e.g. Reed, Simon[10]). This can be generalized to sandwiched semigroup differences. CRITERION 1 Let A, B be two operators self-adjoint, Hilbert spaces 7-/~ and ~/.~. Assumethat 1) e-U(e-t~J - Je,A)e -A is a trace class operator,

bounded below, acting in

Perturbationsfor ContinuousSpectraof Semigroup Generators

349

2) (e-SJ - Je-A)e -A is a compact operator, 3) (J*J - l~tl)e -A is a compact operator, (J J* - ln2)e -B is a compact operator. Then the wave operator fl+(B, J, A) (and also fl_(B, J, A) exists and is complete implying O~c(B) = ~oc(A). ¯ (7) A proof of Criterion 1 is given by van Casteren, Demuth[1]. In the abstract theory of 2-summing operators the trace class property of a product of three operators can be investigated. This can be used for our situation when the Hilbert spaces are L2-spaces. THEOREM 2 Let 7-{~ = L~(#), ~2 = L~(~) be two different be self-adjoint, bounded below in these L2-spaces, such that

L2-spaces. A,B

e -A E B(L2(#), L°~(#))

(s)

e-~ e B(L~(~), L2(~))

(9) (10)

e-uj _ Je-A ~ 13(L°°(t~),

L~(~)).

Then the following assertions are proved by Eder, Demuth[5]: a) The sandwiched semigroup difference operator. Its norm can be estimated by

e-B(e-sJ-

Je-A)e -A is a trace class

(11)

eI[e-t~(e-t~ J - Je-’A)e-A[Itrac

where ke is the complex Grothendieck constant known to be smaller than ~r/2. I]’ I]~,~ denotes the normfrom L~ to ’~. L b) The operator (e-t~J - Je-~)e -A is. compact. Its operator norm can be estimated by ~. ]](e-s J - Je-A)e-A]] _< [[e-Al[~,oo

(12)

II(e-BJ - Je-A)e-Al[2, ~ [le-BJ-

je-Alloo,~ ¯

Note that in both estimates above the L~ - L°° smoothing of the semigroup difference e-BJ -- Je-A plays the crucial role. This seems to be a characteristic feature in manyspectral theoretical considerations. The last theorem is nowapplied to integral operators

in

Demuth

350

COROLLARY 3 Let A, B be self-adjoint, bounded below in 7-/1 = 742 :: L"(~.d). Set J = 1. Assume that A and B generate ultracontractive semigroups. Suppose for their integral kernels: supl(e-A)(x,y)l

< c, supl(e-U)(x,y)]

(13)

X,y

and sup~ /

I(e-d)(x,y)ldy

sup /I(e-’)(x,y)ldy

< c,

(14)

Set D := e-B - e -A. This is an integral operator, too. Denote its kernel by D(,., .). Then ao.c(A) a~c(B) if

f dx

j dy

(15)

ID(z,y)l

For additional information see also Demuth[2].

4

STABILITY

CRITERIA FOR asc

For time dependent completeness criteria one needs more information about the free evolution eirA. Hence we restrict our consideration to A = -A =: Ho in L’z($~d). In case of potential scattering with B = Ho + My, the standard assumption is

f

llMv(Ho + 1)-lM~(l.,ikr)lldr

< oo

(,16)

0

with the multiplication operators (U~f)(x) := ~(z).f(z).

(:17)

The condition in (16) is satisfied IV(:OI <_-’-~ c(1 + Ixl) for some ¢ > 0 and all [z I > R0, R0 large a~c(Ho + My) = ~ follows.

(18) enough. For such potentials

However,the condition in (16) is not applicable for singular perturbations, where B arises by imposing Dirichlet boundary conditions on a closed set F, called obstacle or singularity region. One possible way to include such singular perturbations consists in proving the existence and co~npleteness of ~t+(e-B, e-tt°) and show!rag ran[l~+(e -u, e-"O)] = P~(e-B)N = P~(B)T{,

19)

Perturbationsfor ContinuousSpectraof Semigroup Generators

351

with 7-I = L~(l~d). The following theorem is proved by Demuth,Sinha [3]: THEOREM 4 Using the notation from above assume that the wave operators f~=k(e-B,e -tt°) exist. Assume that we have two projection operators P+,P_ decomposing the Hilbert space and satisfying. P+ + P_ = 1,

(20)

s - lim P+e -it(exp(-H°)) = 0,

(21)

s - lira P_ eit(exp(-H°)) = 0,

(22)

(f~+(e-u,e -u°) --1) ¢(H0) P- is compact,,

(23)

(f~_(e-B,e -H°) - 1) ~b(Ho) P+ is compact,

(24)

(In (23), (24) ¢(.) is a special smooth function with compact support.) Then ran (a+(e -B, e-H°)) = Pc(B)7t and

The strategy of the proof is similar to that of Enss. He used a decomposition by defining P+, P_ via a transformation to the momentumspace. For the result here we defined the decomposition in terms of the spectral representation where ln(Ho) is diagonal. This possibility was studied by Perry [9], too. Howeverwe translated it to functions of H0. For an interested reader let me give the definition of P+, P_: DEFINITION 5 Let F be the Fourier transformation fl’om L2(~:~d,dx) to L2(~’t, dk). Let U be the transformation from L2(IRd, dk) to Le(~, db; L~(Su-~)) such that for smooth functions ~o (UF~(Ho)f)~(w) = ~(e~)(UF ~3 ~ b~t.

S d-1 ,

f

~ LZ(1Rd,dx). Let G be the Fourier transformation

(25) with respect to

Then we introduce P+f := F*U*G*Mx(~>o)GUFf,

(26)

P_ f := F* U’G* Mx(,~
(27)

Clearly, P+ +P_ = 1. The other conditions in (21) - (24) can be verified -xBe -xg° is compact for all A > 0 and if the extension of (e -s -e-H°)e-~°M(z)~+e a bounded operator. Weuse the abbreviation (x) := (1 + Ixl 2 1_

De~uth

352 Hence we can formulate the following theorem. THEOREM 6 Let H0 be the Laplacian in L2(~d). Let B be a sclf-adjoint ator, bounded below, acting in L-~(_~d). Assumethat e -’xB -- e -’xH°,

fl > 0, is

compact.

oper-

(28)

2 1_ Denote (x) := (1 + I )~. Le t th e ex tension of (e -B --

(29)

e-H°)e-H°M(~)l+~

be a bounded operator in L~(~d). Then B has no singularly

continuous subspace, or asc(B) =

¯

This theorem can be applied easily to the case where e -~B forms an ultracontractive sernigroup with a Gaussian estimate. COROLLARY 7 Let e-’xB be an integral (e-AB)(x,y), A > 0. Assume

operator

with the kernel

le-~’(x,y)l _< e A-~_1~_~ e

e~

¯

(30)

Denote the semigroup differences by D~ := e -:~" - e -~H°, A > 0.

(31)

Let dx / dy D:~(x,y)l(1

[y]:)~+~ < ~

(32)

for all A > 0. Then Theorem6 is applicable,

5

i.e. asc(B) =

APPLICATIONS

Werestrict ourselves here to singular perturbations. The Laplace operator Ho is associated with the Wiener process. Denote by E.~ {.} the expectation with respect to the Wiener measure, such that (e-~H°f)(x) -- E::{f(X(X))}

(33)

holds for all f ~ L~(~d). Denote the singularity region by F, where F is a closed set in ~d. Let its complement be denoted by Z = K/d \ F. The first hitting time of F for the Wiener trajectory X(.) (34) Sr := inf{s, s > 0, X(s) F}.

Perturbations for ContinuousSpectra of SemigroupGenerators

353

The family E.~{f(X(~)); Sr >_ A} restricted to L2(E) generates a strongly continuous semigroup in this space. Its generator is denoted by H~. e -’~H~ is an integral operator, the kernel of which is smaller than (e-~g°)(x,y). Dx = e -’xg° - e-~Hn has the kernel

The semigroup difference

Dx(x,y)= E~A{I}- E~’~{Sr>_ A} = E~A{Sr< A},

(35)

hereE.~"x{.} is theconditional Wienerexpectation. Moreover we have

/

IDx(x,y)[dx

< A}.

=

(36)

Thisis thepointwherewe canapplyCorollary 3 andCorollary 7. COROLLARY 8

LetH0,HE be givenas described above.

a) If Ez{Sr < A}dx < ~,

(37)

we get

b) If

aac(He) = aac(Ho) = [0,

(38)

f E~{Sr< ~}(1 I~l~)° <

(39)

+

for some a > 1 and all A > 0, we get

(4o) REMARKS The last results are related to the capacity of the singularity region F. The one-equilibrium potential of F is defined as vr(x) := E~{e-Sr,Sr

< co}

(41)

mapping ~d _+ [0, 1]. Clearly one has E~{Sr < A} _< e~vr(x).

(42)

On the other hand in this context the capacity of F satisfies

cap(I’) =f vr(x)dz.

(43)

354

Demuth

Hence a,.c(Hn) = aac(Ho) if cap(F) is finite. capacity is finite and if additionally vr(x)(1

Moreover, asc(He) is empty if the

(44)

for some 7 > 1. In both cases unboundedsingularity regions F are allowed. They can consist of a union of infinitely manyballs, if the dimension d is large enoughand if the radii of the balls are sufficiently decreasing.

REFERENCES J. van Casteren and M. Demuth, Completeness of scattering systems with obstacles of finite capacity, Operator Theory: Advances and Applications, 102:39 - 50 (1998). 2. M. Demuth, Integral conditions for the asymptotic completeness of two-space scattering systems, Helv. Phys. Acta, 71:117 - 132 (1998). 3. M. Demuth and K. B. Sinha, SchrSdinger operators with empty singularly continuous spectra, Preprint TU Clausthal, 1999. 4. M. Demuth, P. Stollmann, G. Stolz and J. van Casteren, Trace norm estimates for products of integral operators and diffusion semigroups, Integr. Equat. Oper. Theory, 23:146 - 153 (1995). 5. S. Eder and M. Demuth, A trace class estimate for two-space wave operators, Preprint TU Clausthal, 1999. 6. V. Enss, Asymptotic completeness for quantum-mechanical potential scattering, I. Short-range potentials, Comm.Math. Phys. 52:233 - 258 (1977). 7. V. Enss, Asymptotic completenes’s for quantnm-mechanical potential scattering, II. Singular and long-range potentials, Ann. Phys. 119:117 - 132 (1979). 8. T. Kato, Scattering theory with two Hilbert spaces, J. Funct. Anal. 1:342 369 (1967). 9. P.A. Perry, Scattering Theory by the Enss method, Mathematical Reports Vol. 1. Harwoodacademic publishers, Chur (1983). 10. M. Reed and B. Simon, Methods of modern mathematical physics, Vol. III: Scattering theory. Academic Press, NewYork (1979). 1.

Mathematical Study of a Coupled System Arising in Magnetohydrodynamics J.-F. GERBEAU and C. LE BRIS Cermics, Ecole Nationale des Ponts et Chauss~es, 6-8 av. Blaise Pascal, Champs-sur-Marne77455 Marne-La-Vall~e, France

1

INTRODUCTION

This work deals with the mathematical study of a system of partial differential equations related to a magnetohydrodynamic (MHD)problem. The MHDequations we consider govern the behaviour of an homogeneous incompressible conducting viscous fluid subjected to a Lorentz force due to the presence of a magnetic field. More precisely, we study a coupling between the transient Navier-Stokes equations and the stationary Maxwell equations. This model can be considered for example in industrial situations when the magnetic phenomenaare knownto reach their steady state "infinitely" faster than the hydrodynamics phenomena. Many mathematical works have been devoted to the study of MHDproblems. Weonly present here some of them briefly and we refer to [5] and A.J. Melt, P.G. Schmidt [10] for some more detailed overviews. The coupling between the transient Navier-Stokes equations and the transient Maxwell equations (without displacement current) has been studied in G. Duvaut, J.-L. Lions [3] and in M. Sermange, R. Temam[12]. Numerical methods conserving the dissipative properties of the continuumsystem in 2D are presented in F. Armero, J.C. Simo [1]. Less numerous works have been devoted to the fully stationnary MHD equations, namely a coupling between two elliptic partial differential equations (see for example M.D. Gunzburger, A.J. Melt, J.S. Peterson [7], J.-M. Domingez de la Rasilla [2]). Finally, let us mention an interesting alternative viewpoint which consists in considering the electrical current rather than the magnetic field as the main electromagnetic unknown(see A.J. Meir, P.G. Schmidt [9,10]). In the present work, the equations related to the velocity field are the transient Navier-Stokes equations whereas those related to the magnetic field are elliptic (see (2.1)-(2.8)). The difficulty is that the ellipticity of the equation for B depends 355

356

Gerbeauand Le Bris

the velocity field u. Briefly speaking, if the velocity becomestoo large, the system may become ill-posed. Under restrictive assumptions upon the physical data, we can however prove ~hat a strong solution exists and is unique at least on a time interval [0, T*] for sometime T* depending on the data (see Section 4, Theorem1, the result we prove here has been announcedin [6]). For this purpose, we give in Section 2 a presentation of the equations and the functional spaces, and we establish in Section 3 some preliminary existence and regularity results upon~ the magnetic equation. As soon as the magnetic operator is no longer invertible --- which may occur if the velocity becomes too large - we show in Section 5 that we can construct two distinct solutions to the system. This latter observation shows that the model we study here should be used only with great care in numerical simulations.

2 2.1

EQUATIONS

AND FUNCTION

The Transient/Stationary

SPACES

Model

Let fl be a sitnply-coImected, fixed boundeddomainin I1~3 enclosed in a (7°° boundary F. Weshall denote by n the outward-pointing normal to Ft. The transient/stationary problem we shall consider is the following : find two vector-valued functions, the velocity u and the magnetic field b, and a scalar function p, defined on flx [0, T], such that 0,u+u.Vu-r/Au divu 1 -curl (curlb) divb with the following initial

f-Vp+curlbxb

= =

0

= . curl (u x b) =

0

in ~,

in ~t,

(2.1) (2.2) (2.3) (2.4)

and bouudaryconditions : u b.n curlb×n

= = =

ult=o =

2.2

inQ,

in fl,

0 q k×n Uo

on F, on F, onF, in Ft.

(2.5) (2.6) (2.7) (2.8)

Functional Setting

For m _> 0, we denote as usual by Hm(Ft) the Sobolev space H"~(~)) = {u ~ L’~(Ft); D’ru ¯ L~(~),VT,I~l < ,,z}, where ~/= (3’~, 7~, ~/3) is a multi-index and 171 = 7~ + 72 + 7a. The norm associated with H"(~) that we will use is

357

CoupledSystemArising in Magnetohydrodynamlcs 7)7,

"~ ID I

= I1~’11-o.~)

I ~ The subspace of H (~) consisting of functions vanishing on 0f~ is denoted as usual by H~ (~) We shall denote respectively (LP(~)) 3 and (Hm(fl)) 3 by ~(~) and ~(~) when there is no ambiguity, by ~ and ~. Weshall use the Sobolev inequality : for 2 ~ p ~ 6,

II/ll~,
(2.9) Let T > 0 and let X be a Banachspace. The space L~(0, T; X), 1 ~ p ~ ~ is ~he space of classes of Lp functions from [0, T] into X. Werecall that this is a Banach space for the norm

II~(t)ll~ dt

if ~ ~ p < ~, ess supII~(t)llx if

p = ~.

The following trace spaces and norms will also be needed : H~/~(F) {v]r,v ~ H~(fl)},

’, H-t/~(r)= (H~/~(P))

][q[[Y~/~(r) -

inf

..,

:

[[k[]~-,/~(r

) =

inf

sup

Wedenote by C~(~) (resp. C~(fl)) the space of real functions infinitely differentiable with compact support in ~ (resp. ~). Weintroduce the spaces W= {C e (C~(g)) ~, di~ C : 0, C.nloa = 0}, V : {~ e (C~(a)) ~, div~ = 0}, V = {v ~ ~(a),divv

= 0},

W : {C ~ a ~(a),divC

: 0, C.n[o~ = 0},

H= {v e ~ (~), divv = 0, v.’~lo~= 0}. The space V (resp. W)is the closure of ~ (resp. W)in ~ (~) (resp. 1 ( ~)). H the closure of Y (and ~) in ~ (fl). L et u s r ecall t hat u.n makes s ense i n H-~/~(O~) as soon as u ~ L~(~) satisfies divu = 0. For v ~ V and C ~ Wwe denote

One can establish that [I,IIr (resp. I~.11~)defines a norm (resp. W) which equivalent to that induced by Hl(~) on V (resp. W) (cf. G. Duvaut and J.-L. Lions [4]). Thus we have Nr B e W For 2 ~ p ~ 6, this inequality that, for B ~ W

together with the Sobolev imbedding (2.9) imply

As well, Poincarfi inequality and (2.9) imply that, for u ¢

IlullL.¢m _
358

Gerbeauand Le Bris

2.3 Regularity

of the Data

Weshall suppose in the sequel that UoE H~(fl)~q tHI2 (fl), with divuo q ~ C(O,T;H3/2(F)),

(2.10) (2.11)

with frq=O,

k ~ C(O,T; 1/2 (F)),

(2.12)

f ~ L~(0, T; ~ (f~)).

(2.13)

From a. physical viewpoint, it is natural to assume that k is the trace on F of the gradient of the electrical potential : k = aV¢lr.

3

PRELIMINARY

(2.14)

RESULTS

First of all, we notice that we can split the magnetic field b(t) fi IHI~ (~) satisfying (2.6) and (2.4) into the sum of a function Bd(t) that satisfies (2.6) and a function B(t) ~ W. Indeed, we have : LEMMA 1 Let q ~ C(0, T;H~’-I/2(f~)) for k = 1 or k = 2, there exist C ( O, T; Hk ( f~ ) ) anda constantd4 such

JlB’qJc(o,T;~(m)<_&[lqJlc(O,T;H~-’/~(m)’

Bd.n = q on [0,T] × F and Moreover, we can impose that divBd(t)

= 0 and curIBd(t)

= 0 for

[O ,T ].~

PROOF It suffices to define Bd as follows : Bd(t) = V¢(t) where ¢(t) is a solution of the Nemnannproblem -A¢

=

0¢

= q(t)

On Let B(t) = b(t)following one :

Bd(t).

O~u + u.Vu - yA u div u 1 -curl (curl/3)

0

onr.~

Wereplace the original problem (2.1)-(2.8)

= =

din[) f-Vp+curlBxB+curlB

=

curl(u x B) + curl (u Bd)

div B =

0 in

0

xB

(3.1) (3.2)

f~,

in f~,

with

in ~,

(3.3) (3.4)

359

CoupledSystem Arising in Magnetohydrodynamics with the following initial

and boundary conditions u B.n curlBxn

= = =

ult=o

0 0 kxn

on F, on F,

(3.5) (3.6)

onF, in ~2,

= uo

(3.7) (3.8)

Wenow proceed to establish a preliminary existence and uniqueness result for the magnetic problem and two estimates which will be needed in the next section. Wedefine the (non-empty) convex set I~M = {V E L2(O,T; V),

supte[0,T] IIv(t)llv<_ IIOtvIIL=(O,T;L=Ia)) < M}.

The values of the constant Mand T will be fixed later.

Weonly suppose here that

1 M
(3.9)

Let us note that v E /CM implies v ~ C(0, T;V). For v following problem : find B 6 C(0,T; W) 1-curl (curl B) = curl (v x B) + curl (v ~)in f

]~ M, we

consider the

l x [0, T],

(3.10)

divB = 0 in f~ x [O,T],

(3.11)

with the following boundary conditions : B.n = 0

on F × [0, T],

curlB x n = k x n

(3.12)

on F x [0,T].

(3.13)

PROPOSITION1 For v ~ ICM and M satisfying hypothesis (3.9), the problem (3.10)-(3.13) has a unique solution C(0, T; W). Moreover, we ha ve the f ollo wing estimate :

sup[IB(t)llw_< ~ ÷ ~llvll~(O,T;V),

(3.14)

where a~, ~ and 71 are some constants defined below. ~ PROOFExistence

and uniqueness.

We define on W × W the bilinear

a~(C~, C2):_ol £curlCl.CU,’l C~ ax

form

- f.((tl×Cl).CurlC:~dz

and h~(t) ~ W’ such that, for C ~ W, < h~,(t),C

>= ~v(t)

Be(t).curlCdx +

-
xn, C>r

360

Gerbeauand Le Bris

First, let us prove that problem (3.10)-(3.131) is equivalent to find B E C(0, such that av(B(t), =< hv( t), C > (3 .1 5) for all C E W. Let B ~ C(0,T; W)which satisfies /f~ ~curl(curlB)-curl(v(t)

(3.15). Integrating by part, we have

× (B + Bd)).Cdx =< ~(k × n-curlB

for all C 6 W. First we deduce that < curlB x n,C >r=< k(t) yields (3.13). Moreover, since divC = 0, there exists p such that

x n,C >r which

lcurl (curl B) - curl (v(t) × (B + Bd)) The function p satisfies : -/kp Op On

= _

0 in Ft 1 curl (curl B).n - curl (v(t) x (B + Bd)).n a

It is straightforward to check that the normal component of curl (v x (B + Bd)) contains only tangential derivatives of v. Thus, using vie = 0, curl(v x (B Bd)).n = 0. Moreover,(curl curl B).n = -0~, ((curl B n).t~) - 0~((curl B x n).t ~), where 0t, and 0t= denote the tangential derivatives. Then, hypothesis (2.14) yields (curlcurlB).n

Op

= 0. Therefore ~nn = 0, which proves p

C~ ~

and (3.10). ~oonversely, we easily check that a solution of (3.10)- (3.13) satisfies (3.15). Moreover, a.,(., .) is continuous and coercive on Wx W. Indeed

_< ~llcurlC~llc~(n) llcurl (1_ w +d~d.~M)llC~llwIIC.,.ll < and ~7

> (l__d~dallvllv)llCll~,v O" > (~-d2d3M)llCIl~ Therefore, the Lax-MilgramTheorem implies that the variational problem (3.15) has a unique solution B(t) ~ W. The continuity in time of Bd and v implies that

Be c(o, T;W). Estimate in L~(0, T; W). Taking C B(t) in (3. 15), we hav -l £ ]curlB[~ dx =

/.v x (B + B~).curlBdxl- r..

~) IIBIIw+ ~ Ilvl~ (llBIl~ + IIB¢~ll~

CoupledSystemArising in Magnetohydrodynamics

361

Thus IlBllw _< d,~d.~llvllvllBiIw +e,~llvl]vllqllH,.-+dl IlklIH-,/~.Wededuce the estimate : sup liB(t)llw t~[0,Tl

(1 -- d~d3a~lVllL~(O,T;y))

For simplicity, we introduce the constants ~ = d~ [lk][L~(O,r;a-~/~ ) , ~ = C~ a~Iq][L~(O,T;H~/~), 7~ = d2daa, which gives (3.14). In the next section, the vector field B defined above will appear on the right hand side of the Navier-Stokes equation in the Lorentz force curlB x B. Wesee that we need an estimate on u in L~(0, T; IHI1 (~)) in order to prove the coercivity problem(3.10)-(3.13). Such a control on u is typically obtained with strong solutions of Navier-Stokes equations. To define strong solutions, the force term in NavierStokes equations has to belong to L°°(0, T; L:(fl)) (see R. Temam[13]). In scope, the estimate on B in L~(0, T; W)is not sufficient. That is why we establish now a "better" estimate on B. First, we need the following proposition which is a straightforward extension (in the non-ho~nogeneous case) of Proposition 2.1 Saramito [11] (see also Lemma2.1 and Remark2.3 of [11]). PROPOSITION 2 Letm be a nonegative integer and 1 < p < oo. Let g ~ W"~’~ (f~), with divg = 0 and g.n = 0 on F, k ~ Wm+~-WP’P(F),q ~ Wm+2-~/P’P(F). ~+~’p(f~) such that Then, there exists a unique B ~ W curl(curlB) div B B.n curlB x n

= = = .=

g in f~, in f~, 0 q on F, k x n onE,

PROPOSITION 3 Under hypothesis (3.9), the solution of problem (3.10)-(3.13) given by Proposition 1 satisfies -

1 -

where a,~_, 30. and 7"~ are some constants defined below. PROOFLet g be defined

by

g = acurl (v x (B + Bd)) = a(B.Vv - v.VB + Bd.Vv -- v.VBd). Wehave div g = 0, g.n = 0 on F (because v = 0 on F and the normal component of curl (v x B) contains only tangential derivatives of v, as said above). Moreover

_
362

Gerbeauand Le Bris

Thus, Proposition 2 with m = 0,p = 3/2 yields

IIBtlw~,3/=l~ _
Finally, we use (3.14) and the Sobolev inequality

and we introduce someconstants for ease of notation : c~2 = C2dh(llql[L,,o(O,T;W4/a.a/~) +[[k[ILoo(O,T;W,/~.~/=)), f12 : C4dha[lq[lLoo(O,T;H,/=),72 : c4dhoUq[lLoo(O,T;H,/2), which gives (3.16).

4

AN EXISTENCE

AND UNIQUENESS

RESULT

Let M > 0, we define

where ~o = IlfllL=(O,T;~=(n)) and the constants c~i, ~i and 3’i are defined in the previous section. Wealso define the fimctions lq, 1.~ and p:~ by p~(M)~ = 4max (,,UoU~.,

.=(M)= = Iluoll~.

~O(M)~)

3 + 2To,M)2 + pi(M)

,~(M) = e(M)+ c~(M)+ ~,o#,(M)#2(M).

(4.1)

(4.2) (4.3)

The constants Ch,...,cm ~ppear in the following proof ~nd do not dependon the physical data. THEOREM 1 As soon as the physical data Uo, l/q, a, f, q, k, are "small enough" (in a sense madeprecise below), there exists a time T* > 0 such that the MHD problem(~.1)-(~.8) has a unique solution [0, T*]. This solution satis fies u ~

L~(0,T*;~(a))n L=(0,~*;~(a)) .rid b ~ C(0,T*;a’ (a)) PROOF Existence. In the previous Theorem,"small enough" meansthat the data are suchthat the following property holds : There exists 0 < M < 1/7~ such that pi(M) ~ M, i = 1, 2,

(’.4.4)

CoupledSystemArising in Magnetohydrodynamics

363

Note that it is indeed possible to choose the physical data such that (4.4) satisfied : a straightforward calculus showsthat ®’(0) = c5(1 +

0~1)(0~1"~2

+ ~2) -[-

c50~2(0q~/1

+

thus, q, k and ~ can be set small enough such that 0 < O’(0) < 1 and therefore one can choose M > 0 small enough such that O(M) < M. In view of definitions (4.1)-(4.3) of ~i, i = 1, 2, 3, it is a simple matter to check by an analogous calculus that (4.4) holds as soon as Uo, f, 1/~ are small enough too. We define the time T* by T* = min(T, 3/(4c6~(M))), we choose M> that (4.4) holds and we define ~M KM= {v e L~(O,T*;V), suPte[0,T. ]

~ M,

[IV(t)IIV

Ilvll~(O,T*~(m) ~

The set ~Mis clearly convex. Moreover, in view of a classical compactness result (see for instance R. Temam[13], Theorem2.1), ~Mis a compact set of the Banach space L:(O,T*;V). For ~ ~ ~M, we use Proposition 1 to define B as the unique solution of al-curl(curlB) divB B.n curlB xn

= = =

= cm’l(gxB)+curl(gxB in ~, 0 0 on F, kxn onF.

d) ing, (4.5)

According to the estimates (3.14) and (3.16), we [Icurl B ) × BllL~(O,T.;L2

and IlcurlB x BallL~o(o,r,;~2)

Therefore, the force term F = f + (curlB) x (B + d) i s i n L~(0, T *; L 2 ( f~)) a sup ~(~[o,r] +¢~ (O~2 + ~’2/~/’~11 -t- ]~IM_ ~/1/~// -t- ]~2/~1) (1

"t- O~1+ ]~IM’~

_< Then, it is proved in R. Temam[13,14] that there exists a unique solution u G L2(0, T*; ~ (~)) ~q L~(0, T*; ~ (f~)) to the Navier-Stokes equations c9tu+u.Vu-~l/ku+Vp = F infl, divu = 0 inf’, (4.6) u : 0 on F,

I

364

Gerbeau andL~eBris

satisfying

moreover sup Ilu(t)[l~ < #I(M)2 ~, _< M te[o,T*] 2

3) [IF(t)ll2 dt + it, (M)

< p~(a,l)~ < ~. .,’14 Wethen deduce from the Navier-Stokes equations that

We deduce that u e /CM. Let us check the continuity iu L"-(O,T;V) of ~ --~ u. Let ~n be a sequence that goes to ~ in L~(O,T;V), it defines a sequence Bn, solution of (4.5). The force term corresponding to Bn in the Navier-Stokes equations has the required regularity to define a sequence u, bounded in L:(0,T;I~(~)) and such that Otun is bounded in L2(0, T;L~(It)). The sequence Un is therefore compact in L~(0, T; IE~ (~)). The uniqueness of the solution yields that u= goes u corresponding to g. Thus the application i7 ---~ u maps continuously the convex compact set /Ca4 into hi~nself. Therefore, the Schauder theorem ensures that the existence of a fixed point. This yields the existence result. Regularity of b. We have just proved that B ~ C(O,T*;W). We show as in Proposition 3 that B ~ C(0,T*;W~’a(~)) and therefore we have in particular B ~ L~(O,T*;Lq(~)),Vq > O. Using for example that B e Lc~(0, T*;LS(!~)), we easily check that the right-hand side of (4.5) belongs to ~¢ (0, T*; Ls/5 (it)). U lug Proposition 2, we deduce that B ~ L°°(0, T*; W~,S/5(Ft)), which implies B ~ L~(0,T*;L~(it)). The right-hand side of (4.5) is then in Lc~(0, T;L2(~)). Applying one more time the regularity result of Proposition 2, we finally condude that B ~ LC~(0, T*; lI-ll~ (it)). In view of the regularity d, we deduce that b e L°"(O,T*;IHI~(Ft). Uniqueness. Let (u~, p~, B~) and (u2, p2, B2) two solutions of problem(3.1)Wedefine u = u~ - u~, B = B~ - B~.. Combiningthe equations satisfied by (u~, 13~) and (u2, B2), we have Otu + u.Vu, + u~.Vu - rlA u + Vp = curl B x B~ + curl B~ x B, -1curl (curl B) = curl (u x B~) + curl (u~_ x B) + curl (u

(4.7) (4.8)

with u = 0, B.n = 0 and curlB x n = 0 on the boundary. Multiplying (4.7) by u, (4.8) by B and integrating we obtain

2dr

&

~ j~ a

- J2

(4.9)

365

CoupledSystemArising in Ma~net0hydrodynamics Weestimate the right-hand side of this inequality as follows :

where Ce and e are some constant, with e arbitrarily

small.

lcurlBz x )B.uldx ~ [IcurlB~llg~(n)llBllLqn)llull~(n

_

(n~llull~¢n~ + ellcurl~ll[~(n~,

~ x Ba.curlBldz ~ C~ lIB~ II~ll’~ll~(n~ + ~ll~ur~l~(n~, fnb~

In this last inequality, weestimatecur] B with equation(4.8)

~ + d~d~llu~ll~ L~£ ~cu~]~1~dx~ (11~11~ + II~all~)llc~r]~ll~llull~ Using sup~[0,T. ] ~[u~H~ ~ M and the coercivity 7x = d2daa, we deduce that 1

-

")’1M(II~IlIL~÷

assumption 0 < M < 1/~ with

IIBdIIL~)IlUlIL’Z"

(4.10)

Thus, [ue x B.curlB I dx <_ C~ 1 -~IM (IIB~II~ +

IIBall~’)elluell~ IMI[~+

d

(4.9) yields ~11-11~. ~ ¢(t)ll-li~.~ with¢ . I1~(~) (~)(ll~lll~+ll~ll,.~)=llu=ll~ta)) C~(llu, ll~(m+llB2lla~(a)II Oathering th~se inequalities,

estimate

Note that ¢ ~ LI(0, T*). Therefore, by GronwaHlemma~u = 0, and using again (4.10)~ B = 0. This provesthe uniqueness of a sgro~g solugion (lt~ b) of (2.1)-(2.8).

5

REMARK ON THE NON-UNIQUENESS

It has been proven in the previous section that the MHD problem (3.1)-(3.8) has unique solution for small data, at least on an interval [0, T*], T* > 0. The idea of the proof has been to ensure the coercivity Of equation (3.3) by controlling the normofu on [0, T*]. Weexhibit in this section an example(due to P.-L. Lions [8]) non-uniqueness in the case when the operator Tu : B ~ curl (curl B) - curl (u x is not invertible. From now on, we assume for simplicity that k = 0, q = 0, thus we deal with the homogeneous boundary conditions on F : u B.n curlBxn

= 0, = O, = 0.

366

Gerbeauand Le Bris

Let us assume that for so~ne to and some fi = ft(to,x) the opera.tot T,~ : B -~ curl (curlB) - curl (u x B) is not invertible. There exists a divergence-free field /) ~ 0 satisfying curl (curl/)) = curl (fix/)). Note that such a ~i is necessarilly "large enough", otherwise, Ta would be coercive. If we consider the force ] = ~.V/~ - r/A fi - curl/) x/), then (~i,/)) is a (stationary) solution to Otu+u.Vu-~lAu+Vp = ]+curlBxB, divu = 0, (5.1) curl(curlB) = curl(u x B), div B = 0. Next, we define u’ as the solution of

{

Otu+u.Vu-~Au+Vp divu

= ], = 0.

with the "initial" condition u’(t0, .)= fi(t0, Wefinally observe that (’C~, B) and (u’, 0) are different (since /) ~ 0) while both satist}" (5.1) on [to, +ec). Thus, we have two different solutions of the problem with homogeneous boundary conditions.

6

CONCLUSION

We have proved that the MHDsystem (2.1)-(2.8) has a unique solution on interval [0, T*] as soon as the physical data are regular and stnall enough, with T* > 0 depending on the data. Note that the proof may probably be extended to the case of muttifluid equations in two dimensions with constant viscosity and conductivity. Moreover, we have shown that a solution is not unique if the operator B --+ curl (curl B) - curl (u x B) is not invertible. This mayoccur as soon as the velocity becomes too large, but it is an open question to show that the operator do indeed becomenot invertible. The practical conclusion of this study is the following : even if the modelpresented here seems well-suited in some physical situations and even if it is mathematicaly well-posed under restrictive assumptions, it should be very carefuly used in numerical si~nulations since it could be ill-posed as soon as the velocity beco~nes too large.

REFERENCES 1.

2.

F. Armero and J.C. Simo. Long-time dissipativity of time-stepping algorithms for an abstract evolution equation with applications to the incompressible MHDand Navier-Stokes equations. Comp. Methods Appl. Mech. Engrg., 131:41-90, 1996. J.-M. Dominguezde la Rasilla. Etude des dquations de la magndtohydrodynamique stationnaires et de leur approximation par dIdments finis. Thb~se, Paris VI, 1982.

CoupledSystem Arising in Magnetohydrodynamics 3. 4. 5.

6. 7.

8. 9. 10.

11. 12. 13. 14.

367

G. Duvaut and J.-L. Lions. In~quations en thermo61asticit~ et magn~tohydrodynamique. Arch. Rat. Mech. Anal., 46:241--279, 1972. G. Duvaut and J.-L. Lions. Les indquations en mdcanique et en physique. Dunod, 1972. J.-F. Gerbeau and C. Le Bris. Existence of solution for a density-dependent ~nagnetohydrodynamic equation. Advances in Differential Equations, 2(3):427-452, 1997. J.-F. Gerbeau and C. Le Bris. On a coupled system arising in magnetohydrodynamics. Appl. Math. Letters, 1998. M.D. Gunzburger, A.J. Meir, and J.S. Peterson. On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics. Mathematics of Computation, 56(194):523-563, April 1991. P.-L. Lions. Private communication. A.J. Meir and P.G. Schmidt. A velocity-current formulation for stationary MHDflow. Appl. Math. Comp., 65:95-109, 1994. A.J. Meir and P.G. Schmidt. Variational methods Ibr stationary MHDflow under natural interface conditions. Nonl. Anal., Theo. Meth. Appl., 26(4):659689, 1996. B. Saramito. Stabilitd d’un plasma : moddlisation mathdmatique et simulation numdrique. Masson, 1994. M. Sermange and R. Temam. Some mathematical questions related to the MHDequations. Comm. Pure Appl. Math., XXXVI:635-664, 1983. R. Temam.Navier-Stokes Equations, Theory and Numerical Analysis. NorthHolland, 1979. R. Temam. Navier-Stokes Equations and Nonlinear Functional Analysis. CBMS-NSFRegional Conference Series in Applied Mathematics, SIAM, 2d edition, 1995.

A Disease

Transport

Model*

K.P. HADELER Universit~t Tfibingen, Biomathematik, Auf der Morgenstelle D-72076 Tfibingen, Germany ([email protected]) R. ILLNERDepartment of Mathematics and Statistics, University Victoria, British Columbia, Canada V8W3P4 ([email protected])

10,

of Victoria,

P. VANDENDRIESSCHE Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada V8W3P4 ([email protected])

ABSTRACT The Kermack-McKendrick model for the spread of disease in a homogeneous population is combined with a transport equation to yield a new model for the spatial spread of disease. This system provides a more detailed description of the migration and contact processes than the standard reaction diffusion model, which however, is a limiting case. For the epidemic transport model, global existence of solutions is shownby a Kaniel-Shinbrot iteration scheme. It is proved that the population of infectives eventually .disappears and also, using an energy integral approach, that the susceptible population approaches a constant. Key words: epidemic model, transport equation, monotone iteration, tional AMSsubject classifications:

1

energy func-

92D30, 82C70

INTRODUCTION

Transport equations are an established modeling tool in diverse areas such as neutron physics, the theory of dilute gases (and Boltzmann equations), and extended thermodynamics. In mathematical biology they have been used, sometimes under *Research of the first author supported by Deutsche Forschungsgemeinschaft; research of the second and third authors partially supported by the Natural Sciences and Engineering Research Council of Canada. 369

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the name of velocity jump processes, in modeling the motion of bacteria aud of slime mold amoebae, aiming at a better understanding of chemotaxis and pattern formation [21], [13]. Transport equations have two advantages over the traditional reaction diffusion approach. They do not showthe unwantedeffect of infinitely fast propagation and they allow a more detailed description of the transport processes. In Brownianmotion particles are characterized by their position in space, they do not have an individual velocity or direction. As a consequence, diffusion equations show the phenomenonof infinitely fast propagation, as do similar systems with integrals over space modeling long range interaction [12]. To our knowledge, transport equations have not so far been applied in epidemic modelling. The goal of the present paper is to show that transport syste~ns can be formulated as realistic models for the spread of disease. They are generalizations of diffusion models; the latter occur as limiting cases for high particle speeds and large turning rates. An epidemic model based on the diffusion approach depends, with repect to the transport processes, only on the diffusion rates of susceptibles and of infecteds. These rates measure the overall mobility of the two populations. The epidemic transport models presented here depend on two transport kernels for the migration patterns of the susceptible and the infected populations, and a contact kernel, which measures, by comparing velocities, the duration of a contact and thus takes into account that transmission rates depend strongly on the duration of exposition. In this sense transport models are intermediates between diffusion models for overall behavior, and models based on individual behavior. Weunderline that transport systems, being hyperbolic partial differential equations as opposed to parabolic diffusion equations, require different mathematical methods, e.g. the method of characteristics. While parabolic systems have strong s~noothing properties, solutions of hyperbolic systems do not become smooth and therefore a concept of mild solution.is needed. Weshall show that some tools that were originally developed for the Boltzmann equation, work well for epidemic models, due to the product structure of the infection term. This applies, in particular, to monotoneiteration schemes of the Kaniel-Shinbrot type [17]. Weshow global existence of mild solutions to the initial value problem, and show that the class of infecteds goes to zero in the L~ norm, independent of the parameter values and the initial data. Finally we show that the susceptibles approach a spatially constant solution. So far an epidemic threshold theorem has not been proved, nor the final size of the susceptible class determined. Next we describe the modelling approach in some detail. We start from the simplest SI model in the form of ordinary differential equations, for susceptibles u and infecteds w

~ = -~u~ ~b = fluw - aw,

(1.~)

usually attributed to Kermackand McKendrick[19] (actually a special case of their more general model), see also [7]. Here ~ > 0 is the transmission rate and a > 0 is the recovery rate. The basic reproduction number is Ro = ~/a. If this modelis applied to a population structured by so~ne feature such as position in space, age, gender, social group, then the transmission pattern and the motion of individuals in feature space can and must be modeled. First we define, for a given individual an individual susceptibility, and then, for this individual, the infectivity

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to which it is exposed. The infectivity is a weighted integral over the infected population. Second, we describe how individuals move between classes defined by the structure variables. With respect to position in space we can have a variety of migration and diffusion models. Wecan either model the infectivity as an integral over space, or the movementsof individuals in space (or both). Both approaches involve integrals over the space variable. The first approach yields, in the simplest form, a system ut = -~u / k(x - y)w(t, y)dy

(1.2)

where k(x - y) is the infectivity an infected at position y exerts upon an individual at position x. The kernel function k is even and non-negative. In the second approach we keep a simple local transmission law and let individuals moveaccording to an integral equation,

u~= -Zu~o + 9(/g(x- y)u(t,~)du- ~(t, (1.3)

~t = Zu,~- ~w+/)(1/~(~- ~)~(t,~)dy- ~(t, where the kernels are even, nonnegative and normalized to have integral equal to 1. The number D is the rate at which susceptibles change position, and then K(x - y) is the density with repect to y of susceptibles arriving at x, and similarly for b,/~ and infecteds. While the kernel k in (1.2) describes contacts between individuals distant locations, the kernels in (1.3) describe movementsof individuals. Each system (1.2) and (1.3) can be reduced to a system of reaction diffusion equations, using the standard Taylor .expansion approximation and keeping three moments. Following the ideas of [18], (1.2) can be reduced

ut =-flu(koW+k~w.~.~) wt = flu(kow+ k,,w.~:,) where ko = f k(y)dy and k~ = f k(y)y2dy/2. obtain a standard reaction diffusion system

(1.4)

On the other hand, from (1.3)

ut = -~3uw + D~u~.~

(1.5)

Of course, mathematically, the two systems (1.2), (1.~) and also (1.4), (1.5) rather different structures. Modelsof the form (1.2) have been studied in [1], [2], [6]. In [3] a system with diffusion and contact distribution incorporating features of (1.3) and (1.5) was considered, and in [20] a system with a contact kernel and a diffusion kernel that includes both (1.2) and (1.3) as special cases has been studied. In [16] system (1.5) with D~ = 0 has been applied to the spread of rabies. Several authors, e.g., [8] and [23], have studied the joint effects of diffusion and age structure.

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Before studying equations of the form (1.3) with transport operators, we introduce a linear transport equation (without disease) in one space dimension. Lef, u(t, x, be the density of particles at position x with velocity s at time t. The free stream equation gives ut(t, x, s) + su.~(t, s) : O . (1. To incorporate particles turning (i.e., changing velocity), let # > 0 bc the rate constant of turning events, i.e., let turning be governed by a Poisson process with parameter #. Velocities are restricted to lie in the boundedset Y = {s: [s[ _< a}

(:1.7)

for some a ¯ (0, oo). At a turning event the particle chooses a new velocity s from ~) according to someprobability distribution that contains the previous velocity .~ as a parameter. Thus the transport equation becomes, see [4], ut(t,x,s)

+ sux(t,x,s)

where K/# is the probability

= fv K(s,$)u(t,x,$)d~

- #u(t,x,s),

(1.8)

kernel with K(s, ~) nonnegative, bounded and

fv

K(s,$)ds

= p.

(1.9)

If the set Y is replaced by a set of only two velocities, ~,’ = {+7}, with "7 > 0, then the transport equation becomes a correlated random walk in the sense of Goldstein and Kac [15]. In fact, if we write u+(t,x) = u(t,x,"7), u-(t,x) = u(t,x,-’y), then the system reads

ut+ + ~’u~= (#/2)(u- - +) u

+-

-

(1.10)

Then the function u = u+ 2u.~, + u- ~, satisfies the telegrapher’s equation uu + #ut = 7 from which the diffusion approximation can be obtained for large "7 and # [11], 1112]. To study the equation (1.8) on a bounded interval x ¯ I0, £] we need to impose boundary conditions. For a hyperbolic system data can be given along ingoing characteristics; thus at x = 0 data can be given for u(t, O, s) with s > 0, similarly at x = t7 data can be given for u(t,£,s) with s < 0. Wetreat here the case of periodic boundary conditious, which amounts to studying the problem on a circle S1. This is the simplest situation although not very realistic from a biological point of view. However,the proof carries over to the biologically realistic case of reflective (Neumann) boundary conditions, as shown in Section Nowwe formulate our model system by combining the disease transmission and the transport processes. We keep # and K as defined above for the density of susceptibles u(t, x, s) at time t, position x and velocity s, and take #7 and/-(" defined in a similar wayfor the density of infectives w(t, x, s). Thus, for infectives,/~7/#~ is the probability kernel where/5 is the rate constant of turning events, with /’((s, ~) >_ and [((s, ~) = D. (1. 11)

Sv

Disease TransportModel

373

To account for different velocities of infectives, we introduce into the transmission term the contact distribution L(s, ~), which is nonnegative, bounded and satisfies

f

(1.12)

v L(S, ~) ds = 1.

Combiningsystem (1.1) and (1.8) leads to the following system: ut(t,x,s)

+ suz(t,x,s)

= fv K(s,g) u(t,x,g)d~-

#u(t,x,s)

- flu(t, x, s) Iv L(s, ~) w(t, x,

wt(t,x,s)

+ swx(t,x,s)

=/, [f(s,$)w(t,x,$)d$-

ftw(t,x,s)

+flu(t, x, s) f, L(s, ~) w(t, x, ~) d$ - c~w(t,

(1.13) In contrast to equation (1.3), the integral is taken over velocity. Note that a system of the form (1.13) based on the correlated random walk (1.10), i.e., an epidemic model with only two velocities, has been investigated in [9], [10]. As described earlier, the parameters # and/t and the kernels K and/~ describe the dynamic behavior of the migrating populations of susceptibles and of infecteds. Here we can think of a time scale of hours or days. On the other hand, the kernel L measures transmission rates between individuals with different speeds. Wereasonably assume that infected individuals passing through a location at high velocity contribute little to total infectivity as opposed to sedentary individuals. For some proofs we need the assumption that the kernel K is symmetric (this assmnption may be technical). Symmetryin this context says that individuals switch from low velocities to high velocities at the same rate as from high to low velocities. Assume, on the contrary, that individuals choose higher velocities at a faster rate. Then either all individuals would have higher speeds or there would be transitions via intermediate speeds back to lower speeds. In the first case we could perhaps choose another set of velocities from the beginning. In the second case we would need more detailed information about cyclic changes in velocity. This argument amounts to saying that the symmetry assumption is the most natural and simple assumption next to the assumption of a constant kernel producing the uniform distribution. Wehave to specify initial and boundary conditions. Initially the densities, given by u(O,x,s) =_ uo(x,s) and w(O,x,s) -- Wo(X,S), are assumed to be continuous, nonnegative and bounded. For s > 0, the ingoing density at x = 0 is defined by the values of the outgoing density at x = g, and similarly for s < 0 with g and 0 interchanged; thus for all s, u(t, e, s) = u(t, O, s), w(t, e, s) = w(t,

(1.14)

This amounts to the earlier mentioned assumption of periodic boundary conditions. Note that we work on I~ x S~ x V, where V is given by (1.7), and the space variable is taken rood g.

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Welook for mild solutions to the initial boundary value problem formulated above for u and w. Wedefine mild solutions of (1.13.) as a pair of functions u(t, x, s) and w(t, x, s) that satisfy the system ~[u(t,x

+ts, s)] = fv K(s,~)u(t,x

+ts,$)d~

- uu(t,x +ts, s) - S7u(t,z +ts, s) f, L(s,

¯ +ts,

~dt [w(t, x + ts, s)] = fv R(s, $)w(t, x +ts, $) - fi w(t, x + ts, s) + Z u(t, x + ts, s) fv L(s, ~) w(t, ts, $) d~ - a w(t, x + ts, s). This type of solution concept is well-known in the theory of transport equations, see [5] or [17]; one of its main advantages is that it allows solutions of (1.13) for initial data that are not smooth. Indeed, with respect to the space variable, the solution is as smoothas the initial data, i.e., continuous in the present case (or just measurable, if initial data are measurable). Weremark that this concept of mild solution is stronger than the concept of weaksolution in the sense of distributions.

2

MONOTONE

APPROXIMATIONS

Weuse a Kaniel-Shinbrot [17] iteration scheme (see also [5, pp.136-137]) to show that the IVP formulated in the Introduction has a unique Inild solution for all t > 0. Firstly, for u, w, let lower sequences be denoted by {u(J)}, {w(J)}, (j) }, respectively, for j = 0, 1,.... Note upper sequences be denoted by {U(j) }, {W that the independent variables are dropped where no confusion arises, e.g., U(°) = U(°)(t,x, s). All equations are written in the classical form, but solved in the mild sense defined above. The iterative schemeis defined so that each equation is linear in the current variable. For j = 0, take u(°) = w(°) = 0,

with initial condition U(°)(0, m, s) = Uo(~, and periodic boundarycondition as in (1.14), U(°)(t, ~, s) = U(°)(t, With U(°) now given as a solution of the above, W(°) solves (°) +s W~ °) + (f~ + ~)W (°) = .f, /~ W(°) d~ + f~ U(°) iv LW(°) d~, W~ with initial condition W(°)(0, x, s) -- wo( x, s), and periodic boundary condition as in (1.14), W(°)(t, g,s) = W(°)(t,O,

(2.1)

Disease TransportModel For n = 0, 1,..., u(,~+l)

375

take the following iteration scheme

.(,~+~,

÷ (#+~/vLW(n,d~)

) /vKu(n)d~,

"+’) (D+~)w ~"+~) t + sw~ +

=

/v ~ w(~) d~ + ~ u(~) /~ L w(n)

(2.2)

w:

+

+ (b + w

withinitial conditions u(~)(0,x,s)= (j)(0, ,x s ) u( = O, x, s)~ U o( and w(J)(O,x,s)=W(J)(O,x,s)=w(O,x,s)~Wo(X,S) for j=1,2,..., andperiodic boundary conditions ~ in (l.14) foreachtermin ~hesequences. To prove results on this iterative scheme, we make use of the following theorem, which can be proved by use of an integrating factor. THEOREM 2.1

Consider the linear first order equations

dyi(t_____~)+ hi(t)yi(t) = gi(t), i dt where hi(t), gi(t) are nonnegativeand continuous on [0, c~), subject to th e i niti al conditions yi(O) = Yo > O. Then (i) I] h,(t) = h2(t), gl(t ) >_ g2(t). (res p. g,(t ) <_ g2(t)) for [O, oc),then y~(t) >_y2(t) (resp. y~(t) <_y~(t)) for [O, oc). (ii) If g,(t) = g2(t) and h~(t) h2(t) (resp. h~( t) <_ h2(t)) for all t ~ [0,~ ), then y,(t) <_y2(t) (resp. y,(t) >_y~(t)) for [O, oc). Monotonicity of the sequences defined by (2.1) and (2.2) is proved by induction in the following lemmas. LEMMA 2.2 Each starting term in the upper sequences defined above in (2.1) is nonnegative, namely U(°) > 0 and W(°) > O. Proof. Considerthe equation for = u(O, x, s), written as U0, sequence, denoted by {V(1)}, V(°) = O,

U(°) (t, x, s) in (2.1) with initial values (°) (0, x, s nonnegative and bounded. Construct an approximating to U(°) from the scheme

Vt(n+t) + s V.~n+~) +/z V(n+l) =/v K V(r0 d$, with v(J)(O,x,8)

-~- 0

for j = 1, 2, .. ..

(2.3)

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For n = 0, d dt

(v¢’)(t,

x + +, x + st,

Thus V(1)(t, x + st, 8) : Uoe-*tt (°) >_V giving where

Co: IIUoll~. To proceed by induction, suppose that, for a fixed n,

IIV(’~)(t,’)ll~ < co \p=O

-"~, ~j e

where Then from (2.3) d (V(n+l)(t,

x + st,

8)) + tt

V(n+l)(t,x

dt (Clt)P

<_ C~Co

\~,=o

P!

Solving gives V(n+l)(t,x + st, 8) < I~-~ (Clt)P~_

p=O-t~. e-lit,

showing that

IIV(’~+~)(t, ")ll~< CoeC’te for all n, and t _> 0. Also Theorem2.10) showsthat for

v(~+’)(t,.) ~v~")(t,.). Thus {V(j)} is a nonnegative, nondecreasing sequence that is bounded above, and so on any arbitrary but bounded time interval [0, T], lim V(n)(t, .) = V(t, exists, is nonnegative and finite. Integrating (2.3) and letting n -~ oo, it foll[ows that V(t, .) is a mild solution of the first equation of (2.1). To showuniqueness this solution, suppose that V1(t, .) and V2(t, .) are both mild solutions of (2.1), and consider their difference

~’(t,.) =~q(t,.) - v.~(t,

Disease TransportModel

377

By linearity, this satisfies an estimate

e_

x +st, 8))+.e(t, x +st, s)-

dt

with !~(0, ") = 0. Integrating gives t

f0

Ilk(t, ’)11~_ 0. (°) in (2.1). ConWith the above knowledge of U(°), consider the equation for W struct an approximating sequence, denoted by {Y(J)}, to lcV{°) from the scheme y(O) = 0,

Yt(n+~) + sy~(n+~) + (# + a) Y(~+l) = /v[~Y(n)dg + flU(°) , with Y(J)(O,x,s)=w(O,x,s)

for j=1,2,

....

(°) is well-defined for Proceeding as in the proof for U(°), it can be proved that W (°) t E [0, oo) and that >_ 0 . | LEMMA 2.3 For the iteration scheme (2.1) and (2.2) all the iterates defined/or t e [0, oc) by (2.1) and (2.2), and

are well-

0 = u(°) <_ u(~) <_ u(2) <_ ... <_ U(2) <_ U(1) (°), =U 0 --~

W(0)

< W(1)

< W(2)

’<

"’"

< W(2)

= W(1)

(0 = ). W

Proof. Consider first the equation for u(1) given by (2.2). An elementary argument shows that u(1) _> 0 = u(°). The same argument on the equation for w(~) in (2.2), shows that w(1) _> 0 = w(°). Comparing the equations for U(°) and U(t) from (2.1) and (2.2), gives (~) =U(°); si milarly, W (1) = W(°). Since U( °) _>u(° and W(°) >_ w(°), Theorem 2.1(i), (ii) shows that (1)
u (n),

limits

w= lim

w(n),

U= lim

U(n),

W= lim

nW

/or the sequences defined in (2.1), (2.2) exist, and u = U and w = W are global mild nonnegative unique solutions of (1.13) assuming uo(x, s) and wo(x, s) as inital ’values and satisfying the periodic boundaryconditions (1.14).

378

Hadeler et al.

Proof. By monotone convergence of (2.2) (theorem of Beppo Levi, applied to finite time interval [0, T]), u, w, U and Wexist and are at least measurable. They satisfy in the mild sense

(2.4)

where u(O,x,s)=U(O,x,s)

and w(O,x,s)=~V(O,x,s).

Define g = U - u and h = W- w, then their initial (in the mild sense)

values are zero and they satisfy

gt + s g* + #g = /v K g dg - ~g ]~ Lw d$ + ~U ffv Lh dg, ht + s h, + (£ + a)h

--£ if

£

£

h da + ~U Lhda + l~g Lwd~.

(.,.6)

Since g and h are nonnegative, we can obtain an L1 estimate. Weintegrate (2.5), (2.6) over s 6 P and a period in x. Then the terms containing K and /~ cancel with the # and/2 terms because of the assumptions on the kernels, and the second to the last term of (2.5) is nonpositive. The three remaining terms containing/~ can be estimated as in

and similarly for the other terms. Then we obtain d

(llgll ÷Ilhll) a(llgll +Ilhlll)

with c = 2a~3[ln[[~ ([[’u[[~ + [IU[[~ + [[w[[~). Since [Ig(0, ")[[1 = [[h(0, ")[l~ = 0, the above implies that g = h = 0, and so U W= w. Thus the equations (2.4) reduce to the pair of equations (1.13), and u w = Ware global mild solutions of the original problem (1.13). By construction, u and w are nonnegative. A standard argument based on Gronwall’s inequality shows uniqueness. Since the initial data are continuous, u and w are continuous with respect to the space variable and differentiable along characteristics. |

Disease TransportModel 3

ASYMPTOTIC

379

BEHAVIOUR

THEOREM 3.1 For system (1.13),

Ilu(t,’)lll

has a limit and

IIw(t,.)lll o a8 Proof. Integrating equations (1.13) with respect to s (thus cancelling the it and terms), then with respect to x, and adding, gives

d-~

(u + w)(t,s,x)dsdx

w(t,x,s)dsdx,

(3.1)

which is ~ 0. Define A =

(u +w)(t, x, s) ds

which exists and is nonnegative by monotonicity, (3.1) and u ~ 0, w ~ 0. Integrating (3.1) from t to ~ gives (u+w)(t,x,s)dsdx

a+~ ~ fvW(r,x,s)dsdsdr=

(3.2)

for all t > 0. Thus JO J~

for~ su~dently large. Usin~theequRtion forw from(1.13), integrating withrespect to~, ¯ and~ gives w(t, x, s) ds dx

=~

w(O,x, s) ds

~ Lw(r, z, ~) d~ ds d~ dr -

The first integral on the left is bounded(because

w(r, ~, s) ds dx dr.

(a.1) shows tha~

is bounded)and the second integral on the left is given by the initial condition. ~he second integral on the right is boundedas a function of t, see (a.2), hence d~ds ~vU d~ ; dr Lw(r,x,g) is bounded(as a function of t), and as it is monotoneincre~ing in t, it h~ a limit as t ~ ~. Nowintegrate the u equation si~nilarly, giving ~(t,~,s)

dsdz

u(O,~,s)

dsdz-

~ Lw(r,z,g)

dgdsd~dr

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Hadeler et al.

thus the left side has a limit A~L>_ 0. Likewise

j~oe fv w(t, x, s)ds has a limit, namely A,, = A - A~ _> 0. But, because of (3.3) A,, must be z.ero. Thus liT(t, ’)111 -+ 0 as t -~ co, and Ilu(t, ")[11 has a limit as t -+ ~. With additional assumptions on the probability kernel K(s,g), we are able to prove that the density of susceptibles, u(t, x, s), tends to a constant in L~ as t -~ o~. The proof uses an energy functional (c]. the constant of motion for the o.d.e, system (1.1)), as suggested by Hartmut Schwetlick [22]. Firstly, we observe that u satisfies a maximumprinciple if the kernel K(s, ~) is symmetric. THEOREM 3.2 Assume that K(s, ~) = K(~, s) for all s, g ~ Y. Then the solution u of (1.13) constructed in Section 2 satisfies

Ilu(t,’)11~ _
(3.4)

Proo]. Suppose that at time t, So and ~o are chosen such that u(t, Xo + tso, So) = sup u(t, x, s). Then by (1.15) and the nonnegativity of u and

ddt [u(t, xo + tso, so)] <_ Iv I((so,~)u(t,

+tso, so)d~ - # u(t, xo + tsoso)

= (f -~ 0,

where the symmetry and normalization of K have been used. For any e > 0, set u¢(t,x,s) = u(t,x,s) - Thenclear ly d u~(t" xo + tso, So) dt ’ It follows as in [14, Lemma 1] that for all e > 0 and all x, s

As e is arbitrary,

the assertion follows. |

THEOREM 3.3 Assume that the kernel K(s, ~) is symmetric and strictly positive on "~ × Y, and that sup~ ILL(., ~)[[~ is finite. Then [or system (1.13), as t u(t, x, s) converges to a constant and w(t, x, s) converges to 0 ~.

Disease TransportModel

381

Proof. The last assertion is already p~oved in Theorem3.1. Define the positive number fi* by/3" = ~sup~ ILL(., g)lll and the (convex) energy functional E(u, w) = C 4: w + u - ln (u + 5)

(3.5)

where 5 > 1 and the constant C = C(a, ~*, 5) is chosen so that E(u, w) >_ for u,w _> 0. Setting b(u) = ln(u ÷ 5), th e fi rst eq uation of (1. 13) giv (bt(t,z,s)

+ sb.~(t,x,s))

ds = (u(t,z,s)

+5)

(3.6)

x (-;K(s,~)u(t,x,~)d$+#u(t,x,s)+~u(t,x,s)fvL(S,$)w(t,x,$)d$)ds. Using the assumption that the kernel is symmetric,

#= Iv K(s,~)ds

= Iv K(s,~)d~.

Thus the first two terms of the right side of (3.6) can be written

/v/v

K(s,,~)

(u( t, x, s ) - u( t,x, -(u-~,x~-~) d$

(3.7)

Because of the symmetryassumption, the integral in (3.7) is equal to the integral obtained by interchanging s aud $, namely ~/vK(s,~)

(u(t’x,~) - u(t,x,s)) (u(t, x, ~) + 5) d$

(3.8)

Taking the average value, each of (3.7). and (3.8) is equal 1 u(t’x’s) u- (t’x’~)u-~,x-~-~-~ -~/v/vK(s’~) _ u(t,x,s) - u(t,x,~) "~~((~,_x_(_~_~ ] ~ which simplifies to 1

(~(t,~,~)- ~(t,~, (u(t,x, s) +5)(u(t,x, ~)

d$ ds.

(3.9)

The last ter~n on the right of (3.6)

(u(t, x, s) + 5) L(s, ~) w(t, x, ~)

<-Z£ (fvL(s,~)ds) w(t,x,~)d~ <_ ~* [ w(t,x,$)dg,

(3.10)

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Hadeleret al.

by the definition of 3*. Returning to the energy functional given in (3.5), C + w + u - ~;- ln(u

d-~ [[E[[~ = ~ =,

w~+ut+~jb~

ds dx

dsdx.

Usiug (1.13), (3.9) and (3.10), this

d~llEll~N

-s(w~ va 1?vf 4~*

+ u~) - aw - ~sbz ds (u(t’x’s)-u(t’x’$))~ g(s,~) (u(t,x,s) + ~) (u(t,z,$)

+ ~ =

+

w(~,~,~)~

(a~ll)

-s E~ ds dx - ~

4B*

x, s) + 5) (u(t, x, ~) + 5) f?fuK(s, ~) (u(t,(u(t’x’s)-u(t’x’$))~

The first integral on the right vanishes for periodic boundaryconditions, thus d

d~]SEth~ 0. Integrating (3.11) gives E(T)

+

Ilwll~ + ~

K(s,.~)(u(t,x,s)+~)(u(t,x,~)+~)d,~dsdx

~ g(o) ~ 2aeC+ H~o~+ ~o~, whereE(t) = [[g(u(t, .), .))][ Nownote that as E(T) ~ for al l T, thelast estimate impli es ( u( t, x, s ) - u( t, x, ~)

K(s,~)(u(t, x, s) + 5) (u(t, z, ~)+ ~)ds d~~.

f Tvfv

As u is bounded by I[Uoll~ (Theorem3.2), and K(s, ~) > 0 for all s, g, it follows that in the sense of measure u becomes asymptotically (~s t ~ ~) independent s. Specifically, consider time intervals IN, N + 1), then for all e > lira 14 {(t,x,s,g)

N~

~ [N,N 1)x [ 0, g) x V xF;[u(t ,x, s) - u(t,x ,g )l > e} =

Disease TransportModel

383

i.e., u(t, x, s) ~ u(t, x) in this four-dirn6nsional Lebesguemeasure sense. Using boundedness of u again, we see from (1.15) that u will then vary very slowly along most characteristics, and it follows that u(t, x) must asymptotically be independent of t and x as well (with the exception of sets of arbitrary small measure in IN, N 1) x [0, ~)). As boundedness and convergence in measure imply convergence in 1, the assertion follows. | Note that the density of infectives tending asymptotically to zero and the density of susceptibles tending aymptotically to a constant agrees with the result in [8]. For their diffusion modelthey are able to showthat if the initial susceptible density is positive (on a set of positive Lebesgue measure), then the density of susceptibles tends to a positive constant as t tends to ~ ; see [8, Theorem2.5].

4

REFLECTIVE

BOUNDARY

CONDITIONS

Our monotone approximation scheme, the resulting existence and uniqueness theorem for periodic boundary conditions (Theorem 2.4), and the asymptotic behaviour of Theorem 3.1 readily generalize to the case of reflective (Neumann) boundary conditions under an additional symmetry assumption on the kernels. The Neumannboundary conditions read u(t,x,s)

= u(t,x,-s)

Weconsider initial

and w(t,x,s)

= w(t,x,-s),

x = 0,

(4.1)

data satisfying the boundary conditions,

Uo(X,S) = Uo(X,-S)

and Wo(X,S) = Wo(X,-S),

x = 0, e. (4.2)

As is done for the Neumannproblem of the heat equation, we apply a reflection. Extend the initial data to [-~, ~] by uo(x, s) = Uo(-X, -s) and Wo(X, s) = Wo(-X, -s),

-g <_ x < O.

In view of (4.2) the extension is continuous at x = 0. Then we extend the data 2g-periodic functions on the real line. THEOREM 4.1 Consider system (1.13) under the assuznption that the kernel K satisfies K(s, ~) = K(-s, -~) and similar assumptions for [( and L. With 2~periodic boundary conditions as described above, the solution satisfies the symmetry conditions u(t,x,s)

= u(t,-x,-s)

and w(t,x,s)

=w(t,-x,-s).

Proof. If u(t, x, s), w(t, x, s) is a mild solution of this initial boundaryvalue problem, then u(t,-x,-s), w(t,-x,-s) is also a solution. This is checked by considering (1.15) and by using the assumptions on the kernel. As by construction, the initial values satisfy uo(x,s)

= Uo(-X,-S)

and wo(x,s)

= Wo(-x,-s),

384

Hadeleret al.

it follows that u(t,x, s) and u(t,-x,-s) are both solutions for the same initial values. So by uniqueness, the result is proved. | COROLLARY 4.2 Under the assumptions of Theorem 4.1, mann) boundary conditions (4.1) are satisfied.

the reflecting

(Neu-

Proof. Let x = 0 and x = t~ in Theorem 4.1. I If, in addition to the assumption on K in Theorem 4.1, K(s,g) = K(g,s) as imposed in Theorem 3.3, the convergence result of that theorem holds also for Neumannboundary conditions, since the convex functional E is nonnegative, see [5, Th. 8.5.1]. Based on similar results for the Boltzmannequation (see [5]), we expect that existence and uniqueness results based on our monotoneapproximation scheme can also be proved for moregeneral reflection laws (e.g., diffusive reflection) at the boundaries, and also for higher spatial dimensions. However, the elegant reduction to the periodic boundary conditions, as shownabove, is then not possible. Instead, use must be made of a solution procedure that tracks characteristics until they hit the boundary, and then continues solutions by using the boundary conditions. In conclusion, we have formulated a new spatially dependent SIR disease transmission model that allows for infection distance via motion. Our model is related to the diffusive spread models, but leads to finite speeds of propagation because of the bounds on the velocity domain. Wehave addressed certain analytical questions about our model, but others remain. For example, fbr periodic and zero Neumann boundary conditions it is plausible that Ro = ~/a plays the same role as in the ordinary differential equations case. But so far we have not proved a threshold theoreIn for the space-dependent problem. ACKNOWLEDGEMENTS We thank Hartmut Schwetlick for suggesting the energy functional used in Theorem3.3, and Dick Phan for constructive discussions. REFERENCES 1.

2. 3. 4. 5.

6. 7.

D.G. ARONSON, The asymptotic speed of propagation of a simple epidemic. In: W.E. Fitzgibbon, H.F. Walker (eds), Nonlinear Diffusion, p. 1-23, Pitman Research Notes in Mathematics 14, 1977. C. ATKINSON ANDG.E.H. REUTER,Deterministic epidemic waves, Math. Proc. Cambridge Philos. Soc. 80, (~976), 315-330. V. CAPASSO, Global solution for a diffusive nonlinear deterministic epidemic model, SIAMJ. Appl. Math. 35, (1978), 274-294. K.M. CASEANDP.F. ZWEWE~, Linear Transport Theory, Addison-Wesley, Massachusetts 1967. C. CERCIGNANI, R. ILLNER, AND M. PULVIRENTI, The Mathematical Theory of Dilute Gases, Springer Verlag 1994. O. DIEKMANN, Thresholds and travelling waves for the geographical spread of infection. J. Math. Biol. 6, (1978), 109-130. O. DIEKMANN, J.A.P. H~nsTE~tnE~z, ANDJ.A.J. ME,Z, The legacy of Kermack and McKendrick. In (D. Mollison, ed.) Epidemic Models: Their Structure and Relation to Data, p. 95-115, Cambridge University Press 1995.

Disease TransportModel 8.

9.

10.

11.

12.

13. 14. 15. 16. 17. 18.

19. 20.

21. 22. 23.

385

W. FITZGIBBON, M. PARROT,G. WEBB, Diffusive epidemic models with spatial and age dependent heteorogeneity, Discrete and Continuous Dynamical Systems 1, (1995), 35-57. K.P. HADELER, Travelling epidemic waves and correlated random walks. In: M. Martelli et al. (eds) Differential Equations and Applications to Biology and Industry. Proc. Conf. Claremont 1994, p. 145-156, World Scientific 1995. K.P. HADELER, Spatial epidemic spread by correlated random walk, the case of slow infectives. In: R.A. Jarvis et al. (eds) Ordinary and Partial Differential Equations, p. 18-32, Pitman Research Notes in Mathematics 370, 1997. K.P. HADELER, Reaction telegraph equations and random walk systems. In: S. van Strien, S. Verduyn Lunel (eds), Stochastic and spatial structures dynamical systems. Roy. Acad. of the Netherlands, p. 133-161, North Holland, Amsterdam 1996. K.P. HADELER, Reaction transport systems in biological modelling. In: V. Capasso, O. Diekmann (eds) Mathematics inspired by Biology. CIMELectures 1997, Lecture Notes in Mathematics, Springer Verlag 1999. T. HILLENAND A. STEVENS, Hyperbolic models for chemotaxis in l-D, Nonlinear Analysis, to appear. R. ILLNER,Global existence for two-velocity models of the Boltzmann equation, Math. Meth. Appl. Sci. 1, (1979), 187-193. M. KAC,A stochastic model related to the telegrapher’s equation, (1956), reprinted in Rocky Mtn. Math. J. 4 (1974) 497-509. A. KXLLI~N, Thresholds and travelling waves in an epidemic model for rabies, Nonlinear Analysis TMA8, (1984), 851-856. S. KANIELANDM. SHINBROT, The Boltzmann equation. L Uniqueness and local existence, Commun.Math. Phys. 58, (1978), 65-84. D.G. KENDALL, Mathematical models of the spread of infection, Mathematics and Computer Science in Biology and Medicine, p. 213-225, Medical Research Council HMSO,London, 1965. W.O. KERMACK ANDA.G. MCKENDRICK, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc., Al15, (1927), 700-721. P. DE MOTTONI,E. ORLANDI,ANDA. TESEI, Asymptotic behavior for a system describing epidemics with migration and spatial spread of infection, Nonl. Anal. TMA3, (1979), 663-675. H.G. OTHMER, S.R. DUNBAR, W. ALT, Models of dispersal in biological systems, J. Math. Biol. 26, (1988), 263-298. H. SCHWETLICK, Reaktionstransportgleichungen, Dissertation University of Tiibingen 1998. G.F. WEBB,An age-dependent epidemic model with spatial diffusion, Arch. Rat. Mech. Anal. 75, (1980), 91-102.

Blow Up and Hovering in Parabolic with Singular Interactions : Can We "See" a Hyperfunction ?

Systems

GONTERLUMERInstitut de Math~natique et Intbrmatique, UniversitY. de Mons-Hainaut, Mons, Belgium / Solvay Institutes for Physics and Chemistry, Brussels, Belgium.

INTRODUCTION Briefly said and anticipating a little on terminology clarified later (1), the purpose of this paper is to investigate the relation(s) between "signals" and "responses" parabolic systems with interactions represented by distributions or hyperfunctions. First for the momentlet us recall (from [3]) that in the setting of a general Banach space X a system as mentioned above, and in the title, is considered to be governed by an equation of the form (more details in section 1)

(P)

u’ = ~u+ F(t) .(0_) = (si .)(0) B.(t) = ~(t) t

where a is a distribution or hyperfunction with support 0 and values in H C X (see section 1). The crucial ingredients in (P) -- setting F = f = T = 0(~) -- are which we call "signal" (see section 1 and [5]) and the corresponding solution u (P), defined for t > 0, called "response" (to a). Henceforth we thus assume in paper F=f=~=0. For such problems (P), when a is a "true" hyperfunction (meaning "it does reduce to a distribution") there can exist a # 0 with response identically 0; such ~Using the terminology of [3] and [2] or what is recalled of it here below in sections (} and 1, and otherwise explained in these sections. 2(P) is linear and hence by superposition F(t), f, ~o(t), play a well-known standard role.

387

388

Lumer

a are called nondetectable signals (in short nds), see [5].(3). Hence for a given response u the set of a giving rise to that u, which we shall denote E(u) and call "signal set" of u, maycontain in general infinitely manyelements. Of course is of the form {go + n : n any ads, or 0}. As said at the beginning we study the relations between u and E(u) in particular what can we say about E(u) knowing u (for t > 0) It follows already from a result in [3], that if E(u) contains a distribution the latter is uniquely determined and it is not difficult to see that it can indeed be computed from u, so that it is clear what is ~neant by the statements : "u co~nes from (uniquely determined) distribution" or "u does not come from a distribution". In this paper we mainly obtain the following : (a) In section 2 we show that if u comes from a distribution and is not identically 0 there must be blow up, i.e. Ilu(t)ll ~ oc as t --~ 0+. As a consequence, since we also show that there exist responses u ¢ 0 which stay bounded ("hovering responses"), we establish the existence of phenomenapermitting in principle to verify "visually" (film) an event that can only be caused by a true hyperfunction (detecting the latter without going through any analytic conditions difficult to verify). There is indeed a large class of such phenomena. (b) An essential general question is answered in section 3 : can one always recognize from knowing u, whether it comes from a distribution or not. Wegive such a characterization (howevernot of the "directly verifiable" kind of the results in section 2). In section 4, spectral expansions of u(t) are used to exhibit situations with blow much"steeper" than what distributions can produce suchas Ilu(t)ll = ON/’/z (while it is easily seen that distributions lead at worst to O(1/t~), integer _>0). up

1

ASSUMPTIONS. TERMINOLOGY

RECALLING

KNOWN FACTS,

NOTATIONS,

First we clarify assumptions, recall knownfacts, about (P). ~ is the "free operator" associated to A generator of a bounded irregular analytic semigroupQ(t) on X, [31, [4], [11, via: D(ft)=D(A)®H

.4 :H~0 ,

(1)

Ac J,.

H c X is the space of "harmonic elements" used as "boundary~ values" (the intuitive image is : for X = C(~), associate to C(O~) via harmonic (A-harmonic’ extension H C C(fl) = X)i the connection with the boundary operator B is B~ = B : X onto H, ker B = D(A). (here D(A) ~ X except trivial Also :

(2)

situations.) A-1 is compact.

3Unlike [5], here and henceforth, unless we specify "classical nds".

we shall

not insist

(3) on an nds occuring

"in classical

A context"

Blow-Upand Hoveringin Parabolic Systems

389

For the (asymptotic) definition and construction of solutions u of (P) we refer [3], [2]. Werecall that we can alwayswrite [3],. [5]:

j----l

j----1

and that (P) (as here with F = f = ~ = 0) has always a unique solution given

u(t)=-~Q(J)(t)c~

for

t>0.

(5)

The assumptions above are sometimes a little stronger than in [3], for a more efficient development, since they remain fully realistic, in line with what is really satisfied in the classical contexts. Werefer to [3], [2], [5], for all that is not indicated explicitely above or elsewhere in this paper.

2

BLOW UP AND HOVERING

THEOREM 1 If u # 0 comes from a distribution

then [[u(t)[]

-~ cx) as t -+

PROOF If u comes from a distribution then it is given by (5) with all cj = 0 for j > some N integer >_ 1. Weshall show that under these circumstances if Ilu(t)ll does not tend to oc as t ~ 0 all cj = 0 thus reaching a contradiction (with u ~ 0). Werecall (see [3], [1]) that since Q(t) is boundedirregular analytic the following holds: Vt > 0 Q(t) : X -+ D(A); Q(t) -~ 1 (strongly)

on D(A) as t -~ 0,

(6)

and that in view of (2) we have D(A) N H = 0.

(7)

Weproceed nowin several steps : (i) First suppose there exists a sequence (in of positive numbers t~ tending to 0 such that A-Nu(tk) -+ some limit g. Then N

-A-~Vu(tk)

N

= E A-~VQ(J)(tk)cj j=l

= E A-~VAJQ(t~)cJ j:l

N

N-1

= e(t~)~A-(X-J)ci

= Q(t~) E A-(N-~)cJ

j=l

+ Q(tk)c~v.

(8)

j=l

In view of (6) the first of the just preceding 2 summandshas by itself a limit, hence Q(tk)cN -~ some limit gN. And again by (6) we have eN ~ D(A). But then A-~t?N = limA-~Q(tk)cg (since A-~CNe D(A)), so

= limQ(ta)A-~cN

(9) = A-leg again using (6)

eN = CN ~ HN D(A) = 0 (by (7)).

390

Lnmer

If (in case N >_ 2) we then also knowthat A-(N-~)U(tk) has a limit we can conclude that CN-1 = 0, and so on, up to c~. (ii) Supposethat for tk as in (i) Ilu(tk)ll remains bounded. Then A-s being compact (by (3)) for j = 1, 2,..., N, we can assume that A-Ju(t~) has a limit, so that from (i) abovefollows cj = 0 (all j _> 1, i.e. all j). (iii) If Ilu(t)l[ does not tend to co as t ~ 0, then there exists a sequence k) t k as mentionedin (ii) above and we get cj = 0 for all j. This concludes the proof. ¯ COROLLARY 2 If a response u is non zero and remains bouuded for t > 0 it can not come fi’om a distribution. Notice that it can be shown (using amongother things (3)) that always u(t) -~ as t -~ 0c, and "bounded for t > 0" is equivalent to "bounded near 0". Now, are there (many) responses u as mentioned in Corollary 2 ? The answer yes. Indeed, if we examine the construction of large classes of nds a in [5] (these being classical nds in RN for all N), we see that it is based on obtaining entire oo functions f(z) = ~ 7jz j ~ -~(~) of finite order p, 0 < p < 1, whose set of zeros j--1

contains the eignevalues of A (or rather of its selfadjoint extension A in that. A context - see [5] and (10) below), combined with finding appropriate corresponding g ~ H, such that setting w = ~ 7i(f(J),

the a in question are w ®g. Going fl~rther

into details recalled fi’om [5] (details not only needed in the present discussion but also in the next section), we recall that in the context of [5] spectral computations can be made adding to the assumptions in section 1, the following : X ~ 7-/a Hilber space, .4 C A in 7/selfadjoint <_ 0, A-~ compact. The eigengalues of .~ are denoted 0 > At _> :~ _> ... >_ ~, _> ..., the corresponding eigenvectors being e~, e2,..., e,,.... See [5] section 1, (4), (5) of [5]. Withthis, can be computedin 7/, see [5] section 3, (8) of [5], for a = w ® g as mentioned few lines above u(t)

= E f(~)e~t/J"e"’

(:11)

where f(z) is as mentioned before ~ 7jz j, g = ~ /~ne~.. j=l

n=l

Nowconsider one such classical nds a = w ® g as constructed in [5] and considered above, its corresponding f and, say, A~. Then f is of the form f(A) 0(3

with f*(£l) # O, and .f* also of order p, hence (see [5]) xw’iting .f*(A) and w* = k 7~5(j), j=l

we obtain a* = w* ® g where the latter

~; ,~ J j=l

is an admissible signal

Blow-Upand Hoveringin Parabolic Systems

391

(see in [5] the last 2 lines on p.735) for (P) bu~ not longer ands. Indeed applying now (11) with f* instead of f, we get for tile new response

(12) just one example belonging to the obviously very large class of bounded non zero responses (hovering responses) of which Corollary 2 speaks. Hence we can state THEOREM 3 There exist infinitely many responses u to which Corollary 2 applies, already in the classical A, R~v, contexts, for all N, hence permitting "visually" to verify that they can only come from true hyperfunctions.

CAN WE ALWAYS "SEE" FROM u WHETHER IT A DISTRIBUTION OR NOT ?

3

COMES FROM

For orientation see (b) of Section THEOREM 4 A continuous u :]0, cc[~ X is a solution distribution iff : ~N integer >_ 1, c* E X, such that : (i)

A-Nu is bounded and exponentially for all Re z > 0;(s).

(ii)

[A(1 - B)]Jc* is defined for j = 1,..., belongs to H.

of (P) (4) coming frown

decaying, and (A-Nu)~(z) = -R(z, A)c* N - 1 and for the latter

value, N - 1,

PROOF That tile condition is necessary follows starting along lines similar to the proof of theorem 1 above; N being as defined there, we have as in (8) -A-Nu(t) = Q(t)c*; c* = A-(N-1)cl + ...

A-lcN_I + ClV,

(13)

the exponential decay follows from (3), and we take Laplace transform (see [3]). Finally, using the equality on the second line of (13), applying repeatedly the operators B and A (recalling that all cj ~ H, while by (2) B D(A) -~ O, H -~ H,)and computinga little, we find the formulas of (15) below, fi’om which the necessity (ii) results. Succinctly, this goes as follows

A(c* - Bc*) = A-!N-2)C1 +... B[A(1 - B)]c* CN-~, et c.

+ c/v-l,

For the converse, (i) tells of course at once by uniqueness of Laplace transform that

--A-Nu(t) = Q(t)c* for t > 4werecall that here (P) is with F = f = ~ = 5"~"denotes "Laplacetransformof", R(., A) denotesthe resolvent.

(14)

392

Lumer

and now by (ii)

we can define cj E H, j = N - k,k = 0,...,N*EH fork=0,...,N-2, = B[A(1-B)]kc = [A(1- B)]N-~c * ~ g

cg-k cL Somelittle

1, via :

(15)

algebraic manipulations will then show that :

A-(N-I)Cl

h- A-(N-2)c2

+ "" CN =

A-(N-1)[A(1 _ B)]N-tc * + A-(N-2)B[A(1 _ B)]N-2c * +...

(16)

A-~B[A(1 - B)]c* Bc* = c* To obtain (16), start by replacing the first summandvia

A-(N-~)[A(1 A-(N-~)[A(1

B)]N-~c* = A-IN-~)(1 - B)[A(1 _ B)]N-:c* - A-(N-~)B[A(1 - *,

B)]N-~c * =

(17)

which added to the second summandof (16) leaves A(N-e)[A(1 - B)]N-’~c *, and then do again as in (17) now with N - 2 instead of N - 1,..., (16) reducing finally to A-~[A(1 - B)]c* + Bc* = c*. Hence (14) tells that with the c~ ~ H defined in (15),

-A-N -N : A

N

N j:l

proving (see (4), (5), above) that u is indeed a solution of (P) coming N

(H-valued) distribution

4

a

A SIMPLE EXAMPLE OF SPECTRAL COMPUTATION OF BLOW UP AT RATE HIGHER THAN POLYNOMIAL IN

For orientation see the final paragraph of the introduction. The exampleis spectral estimation based on (11) above ((10) being satisfied). Wetake for the classical A, R~, context, f~ =]0, lf, X = C(~), as described in detail 4 of [5]. Here ~, = -Tr~n2; en = en(x) = x/~sinTrnx,, x ~ ~, g =

typical of simplicity in section g(x) =

Ifl,,I = v/’~/7rn. Wetake (with the notations of section 2, around (10) and (11)) w defined as follows :

393

Blow-Upand Hoveringin Parabolic Systems -’: ~r ’~2t n ~’~e

(19)

Moreovere -~2n~t >_ 1/2 iff n~u2 S ~, and the largest n (~ 1) satisfying the latter condition is of the form

n=-

1

V~

-0,0-<0<1, Y.

fort_<

rr--

in 2

(20)

Using (20) in (19), and using (11), we conclude that in sup-norm and 7~-norm L~(f~)-norm) we have for appropriate M, and 0 < t _< some to > 0,

Ilu(t)ll_>Ilu(t)llc,¢m

(21)

This illustrates a typical sort of blow up of a response u which alternatively to Corollary 2 exhibits a phenomenum that also cannot come from a distribution, this time because the blow up is too steep 6. (It is of course well knownthat for Q(t) boundedanalytic -- irregular or not -- [1], [4], [7], ItQ(~l(t)[I = O(1/tj) so for a distribution we have the (1/t)-polynomial growth mentioned in the introduction.)

REFERENCES 1. 2. 3.

4.

5.

6.

7.

G Da Prato and E Sinestrari, Differential operators with non dense domain, Ann. Scuola Normle Pisa, 14, 285-344, (1987). G Lumer, Transitions singuli~res gouvern~es par des ~quations de type parabolique, C.R. Acad. Sci. Paris, 322, sfirie I, 735-740, (1996). G Lumer, Singular interaction problems of parabolic type with distribution and hyperfunction data, in "Evolution equations...", Mathematical Topics vol.16, (Advances in PDEs,) 11-36, Wiley-VCH,Berlin, (1999). G Lumer, Semi-groupes irrfiguliers et semi-groupes int~grfis: application l’identification de semi-groupes irrfiguliers analytiques et r5sultats de gfin~ration, C.R. Acad. Sci. Paris, 314, stifle I, 1033-1038,(1992). G5GLumer and F. Neubrander, Signaux non-d~tectables en dimension N dans des syst~mes gouvern~s par des ~quati0ns de type parabolique, C.R. Acad. Sci. Paris, 325, s~rie I, 731-736, (1997). S Ouchi, On abstract Cauchy problems.in the sense of hyperfunction, in Hyperfunctions and pseudo-differential equations, Lect. Notes in Math. 287, SpringerVerlag, 135-152, (1973). A Pazy, Semigroupsof linear operators and applications to partial differential equations, Springer-Verlag, (1983).

SThelatter however doesnot lend itself to easyaccurate"visual"verification.

Some Asymptotic

Problems in Fluid

NADERMASMOUDI CEREMADE(UMR 7534), Universite Place de Lattre de Tassigny, F-75775 Paris cedex 16

1

Mechanics de Paris-Dauphine,

INTRODUCTION

In this note, we summarize some results concerning some asymptotic problems coming from fluid mechanics. These asymptotic problems arise when a dimensionless parameter ~ goes to zero in an equation describing the motion of some type of fluid (or any other physical system). Physically, this allows a better knowledgeof the prevailing phenomenonwhen this parameter is small. This small parameter usually describes a physical reality. For instance, a slightly compressible fluid is characterized by a low Machnumber, whereas a slightly viscous fluid is characterized by a high Reynolds number (which means a low viscosity). In manycases, we have different small parameters (we can be in presence of a slightly compressible and slightly viscous fluid in the same time). Depending on the way these small parameters go to zero, we can recover different systems at the limit. For instance, if ~, ~i, ~,, ~ << 1, the limit system can depend on the magnitude of the ratio of ~/(f .... These asymptotic problems allow us to get simpler models at the limit, due to the fact that we usually have fewer variables or (and) fewer unknowns. This simplifies the numerical simulations, in fact, instead of solving the initial system, we can solve the limit system and then add a corrector. Whenwe try to pass to the limit, we: encounter many mathematical problems due to the change of the type of the equations, the presence of manyspatial and temporal scales, the presence of boundary layers (we can no longer impose the same boundary conditions for the initial system and the limit one), the presence of oscillations in time at high frequency .... In the array of examples, we are going to consider we will try to answer different type of questions: 1) Whatdo the solutions of the initial system (S~) converge to ? Is the convergence strong or weak ? 2) In the case of weakconvergence, can we give a more detailed description of the sequences of solutions ? (can we describe the time oscillations for instance ?) 395

396 2

Masmoudi THE NAVIERoSTOKES

EULER LIMIT

The zero-viscosity limit for the incompressible Navier-Stokes equation in a, bounded domain, with Dirichlet boundary conditions is a challenging open problem due to the formation of a boundary layer satisfying the Prandtl equations, which seem to be ill-posed. The whole space case was performed by several authors, we can refer for instance to Swarm[30] and Kato [17]. In an other work, Bardos treats the ,case of a bounded domain with a boundary condition on the vorticity, which does not engender any boundarylayer [3]. In the sequel, we will study this limit in the case we consider different vertical and horizontal viscosities. Weconsider the following system of equations (NS,,,) Otu’~ + V(un ® un) - ~,O~zu n - ~A~,yU~ = -Up in gt V.u n =0 in

(1)

~t

(2)

ua = 0, in 0gt

(3)

with V.u~~ = 0 (4) 2, 2, where 9 = w x [0, hi, or gt = w × [0, co[, and w = V or ]R ~, = ~,~, r/= y,~. When ~, ~, go to 0, we expect that u’~ converges to the solution of the Euler system

~-(0)~o ~’~

Otw + V(w ®w) = -Vp V.w = 0 in w.n=±wa=O on O~,

(5)

w(t =O)=°.

It turns out that we are able to justify this formal derivation under an additional condition on the ratio of the vertical and horizontal viscosities. THEOREM 2.1

Let s > 5/2, and w°~Hs(l~)~,

V.w °=0,

w°.n=O

on 0~.

We assume that U"(O) converges in L2(~), ° and ~,~?,~/~? go t o O, t henany sequence of global weak solutions ( la Leray) ~ of ( 1- ~ ) s atisfying t he e ne~yy inequality satisfies u’~ - w -~ O in L~°(O,T*,L2(~2)), n -~’ 0 in L2(0, T*, v/-~Vx.~un, V~OzU where w is the unique solution of (5) in ~) L~°(0,T*; HS(~) Wegive here a sketch of the proof and refer to [26] for a complete proof. The existence of global weak solutions for (NS~,,), satisfying the energy inequality is due to J. Leray n 2

n 2

(6)

AsymptoticProblemsin Fluid Mechanics

397

This estimate does not show that un is bounded in L2(0,T; 1) and h ence i f we extract a subsequence still denoted un converging weakly to u in L’~(O,T;L2), we can not deduce that un ® un converges weakly to w ® w. If we try to use energy estimates to show that un - w remains small we see that the integrations by parts introduce terms that we cannot control, since u~ - w does not vanish at the boundary. Hence, we must construct a boundary layer which allows us to recover the Dirichlet boundary conditions : /3 will be a corrector of small L2 norm, and localized near 0f~

{/3~(z= O)+ ~(z= O) div(13 n)=O,

Bn-~0

~(z = c~) = in

2) L~(O,T*;L

a possible choice is to take B of the form

~ = -~,(z = o)e-~Z~, .. . where ~ is a free parameter to be chosen later. Wewant to explain now the idea of the proof. In stead of using energy estimates on un - w, we will work with v n = un - (w + B~). Next we write the following equation satisfied by w~ = w + ~ O~wu + wU.Vw~ _ ~O~w~ - ~A~,~wu = 0~ + ~.Vw ~ + w.VB - ,O~w~ - ~A~,~w ~ - Wp

(7)

which yields the following energy equality

[[w~(0)l[~ w~.[Ot~ + w.vB - ~O~] ~w~ - nA~,~w Next, using the weakformulation of (1), we get for all

Then adding up (6),(8)

and subtracting (9),

Finally, using that f(u.Uq)q 0, we get

(8)

398

Masmoudi

Now, we want to use a Gronwall lemma to deduce that Ilv(t)ll~ remains s~nall. By studying two terms amongthose occurring in the left hand side of the energy estimate (10), we want to show why we need the condition v/rl -~ 0. In fact

2

where, we have used the divergence-free condition O~va = -O~v~ - O~vz. Wesee from this term that we need the following condition to absorb the first term by the viscosity in (10) cCllwll~~ ~, On the other hand, the second term can be treated as Nllows

Iv f O~Bv /~

2 _<~llO~,~ll~ + ,,llO~ll~

< ~lla~llz~+~11,~11~,~ The second term of the left hand side must go to zero, this is the case if we have u/¢ ~ O. Finally, we see that If v- ~ 0 then is a possible choice.

3

COMPRESSIBLE-INCOMPRESSIBLE OF A VISCOUS FLUID

LIMIT

The general set up for such asymptotic problems, is a straightforward adaptation of the one introduced by S. Klainerman and A. Majda [18], [19] in the inviscid case (Euler equations). Werecall the compressible Navier-Stokes equations in ~he so-called isentropic regime (even though Our results are trivially adapted to the case of general barotropic flows i.e. whenthe pressure is a function of the density only). The unknowns(/5, v) are respectively the density and the velocity of the fluid (gaz). From a Physics view-point, a compressible fluid should behave (asymptotically) like an incompressible one when the density is almost consta!~t, the velocity is small and we look at large time scales. More precisely, we scale p and v (and thus p) in the following way

# =o(x,et), v =eu(x,

(]1)

and we assume that the viscosity coefficients p, ( are also small and scale like/5 e~, ~ = e( where e e (0, 1) is a "small parameter" which is called the Math number.

399

AsymptoticProblemsin Fluid Mechanics

Wealso assume that we have # > 0 and # + ~ :~ 0. With the preceding scalings, the compressible system reads

{

-~+div(pu):0,

p>_0, (12)

Opu ~ + div(pu

®u) - #Au- ~Vdivu

a + ~Vp ~ 0.

The domain ~ can be the whole space ~N, the torus TN or an open domain of ~N (with different possible boundary conditions). In fact the proof, we are going to sketch is a local one and only uses the local energy estimates. Wealso recall the incompressible Navier-Stokes system 0~ + div(u

@ u) - ~Au + Vr = 0 div(u)

= 0

Wethen consider a solution (p~, u~) of the compressible Navier-Stokes equations (12) and we assume that L:(O,T;H ~) for all T L~/(~+~)~ i.e. is continuous with respect to t > 0 with values in L:~/(~+~)endowed with the weak topology. Werequire (12) to hold in the sense of distributions. Finally, we prescribe initial conditions :p~ , p~u~

:m~°

(14)

~ ~, ~L o e L~/(~+~), m~o : 0 &.e. on{p~: 0} andp~lu~[ where p~ ~ O, p~ ~ L~, m~ o = 0 on {p~ 0}. m~on {p~ > 0}, u~ denoting by ou~=p~ = Furthermore, we assume that

2 ° ~ u~o converges weakly in L to some u and that we have p~l~l P~ =

+~ (P~)’-70~(~)’-’+(7-1)(~)"

(2.)-" f p~0

(15)

1,

where, here and below, C denotes various positive constants independent of e. Let us notice that (15) implies in particular that, roughly speaking, : indeed, we just need to rewrite (P~)~- 7P~ (~)~-~ and recall that (t ~ ~) i s c onvex on [ 0, ~ ) s ince 7~ 1. Our last requirement on (p~, u~) concerns the total energy : we assume that we have E~(t)+

D~(s)ds~E~

~hereE~(t) =

n~l~l~(~)

(divu~)~(~)andE~= i 1 o

a.e.t,

+D~0

in

~’(0,~)

.f +

U(nT)

(16)

400

Masmoudi

Wenow wish to emphasize the fact that we assume the existence of a solution with the above properties, and we shall also assume that 3’ > N ~-. Andwe recall, the

’N

results in [22] which yield the existence of such a solution precisely when 3" > -2 9 3 and N _>4, 3’_> ~ and N=3, 3’_> ~ and N=2.

THEOREM 3.1 (with P-L.Lions) In addition to the above notations and condiN tions, we assume that 3" > -~. Then, p~ converges to 1 in C([0, T]; L~) and u~ is boundedin L2(O, T; H1) for all T e (0, oc). In addition, for any subsequence of (still denoted by u~) weakly converging in L2(O, T; l) (VT e(0, cx 3)) to some u, is a solution of the incompressible Navier-Stokes equation (13) (as defined above) corresponding to the initial condition u° °. = Pu For the proof of this theorem in the periodic case, we refer to [23]. The argurnent presented there uses the group method introduced by Shochet [29] (see also [].4]). However, we want to point out that the group method does not apply to all type of boundary conditions. Wegive here a sketch of proof based on a local method, which can apply to more general cases (see [24]). Locally, we can decomposep~u~ into a gradient part and a divergence-free one (p¢u¢ = Qp¢u¢+Pp¢u¢). This decomposition can be made by taking a ball B and imposing in some sens that Ppu.n = O. We can show(see [23]) that to pass to the limit and recover the incompressible NavierStokes system, we only have to show that div(Qp~u, ® Qp¢u¢) converges weakly to a gradient Vp. The proof of [24] is based on the following lemma and q~ are such that ~ e L°~(O,T;L~o~(f~)), LEMMA3.2 L~(O, T; H~o¢(a)) and satisfy Oqv~ _ OVq~ 1 Aq~ + F~, Ot Ot ~ e 2 where ~F~ ~ 0 in L and ¢G.~ -~ 0 in L then

I Vqoe + ~e

Vq~,

(17)

div(Vq~ ® Vq~) -+ Vp weakly Finally, we point out that in the case of a bounded domain with Dirichlet boundary condition we can show strong convergence due to the presence of a boundary layer [10] and also mention a result in the Whole space case [9] where the local strong convergence is shown using the dispersion in the wave equation.

4

STUDY

OF ROTATING

FLUID

AT HIGH

FREQUENCY

Weconsider the following system of equations e3 × u" _ VPin

V.u"

= 0 in fl

f~

(is) (19)

AsymptoticProblemsin Fluid Mechanics ~tn(0)

=

401

~ with V.it~’

= 0

u n = 0 on 0~t

(20) (21)

where ~ = ~1"2 x]0, hi, ~ and r/are respectively the vertical and horizontal viscosities, whereas e is the Rossby number. This system describes the motion of a rotating fluid as the Ekmanand Rossby numbers go to zero (see Pedlovsky [28], and Greenspan [13]). It can model the ocean, the atmosphere, or a rotating fluid in a container. Whenthere is no boundary (~t = ~3 for instance) and when t, = ~/= 1, the problem was studied by several authors ([14], [5], [1], [11]...) by using the group method [29]. The method introduced in [29] fails when f has a boundary (excepted in very particular cases where there is no boundary layer, or where boundary layers can be eliminated by symmetry [4]). In domains with boundaries, only results in the "well-prepared" case (which means that there are no oscillations in time) were known( [7], [15], [26] ). Here, we have a domain with boundary and want to study the oscillations in time and also show that they do not affect the averaged flow. To study this system, we introduce a space V~ym consisting in function in Hs with some extra conditions on the boundary (see [27]). Wealso set Lu = -P(e3 x u), where P is the projection onto divergence-free vector fields such that the third component vanishes on the boundary and E0-) = e rL. Let us denote w the solution in L~(0, T*, VsSy,~) of the following system Otw + Q(w, w) - A~,yW 7S(w) = -Vp in V.w = 0 in fl, w.n=±wa=O on Off,

(22)

w(t = 0) = °.

where Q(w, w), S(w) are respectively a bilinear and a linear operators of w, given by

=

b(t,

(23)

~(t)+~(m)=~(~)

where the N~ are the eigenfunctions of L and iA(k) are the associated eigenvalues, c~tm~ are constants and A(l,m) : {l + m, Sl + m, l + Sm, S1 + Sm}, (Sl (l~,12,-In)) is the set of possible resonances. The bilinear term ~ is due to fact that only resonant modesin the advective term w.Vware present in the limit equation. -~(w) = ~ ~(D(k) ~-iI(k))b(t,k)yk(x) k

where

In fact ~(w) is a dampingterm that depends on the frequencies A(k), since D(k) >_ It is due to the presence of a boundary layer which creates a second flow of order e responsible of this damping (called damping of Ekman).

402

Masmoudi

THEOREM 4.1 Let s > 5/2, and w° E V~ym(12)3, ~7.w ° = 0. We assume that u~ converges in L2(f~) to w°, ~l = 1 and ~, v go to 0 such that V~e -~ 7. Then any sequence o] global weak solutions ( la Leray) n of ( 18- 2 1) s atisfying t he energy inequality satisfies un - £(~)w ~ O in L~(O,T*,L2(f~)), V.,y(U n - ~(~)w), V~Ozu’~ ~ 0 in L~(O, T*, where w is the solution in L~(O,T*, V~v,n) of (22) Wealso showthat the oscillations do not affect the averaged flow (also called the quasi-geostrophic flow). Wesee then that ~ (the weak limit of w) satisfies a Navier-Stokes equation with a damping term, namely 0~ + ~.V~ - A.~,y~ + V/~’-~-~ "2, V.~=0in ql

{

where S is the projection ~(t,x,y) = $(w) = (I/h)f:

= -Vp in T (24)

°) °, = o) = S(w =w onto the slow modes, namely that do not depend on z, w(t,x,y,z)dz.

In [27], we also deal with other boundary conditions, and construct Ekmanl~zers near a non flat bottom f~6 = {(x,y,z),

where (x,y)

~ ~, a nd 5f(z,y)

with the following boundary conditions u(x, y, (if(x, y)) Wealso treat the case of a free surface,

=

y)

where a describes the wind (see [28]).

REFERENCES 1. -A. Babin, A. Mahalov, B. Nicolaenko : Global splitting, integrability and regularity of 3D Euler and Navier-Stokes equations for uniformly rotating fluids.Euwpean J. Mech. B Fluids 15 (1996), no. 3, 291-300. 2. A. Babin, A. Mahalov, B. Nicolaenko : Regularity and integrability of 3D Euler and Navier-Stokes equations for rotating fluids. Asymptot. Anal. 15 (1997), no. 2, 103-150. 3. C. Bardos : Existence et unicit de l’quation l’Euler en dimension deux. Journal de Math. Pures et Appliqudes 40(1972), 769-790.

AsymptoticProblemsin Fluid Mechanics 4.

5. 6. 7. 8.

9. 10.

11.

12. 13. 14. 15.

16. 17. 18.

19. 20.

21. 22.

403

T. Beale, A. Bourgeois : Validity of the qfiasigeostrophic modelfor large scale flow in the atmosphere and ocean, SIAMJ. Math. Anal., 25 (1994), 1023 1068. J.-Y. Chemin : A propos d’un probl~me de p~nalisation de type antisym~trique, C.R.Acad.Sci.Paris Sdr. I Math 32_~1 (1995), 861 - 864. J.-Y. Chemin : Apropos d’un problhme de pfinalisation de type antisymfitrique, J. Math. Pures Appl. (9) 76 (1997), no. 9, 739-755. T. Colin, P. Fabrie : Rotating fluid at high Rossby numberdriven by a surface stress : existence and convergence, preprint, 1996. Colin, P. Fabrie : quations de Navier-Stokes 3-D avec force de Coriolis et viscosit verticale vanescente. C. R. Acad. Sci. Paris St. I Math. 324 (1997), no. 3, 275-280. B. Desjardins, E. Grenier, LowMach number limit of compressible viscous flows in the whole space, in preprint. B. Desjardin, E. Grenier, P.-L. Lions, N. Masmoudi: Compressible incompressible limit with Dirichlet boundary condition, to appear in Journal de Math. Pures et Appliqudes. P. Embid, A. Majda : Averaging over fast gravity waves for geophysical flows with arbitrary potential vorticity, Comm.Partial Differential Equations, 21, (1996), 619 - 658. V.W. Ekman: On the influence of the earth’s rotation on ocean currents. Arkiv. Matem., Astr. Fysik, Stockholm 2 (11) 1905. H.P.Greenspan : The theory of rotating fluids, Cambridge monographs on mechanics and applied mathematics ,1969 E. Grenier, Oscillatory perturbations of the Navier Stokes equations. Journal de Maths Pures et Appl. 9 76 (1997), no. 6, p. 477 - 498. E. Grenier, N.Masmoudi: Ekmanlayers of rotating fluids, the case of well prepared initial data, Comm.Partial Differential Equations , 22(5-6),(1997) 953-975 T.Kato : Remarkson zero viscosity limit for nonstationary Navier-Stokes flows with boundary T.Kato : Non-stationary flows of viscous and ideal fluids in R3, J.Functional Analysis 9 (1972), 296-305. S. Klainerman, A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm.Pure Appl. Math. 34 (1981)., no. 5, p. 481 - 524. S. Klainerman, A. Majda, Compressible and incompressible fluids. Comm. Pure Appl. Math. 35 (1982), no. 5, p. 629 - 651. J.-L. Lions, R. Temam, S. Wang:’ ModUles et analyse math~matiques du syst~me Ocean/Atmosphere C.R.Acad.Sci.Paris Sdr. I Math 316 1993 113 - 119, C.R.Acad.Sci.Paris Sdr. I Math 316 1993 211 - 215 C.R.Acad.Sci.Paris Sdr. I Math 318 1994 1165 - 1171. P.L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 1 Incompressible Models, Oxford University Press 1998. P.L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 2 Compressible Models, Oxford University Press 1998.

404

Masmoudi

P.L. Lions, N. Masmoudi,Incompressible limit for a viscous compres~’~iblefluid. J. Math. Pures Appl. 77 (1998), p. 585-627. 24. P.L. Lions, N. Masmoudi, work in preparation 25. A. Majda, T. Esteban :A two-dimensional model for quasigeostrophic ttow: comparison with the two-dimensional Euler flow. Nonlinear phenomena in ocean dynamics (Los Alamos, NM, 1995). Phys. D 98 (1996), no. 2-4, 515522. 26. N. Masmoudi, The Euler limit of the Navier Stokes equations, and rotating fluids with boundary, Arch. Rational Mech. Anal 142 (1998) p. 375 - 394. 27. N. Masmoudi,Ekmanlayers of rotating fluids, the case of general initial data, subbmitted. 28. J. Pedlovsky : Geophysical fluid dynamics, Springer, 1979. 29. S. Schochet : Fast singular limits of hyperbolic PDEs.J. Diff. Equ. 114 (1!)94 476 - 512. 30. H.Swann, The convergence with vanishing viscosity of non-stationary NavierStokes flow to ideal flow in R3 Trans. Amer. Math. Soc 157 (1971), 373-397. 31. G.I. Taylor : Experiments on the motion of solid bodies in rotating fluids, Proc. Roy. Soc. A 104 (1923), 213 - 218 23.

Limits to Causality and Delocalization in Classical Field Theory T. PETROSKY and I. PRIGOGINEInstituts Internationaux de Physique et de Chimie, fond6s par E. Solvay, Bd du Triomphe, CampusPlaine ULB, CP 231, 1050 Brussels, Belgium / Center for Statistical Mechanics and Thermodynamics, The University of Texas at Austin, Robert Lee Moore Hall, Austin, Texas 78712, USA

1

INTRODUCTION

Weare very happy to dedicate this paper to our friend Professor Giinter Lumer whose fundamental contributions in physics we greatly admire. Since the work of Hegerfeldt [1] we knowthat positivity of the Hamiltonian leads to delocalization and superluminal effects in quantumtheory both relativistic and non-relativistic. Whatabout classical field theory? Certainly we cannot expect any effect for the free fields. Therefore we have to consider interacting fields or fields coupled to particles. Weshall consider here the case of a scalar field interacting with a harmonicoscillator. Weuse as a field modela classical analogue of the Friedrichs ~nodel (but including virtual processes). As we have shown recently this model [2] is exactly soluble by an appropriate Bogoliubovtransformation. Weeliminate, as in the Friedrichs original solution, the particle and verify the positivity of the Hamiltonian. Wecenter our presentation on the following problem: a wave packet is approaching a harmonic oscillator at rest at the origin of coordinates (x = 0). Weshowthat the oscillator is excited before the wavepacket touches it. Wehave therefore, as in quantummechanics, a violation of Einstein’s causality. It is interesting to compare our results with traditional classical field theory based on retarded potentials in which Einstein’s causality is accepted a priori [4]. Our model includes no violation of relativity as the particle is described by a classical non-relativistic equation. Webelieve however that going to a complete relativistic model such as the model discussed in [5] manyof our conclusions may rema;a valid. Our paper, a preliminary report containing the details of the calculations as well as other examples, is in preparation. The main conclusion is that localization leads 405

406

Petroskyand Prigogine

to difficulties for Einstein’s causality even in classical field theory. Therefore one of the possibilities proposedby Hegerfeldt, in whichit is suggested that the effect is due to zero point fluctuations, is excluded. Wecan also study the interaction between dressed particles and the field. Wehave done it for quantummechanical systems [3]. Deviations from causality appear as well. Of course nobody has yet observed such deviations from causality. Therefore there are two possibilities: Einstein’s causality is only an approximation or there is something wrong in the usual approach to field theory. These are so fundamental problems that we cannot guess in which direction future will decide.

2

HAMILTONIAN

AND BOGOLIUBOV

TRANSFORMATIONS

Weconsider the one-dimensional extended Friedrichs model including "virtual processes;’ H : t.ola~a 1 -~ 2 ("dkblbk

+ "~ 2 Vk(a~ + al)(bl

nc b~:)

(1)

where A is the dimensionless coupling constant and 6Ol > 0, wk > 0. The quantities al,a~ refer to the particle and b[., bt¢ to the field ¢(x)

1[

= Z bl :e+ik’~ k

+ b~e-ik:c

]

(2)

The star means complex conjugation. Weput the system in a large oue-dimensional box of size L with a usual periodic condition. Then the spectrum of the field is discrete, i.e., k = (2rc/L)j with integers j. Later we shall take the continuous limit L ~ c~. Weassume that Vk " O(L-~/~) to obtain a consistent interaction in the continuous limit. Wemay give a classical interpretation to (1) and (2). Instead of operators deal then with functions,

(3) (4) and the quantities at, a~ and bk, b*~ satisfy the Poisson bracket. In the bracket the derivatives are taken with respect to qa,p~, q~ and p~,. The Bogoliubov transfor~nation for the Hamiltonian (1) as given by [2]

a~ = _ Z A V~, [(Wk + wa )G- (wk )B~ - (Wk - w~ )G+(wk ]

(5)

k

b k = B~ - 2w~AV~ AVt G-(wt) wt - wk - i¢ t

Bt wt + Wk

(6)

Limitsto CausalityandDelocalizationin Field Theory

407

and its inverse B~ = bk + 2WlAl/~G-(w~)

E AVk wl - w~, + ie l

where ¢ is a positive infinitesimal, i.e.,

(~)

e ~ 0+. Wedefine

~(~)= ~(~¯ i~)

(s)

with

(9)

Weimpose the condition

w~- ~

/10)

~? > 0 l This condition ensures that the particle becomes unstable due to the interaction. Weobtain in this way a diagonal positive Hamiltonian H= Ew~B;Bk

(11)

Asthe Bogoliubov transformationsconservethe algebra, the B~and B~satisfy also the Poissonbracketrelations. Hencethe solutionof the equationof motionis given by Ba.(t) = e-i~tBa(O)

(12)

This is the extension of the well-knownFriedrichs solution eliminating the particle (in traditional theory, one tries to eliminate the field, thus leading to the Darwin Hamiltonian [4]). Note that the field (11) "guides" the particle according to (5). Let us first find the Zeroth’s of G-~ (z). To order ~ we h ave

~ = ~ ~ - ~ (~ + ~)(~=~, ¯ ~e) + °(~4)

03)

with ~1-~o~

wtV~ 2 p 1 =-2A 2Ewt+w~ w~-w~ l

where P stands for the principal part. Therefore for weak coupling 1 G~ (w~.)

(~. + ~)(w~- ~ ¯

Let us consider the evolution of the particle (the harmonicoscillator).

(14)

(16)

408 3

Petroskyand Prigogine EVOLUTION

OF THE PARTICLE

We~ssumethat at t = O, there is a non-vanishing field while the particle is at, rest

(ql = pl =0) hi.(0)

¢ 0, bk(0) ~ 0 and a~(0) = a~(0)

(17)

Using (6), (7), (12), we obtain a~(t) = - E AVa I (wk +

(18)

~l)~--(~k)

-(w~, - ~,)~+(~) ~l

I

.

This formula is greatly simplified when we consider weak coupling (A << 1) and short time in respect to the lifetime 1~ < ~ < 1 with

1 ~ A_

(19)

Then with a simple transformation using also (16) we obtain

a~(t)~d~’ E ....

-XU~[~ (~o)

+ eiW~t ~ ~k + ~1 }~ nowconsider a rectangular wave packet approaching the oscillator That leads to

(see Fig. 1)

where O(x) is the step function, Co is a normalization and we assume xo <0,

(22)

ko >0

wave packet

xo - d

xo

xo + d

Figure 1 Wave packet approaching the particle

oscillator ~ 0

x

Limitsto CausalityandDelocalizationin Field Theory

409

Through a Fourier transformation we o~)tain for bk(O) = b~(ko, 0) + bk(-ko, 0) that "x ~ o+d I

axe

l

(23) eL

-i(k + ko) where ~ = L/2~r. Wehave therefore

a~(t) = ei~’t[Ii(t) + I2(t)]

(24)

whenI~(t) and I2(t) correspond respectively to the two terms in (20). Putting as usual

v~ =Vk ~

(2~)

and introducing a dispersion relation appropriate to a "photon" field we get (c = 1)

(26)

~ = I~1 Weobtain (with xo~ = ~o + d) for I~(t) = I~(ko,t) + Ii(-ko,t) I~ (ko, t) =

~

aCo

dkik t

F

that

__ 031 __ ie

~ x/~v~

~

(27)

and

Z2(k0, t) -

(28)

Wecan nowestimate the short time influence of the field on the particle.

4

SHORT

TIME

BEHAVIOUR

OF THE PARTICLE

Contrary to what happens for the contribution of the pole (see [3]) there is indication that the light cone of relativity plays any role in the integrals (27), (28). This is confirmed by the evaluation of the integrals in the acausal region. Decomposingthe integration over k into the two parts k > 0 and k < 0, we have I, (ko, t) = Ii+(ko, t) + IT(ko,

(29)

410

Petroskyand Prigogine

with

(30) and I~(ko,

t)-

iACo ~.~ dk V~v~. -~-~, ’ k-kok-w~-i¢ (31)

Wenote that k = k0 and k = w~ are regular points. Weconsider the acausal region

xg+ t
(32)

The expressions (30), (31) are evaluated by changing the contour to integrals the imaginary axis from -ioc to i0.* Then it becomeseasy to derive an asymptotic formula from far from the boundary of the light cone with IXo + d+ t I >> 1

(33)

kolzo± d + tl >>1

(34)

we obtain

’35) ’2 ~- v/~ ko ( t + xg + d) 1

.l ] ] [e~0(~o+d) _ e_~o(.~0+~)

(t 1

~ (t + xo - d) where we have introduced the form factor f(k) = V~/Vr~ and assumed that f(k) regular at k = 0. Therefore before the edge of the wavepacket reaches the particle at the origin, there appears already a non-vanishing effect. This effect comes from the integration over k (0 to ~) and originates fl’om the positivity of the Hamiltonian.

CONCLUDING

REMARKS

To conclude we want to mention that the problem of nonlocality is related to the problem of Hegerfeldt [1] on the relation between causality and positivity of Hamilton. However the Hegerfeldt approach is limited to quantum theory, while our approach shows a similar problem which exists even in classical dynamics. The relation with Hegerfeldt’s problem will be treated in a separate paper. *Similar calculation

can be performed for 12(t).

Limitsto CausalityandDelocalizationin Field Theory

411

ACKNOWLEDGMENTS We would like to thank Dr. I. Antoniou, E. Karpov, B. Misra and G. Ordonez for their interesting discussions. Weacknowledge the European Commission, the "Loterie Nationale de Belgique", the "Communaut~ fran~aise de Belgique", the U.S. Department of Energy Grant No. DE-FG03-94ER14465 and the Robert A. Welch Foundation Grant No. F-0365 for support in this work.

REFERENCES 1. 2.

Hegerfeldt, Phys. Rev. Lett. 72, 596 (1994). Karpov E., Petrosky T., Prigogine I. and Pronko G., submitted to J. Math. Phys. A. 3. - T. Petrosky, G. Ordonez and I. Prigogine, to be submitted to Phys. Rev. 4. Landau and Lifshitz "Classical Theory of the Field" (1962) 5. Antoniou I., Gadella M., Prigogine I. and Pronko G., J. Math. Phys. 39, 2995 (1998)

Remarks to the Blow-up Rate of a Degenerate Parabolic Equation BURKHARDJ. SCHMITT and MICHAEL WIEGNER Department matics I, RWTHAachen, Aachen, Germany

1

of Mathe-

INTRODUCTION

This is a continuation of the study of the second author (compare[6], [7], [8]) of the borderline case of a degenerate parabolic equation with a powertype source, namely

u~ = uP(Au+ u), u(x,0) = ~(x), ulo~= 0 Here p > 1, Y~ C Rn is a smoothly bounded domain and ~ is smooth enough with 0 < Co <_ ~(x)dist(x,Of~) -1 <_ c~ < oo A proof that the solutions blow up in finite time, if ~l(f~) < 1, with ~l(fl) denoting the first eigenvalue of the Dirichlet-Laplacian was given by Friedman, McLeod[2] for p = 2. Subsequently further progress concerning the blow-up rate, the blow-up set and the blow-upprofile was made, see Wiegner[6], [7], [8] and Tsutsumi, Ishiwata, resp. Tsutsumi, Anada and Fukuda [3], [4], [5], [1]. The most complete and instructive results were obtained in the one dimensional symmetric case. From Wiegner [8] we infer the following result. PROPOSITION 1 Let

" a> y,~:

I-a, a] ~ R a C3-function with

(i) ~(x) > 0,~(x) = ~(-x) x~’( x) < 0 on(- a,a ), ~(=l= a) = O. (ii)

~o’(x) ÷ ~(x) >_ 0 x. ( ~p’"(x) + ~’(x)) <_ 0 for x e (- a, a) , ~o’(+a (an exampleis V(x) = cos (2m~x)).

Then there is a finite time T > 0, such that a unique (smooth) solution u ut = u~(uzx + u) on (-a,a)

x (0,

exists with u(:l=a, t) = O, u(x, 0) = ~(x). It has the properties 413

414

Schmitt and Wiegner u(x) > O, ut(x,t) >_ O, xu..(x,t) on (- a,a) × (O,

¯ u(x,t)

= u(-x,t),u(O,t)

= maxu(x,t),ux(O,t)

. pu~(O,t)(T-t)_>lfort
decreasing

for t ~ T with a

For the last item part (ii) of the assumption was used. But there is a difference depending on thesize of p. Weshowed moreover that: I. If p ~ 2, then . ~(p u(O,t)~(T-

t))-’

0 for . If S = {x~u(x,t)

~ _

_

~ fort ~ T}is the blow-up set, then ~ = [ -~ , ~].

On the other hand: II.

Ill


. t!~(p u(0, t)P(T - t))-’

= ~ ~ (0, 1

L(x)= ~ w6(x) for ~z~ ~a5 where w5 is the unique solution of 1-p w~’ + w6 = Sw 6 ,w~(0)=0,

w6(0)=l

and a6 its first zero. a~ is increasing with 5 fl’om a0 = ~ to a~-p/2 -- ~

.

=

The size of 5 (and hence L) depends now on the initial

value

REMARK The proof that 5 > 0 in case II was not given in [8]; it can be found e.g. in [1], and for completeness we give the argument in Corollary 4 below.

2 THE RESULTS Lookingcloser to these two types of behaviour we see that the blowiup rate in case I is unknown.Especially there are no estimates for u(0, t) from above. Nevertheless already Friedman-McLeodconjectured, that for p = 2 u(0, t) v ~ (T- t) -1 In[ln(T- t)[ for t-~

415

Blow-upRateof a DegenerateParabolic Equation

and based on numerical 2-D studies, Ishiwata and Tsutsumi conjectured this behaviour also for p >_ 2 and arbitrary domains. Similarly the behaviour of u(~, t) for t -~ T is of interest - we know, that u(O,t) --4 0, but there might hold u(.~,t) -~ ~x~ for t ~ T with a different blow-up rate. intend to give someresults into this direction. THEOREM 2

Let p = 2. Then, for ¢ > 0, there is some C~ > 0, such that (2(T-

t))-½

-½-~, < C~(T-t)

< u(O,t)

Crucial for the proof is the following lemma: LEMMA 3

Letx0 1. Forp#2,1et 1 /u(x,t) 2-p

L(t)

2-v dx,

0

for p = 2 let =

L(t)

/In u(x,t)dx. 0

Then L"(t)

=

2 j uV(x, t) dx + u(xo, t)u‘Tt(Xo, t) - ut(xo, t)u‘T(Xo, 0 ~o

<

f 0

PROOF We get ¯ o

,To

f ul-Putdx = f u(uz‘T + u)dx, 0

0

hence L"(t) Integrating the last term twice by parts gives ‘To

0

.To

0

with \

~t

416

Schmitt and Wiegner

By [8], proof ofTheorem 1, we see that f(x,t) <_ u.,t(0, t) = 0, we have f(0, t) = 0, hence the claim. COROLLARY4 Letl
As u.(O,t) =_- O, implying

Then

~i~n~(pu(O,t)P(T - t))-’ = PROOF The existence of the limit is known([8]).

By [6] f(a,t)

= 0, as

lu~(zo,t)l _
xo = a in Lemma3 gives L"(t)=2/~dx. o

For p < 2 we get by HSlder’s inequality L’(t) 2 ~ ~n"(t)n(t), which implies that L’(t) La(t ) _> Co > 0 for a = 1/(1 - p/2) > 1. Thus T

co(T-t)

<_ / -~s k, ~-’d ] --a 1__ 1 Ll-a (t),

as L(t) -+ oo for t-+ T. Together with a- 1 = ~ and L(t) ~ cu(O,t) 2-~, one gets u(O, t)e(T - t) ~ theclai m. We need another preparing lemma. LEMMA 5

Let p = 2. Then T

u(O,t) u -~,s ds ff -~ fort t

PROOF 0

Sin(y-x)~ax2 u (x, t)

< o y

= /sin(y- x)(u=+ u)&= ~(y,t) - ,,(O, tlcosy. o

417

Blow-up Rate of a Degenerate Parabolic Equation Thus q(x, t) = ~(0,t) = cos x + By integration

¢(x, t) with 0 _< e(x, t) "~ 0 uniformly

[0, ~] by

for y - ~

ut = (x’s) /sin(~-x)/

u2 (x, s) dsdx

0

t 2

f

COSX .

o Thus T u(O,t)

u 3’s

f

ds

o

t

COSX q(x,t)

cos x +

~

o PROOF of Theorem 2 Let p = 2. Because of~ = [-,~,~], we may takexo = .~(l+2~) in Lemma3 and get someconstant~ > 0, suchthat 0 _~ -f(xo,t) ~_ IVI~ for t < T. Thus L’(t) "2 =

Let Q(t) := L(t)

--dx

~ L"(t)

+ C~.

- C] t. Then

+

< 0"m.

Note, that Q~ is positive for t ~ t0: For xo = a, L~ is monotonicMly increasing to infinity as L" ~ 0, and L~ bounded ~ would imply the s~me for L, ~ contradiction. Now stays bounded on [xo,a], T0

therefore

f ~dz ~ ~, hence o

Therefore by integration

same for

(with various

lnQ’>_ Integration

the

xo

constants

2__Q_C~Q,(e_~

depending on e) Q) >c>O.

from t to T implies with Q(t) -+ oo for t -~ T that 2 ---Q(t)

>_ ln(T- t) -

X0

hence 2 --L(t) xo

<_ -ln(T-

t) + c.

418

Schmltt and Wiegner

As 2

L(t) >_ fln((u(0,t)cos )d 0

~r ln(u(0, t)) - c, we conclude

u(0,t)=o(T- t) _
<_ co(T

The estimate from below is given in Proposition 1. By Lemma5 we have u(0, t)u (-~,t) (T - t) <_ -~ because of ut >_ O. Thercfore u (-~,t) - t)22-~ (T 1 _< ~. A bound for u (~,t) from below now follows by noting T

that -~ <_ u(O, t)f u(O, s)q (-~, for t _> t o,again due to Lemm a 5. As q (x, t) decreasing,

This finishes the proof of the theorem. CONCLUDING REMARKS It is tempting to take Xo = ~. But then f (~, t) has to be estimated, which looks as complicated as an estimate for u(0, t). Also the case p > 2 bears somedifficulties, and we shall address this question in a later paper.

REFERENCES 1.

2. 3.

K. Anada, I. Fukuda, N. Tsutsumi. Regional Blow-up and Decay of Solutions to the hfitial-Boundary Value Problem for ut = uu.~,~ - ~,(u.~) ~ + ku~. Funkc. Ekvacioj 39: 363-387, 1996. A. Friedman, B. McLeod. Blow-up of Solutions of Nonlinear Degenerate Parabolic Equations. Arch. Rational Mech. Anal. 96: 55-80, 1986. T. Ishiwata, M. Tsutsumi. A Numerical Study of Blow-up Solutions to ut = u~(Au + pu). Proceedings of the 4th MSJInternational Research Institute, Sapporo, 1995. GAKUTO Int. Set., Math. Sci. Appl. 10: 195-207, 1997.

Blow-upRate of a DegenerateParabolicEquation 4.

5.

6. 7. 8.

419

T. Ishiwata, M. Tsutsumi. Numerical studies of blow-up of solutions to some degenerate parabolic equations. To appear in GAKUTO Int. Ser., Math. Sci. Appl. 11. M. Tsutsumi, T. Ishiwata. Regional blow-up of solutions to the initial boundary value problem for u~ = u~(Au + u). Proc. Roy. Soc. Edinburgh Sect. A 127: 871-887, 1997. M. Wiegner. Blow-up for Solutions of Some Degenerate Parabolic Equations. Diff. and Integral Equat. 7: 1641-1647, 1994. M. Wiegner. A Degenerate Diffusion Equation with a Nonlinear Source Term. Nonlin. Anal. TMA12: 1977-1995, 1997. M. Wiegner. Remarks to the Blow-up Profile of a Degenerate Parabolic Equation. In: H. Amann,C. Bandle, M. Chipot, F. Conrad, I. Shafrir, eds. Progress in partial differential equations Vol 1 8z 2. Papers from the 3rd European Conference on Elliptic and Parabolic Problems, Pont-/~-Mousson, 1997. Pitman P~esearch Notes in Math. 383/384, Harlow: Longman1998.

Stability

in Chemical Reactor Theory

K. TAIRA Institute Japan

of Mathematics, University of Tsukuba, Tsukuba 305-8571,

K. UMEZU Maebashi Institute

1

INTRODUCTION

of Technology, Maebashi 371-0816, Japan

AND RESULTS

Let D be a bounded domain of Euclidean space l~ N, N _> 2, with smooth boundary 0D; its closure D = D t2 ODis an N-dimensional, compact smooth manifold with boundary. In this paper we consider the following semilinear elliptic boundary value problem arising in chemical reactor theory (cf. [3]): U

(*)~

Ou Bu := ~nn + b(x’)u =

on 0D.

Here: (1) A ~v. = ~ + ~ + ...+

~ is the usual

Laplacian

in R

(2) c(x) e C~(-~) and c(x) _> 0 in (3) A and v are positive parameters. (4) b(z’) e C~(OD)and b(x’) >_ on0D.

(B)

(5) n = (nl,n2,... ,nN) is the unit exterior normal to the boundary OD. In chemical reactor theory the nonlinearity

f(u) = exp describes the temperature dependence of reaction rate for exothermic reactions obeying the simple Arrhenius rate law in circumstances in which heat flow is purely 421

422

Taira and Umezu

conductive, and the parameter ~ is a dimensionless inverse measure of the Arrhenius activation energy. The equation

represents heat balance with reactant consumption ignored, where u is a dimensionless temperature excess and the parameter A is a di~nensionless rate of heat production. On the other hand, the boundary condition Bu = -~n + b(x’)u = represents the exchange of heat at the surface of the reactant by Newtoniancooling. In chemical reactor theory, the coefficient b is called a Blot numberif it is a positive constant. A function u(x) is called a solution of problem (*)~ if u(x) ¯ C2(-~) and satisfies problem (,)~. Moreovera solutiou u(x) of problem (*)~ is said positi ve if it is positive everywhere in D. The existence, uniqueness and multiplicity of positive solutions for problem (*)~ have been studied by manyauthors (see [4], [5], [6], [9], [11], [19], [20], [17]). In particular, under condition (B) Wiebers [20] proved rigorous connection between the positive solution set of problem (,)~ and that of simple scalar equation (see equation (2.2) in Section In order to formulate our existence and multiplicity theorem of positive solutions of problem (,)~, we introduce a function ~(t), called Semenov approximation, as follows.

= IN’t > 0. It is easy to see that if 0 < e < 1/4., then the function u(t) has a unique local maximumv(t~ (¢)) 1 - 2e - v/f - 4e

t~ (e)

2e 2

and a unique local minimum,(t2 (¢)) t=(e)

1 - 2e + v~- 4e 2d

Weremark that, by a direct computation, the local maximnmv(t~ (e)) is positive near e = 0, while the local minimumv(t~(e)) tends to 0 as e $ 0. The graph of the function v(t) is shownin Figure 1.1. Let ¢(z) ¯ C~(~) be a unique positive solution of the linearized problem A¢=I inD, Be 0 on 8D.

(1.1)

Our starting point is the following existence theorem of positive solutions of problem (*)a due to Wiebers [19, 20] and Taira-Umezu[17]:

423

Stability in ChemicalReactorTheory t

1/41

~t t~(e)

t~(e)

Figure 1.1

THEOREM 1 Assume that condition

(B) is satisfied

and that

c(x) ~. 0 in D if b(x’) = 0 on OD.

(H)

Then we have the following: (i) Problem(,)x has at least one positive solution for every A > O. (ii) If ~ >_ 1/4, then problem (,)~ has a unique positive solution u~(x) for every ~>0. (iii) If 0 < e < 1/4, then problem (*):~ has unique positive so lution u~ (x) fo all small )~ satisfying the condition

0<~<

2 4~

Here ~ > 0 is the first eigenvalue of tile eigenvalue problem

B~o = 0 on OD.

(1.2)

Furthermore, there exists a constant A > O, independent of e, such that problem (*)~ has unique positive so lution ux (x) fo r al l A > (iv) There exists a positive constant fl, independent of ~, such that if ~ > 0 is so small that

~(t2(~)) ,4t1(~))

fl max~ ¢’ then problem (*)x has at least three distinct positive solutions for all A satisfying the condition

~(t~(e)___A) <~<~(t~ (~)___~) fl REMARK 1.1 In part (iii) 1] we can prove that

max~ ¢’

of Theorem 1, arguing as in the proof of [7, Theorem

u~(x)~¢(z) as,~ J,

424

Taira and Umezu

Figure 1.2

ux(x),,-Aexp[!]¢(x) More precisely,

we have ~¢(x) u~(x)~

inC 1(~)

~exp[~]¢(x)

asA$O, inCl(~)as,~

The positive solution sets for e _> 1/4 and for 0 < e << 1/4 in Theorem1 may be represented respectively as in Figures 1.2 and 1.3. Nowwe are concerned with the study of the asy~nptotic stability of the positive solutions. To do so, we consider the following semilinear parabolic initial boundary value problem: Ov

v

in D x (0,

VI$----O-- "~$0

in D,

Ov ~ Bv:= ~nn +b(x)v=0

on ODx (0,

A function v(x, t) is called a global solution of problem(**)~ if v(., t) E Cl([0, v(x, .) e C2(-~) and if it satisfies problem (**)x. Since we have

l = f(O) <_ f(t)

< exp [!] > O,

(1.3)

we can show, by using a monotone iteration scheme for semilinear parabolic problems, that problem (**)x has a unique nonnegative, global solution v(x, t) for any nonnegative initial data no(x) ~ C2(-~) (see [13], [14, Theorem2.3.2], [2]).

425

Stability in ChemicalReactorTheory

i

i

<< 1/4

max,-¢ Figure 1.3

A positive solution ~(x) of problem (*)~ is said to maximal if u(x) <_ ~(xon ~ for any positive solution u(x) of problem (*)~. Similarly a positive solution u_(x) of problem (*)~ is said to be minimal if u(x) _< u(x) on D for any positive solution u(x) of problem (*)~. It is worth pointing out here (cf. [1]) that problem (*)~, the maximal and minimal positive solutions for each A > 0. The stability of positive solutious of problem (*)~ was studied by Wiebers [19] (cf. [10, Chapter 5]). He considered the linearized eigenvalue problem at a positive solution u of problem (*)~

{

Aw- Af’(u)w Bw = 0

=

inD, on OD,

and proved that the first eigenvalue #~ (u) is positive if u is unique ([19, Theorems 2.6 and 2.9]), and further that the first eigenvalues #~ (~) and #~ (u) are nonnegative ([19, Corollary 1.4 and Proposition 1.2]). The purpose of this paper is to study the asymptotic stability of maximal and minimal positive solutions of problem (*)~ in terms of the size of initial values Uo of problem (**)~ with respect to the solution ¢(x) of problem (1.1). Webegin by stating a fundamental asymptotic stability theorem for the unique positive solution u~(x) of problem (*)~: THEOREM 2 Assume that conditions (B) and (H) are satisfied. If uniq ueness of positive solutions of problem (*)~ holds, then the positive solution ux(x) is asymptotically stable in the following sense: For any global solution v~(x, t) problem (**)~ with an initial value Uo(X) E C:(-~) satisfying the condition

0 < u0(x) < c¢(~) on D with a constant C >_ Aexp[1/e], we have

ma__xlv~(x,t)- u~(x)] -~ 0 t ~oo. x~D

Our first result asserts the asymptotic stability of minimal positive solutions of problem (.)~ in the case where A is small (cf. [10, Chapter 5, Theorem4.3]):

426

Taira and Umezu

THEOREM 3 Let 0 < e < 1/4, and assume that conditions (B) and (H) satisfied. Thenthe minimalpositive solution u~ (x) of problem( , ):~ asymptotically stable if A is so small that

o
value

t~(z) ¢’z) o < ’uo(x) < ma-~~ onD, we have

m~lv~(~,t)- ~(~)l --~ 0 as -- ~ ~. Secondly we study the asymptotic stability of maxirnal positive solutions of problem (,):~ in the case where A is large. To do so, we remark that min¢ > 0. Indeed, if condition (B) is satisfied, then it follows from an application of the strong maximumprinciple and the boundary point lemma(see [12], [15]) that the solution ~b(x) of problem (1.1) is positive on Then we have the following asymptotic stability of maximalpositive solution.s of problem (*)~ ibr A large (cf. [10, Chapter 5, Theore~n4.3]): THEOREM 4 Let 0 < g < 1/4, and assume that conditions (B) and (H) are isfied. Then the maximal positive solution ~(x) of problem (* ):~ asymptotically stable if )~ is so large that

:~ > ~(t~(~)) - mince "

Moreprecisely, for any global solution vx ( x, t) of problem(**)), with an initial value uo(x) e C~(-~) satisfying the condition

on ~, uo(x)-> t~(s) ¢(~) mince " we have

ma___xlv~,(x,t) - ~,(x)l---~ast -+cx3. Finally we consider the case where the Arrhenius activation energy is so ihigh that 0 < ~ << 1/4 as in part (iv) of Theorem1. By combining Theorems3 and ,4, we can obtain the following asymptotic stability of maximaland minimal positive solutions of problem (*)~ for A in an interval as in Figure 1.3:

427

Stability in ChemicalReactorTheory U

t~_~ ~h min~~b"~ maxD.¢ w 1/41 0[

I

~(t~(£~

~(t~(~))

min-ff ~

max,- ¢

Figure 1.4 THEOREM 5 small that

Assume that conditions (B) and (H) are satisfied.

If ~ > 0 is

~(t2(~)._.__~) <~(t, min~ ¢ max~ ¢’

then, for each A such that ~(t2(¢)) < A < u(tl(¢)) ming¢ - - maxg¢’ any global solution vx(x, t) of problem (**)x with an initial satisfying the condition

value Uo(X)

on D

O
converges uniformly to the minimal pDsitive solution ux (x) of problem (,)~ as On the other hand, any global solutions v~(x, t) of problem (**)~ with an initial value Uo(X) C2(~) satisfying th e co ndition

uo(~:) > t~(~) ¢(z) - ming ¢ converges uniformly to the maximal positive solution ~ (x) of problem (*)~ REMARK 1.2 For condition (1.4), we can estimate the constant/3 in part (iv) Theorem1 as follows (see estimate (a.2)):

~,(t~(s)~) <~,(t~@)) ~ The situation Figure 1.4.

mince

"

of Theorems 3, 4 and 5 may be represented

schematically

by

428

Taira and U:mezu

The rest of this paper is organized as follows. Section 2 is devoted to the proof of Theorems 2, 3 and 4 by using Sattinger’s stability argument. In Section 3 we estimate the constant fl in part (iv) of Theorem1 in terms of the function ¢(x). The research of the first author is partially supported by Grant-in-Aid fc, r General Scientific Research (No. 10440050), Ministry of Education, Science and Culture Japan.

2

PROOF OF THEOREMS 2,

3 AND 4

In this section we prove Theorems2, 3 and 4, by using Sattinger’s stability theorem (see [13], [14]). h nonnegative function ¢(x) E C2(~) is called subsolution of problem(,)x if it satisfies the conditions A¢_
inD, on OD.

Similarly, a nonnegative function ~ ~ C2(~) is called supersolution ofproblem (,)~ if it satisfies the conditions

{A~>_ ~f(w)i~ B~ >_ 0

on OD.

Our proof of Theorems 2, 3 and 4 is based on the following criterion for the asymptotic stability of positive solutions of problem (*)x (cf. [14, Theorem2.6.2], [10, Chapter 5, Theorem4.4]): THEOREM 2.1 Assume that conditions (B) and (H) are satisfied. Let ~(x) positive solution of problem (*)~ and let ~(x), ¢(x) be respectively a supersolution and a subsolution of problem (*)~ such that ¢(x)_<~(x)_<~(x)

on D.

Then any global solution v~(x, t) of problem (**)~ with an initial value uo e [¢:. ~] converges uniformly to ~(x) as t ~ oc if and only if the uniqueness of positive solutions of problem (*)~ holds in the order interval

[¢,~1= {~ e c~(~):¢(z) < u(.~.) < v(.~:) on~}. 2.1

Proof

of Theorem 2

By inequalities (1.3), it follows that the function ¢(x) = 0 is a subsolution of problem (*)~ and further that the function C¢(x) is a supersolution of problem (,)x if C >_ £exp[1/~]. By using the super-subsolution method, we can find unique positive solution u~(x) of problem (,)~ such that O <_ u~(x) <_ C¢(x) onD.

Stability in ChemicalReactorTheory

429

Therefore Theorem2 follows from an application of Theorem2.1 with ~(x) := u~ (x), ¢(x) := 0 and ~(x) := C¢(x).

2.2

Proof

of Theorem 3

In order to construct a supersolution of problem (,)x, we let

max~ ¢ If A is so small that 0 < A < ~(tl(e)) - max~¢’

(2.1)

then we have tl(E)

A~ - Af(~p)

max~ ¢

u(tl(¢))f (tl(8) max~¢ \max~¢ \maxg¢

1max~¢ >_0 inD,

since the function f(t) is strictly increasing for all t >_ 0. This implies that 99(x) is super-solution of (,)~ if )~ satisfies condition (2.1). Hence, by using the super-subsolution method we can construct a positive solution ~(x) of problem (,)~ that 0_< ~(x) < t~(~----~-) max~ ¢

¢(x)

Nowwe remark that the scalar equation t A = A;u(t) = )~ ~--~

(2.2)

has at most one solution in the interval [0, t~(e)] for 0 < A < Aly(tl(¢)) (see 1.1). A similar assertion holds for problem (,)x. In fact, we have the following (see Figure 1.4): LEMMA 2.1 Let 0 < s < 1/4. Then problem (*)~ has at most one positive solution in the order interval [0, t,(t)] : {u e C’~(~) : 0 _< u(x) _< on-~} for A > O. PROOFLet u~(x) and u2(x) be two positive the condition 0 ~ ~t I(x),

~t 2 (X)

solutions of problem (*)~ sutisfyin~ ~< 1(~)

on

D.

430

Taira and Umezu

Then we have, by Green’s formula, (2.3)

where da is the surface element of OD. Since we have ~2

~n’U2-ul-~n

=0 onOD,

we obtain froIn formula (2.3) that

\ ul

i=

u20xi

1 OZi

dx

>0. This implies that ua (x) ~ u2(x) on D, since the function f(~)/~ is strictly for all 0 < t _< tl(~). The proof of Lelnma 2.1 is complete. []

decreasing

Lemma2.1 implies that the positive solution ~ (x) is unique in the order interval

max~¢ ]’ in particular it is minimal. Therefore it follows from an application of Theorem2.1 with ~(x) := u~(x), ~b(x) := 0 and ~o(x). := (~a(e)] max~b)~b(x)that positive solution u_u_~(x) = ~a(x) is asymptotically stable. The proof of Theorem 3 is now complete. []

2.3

Proof of Theorem 4

The proof is essentially the same as that of Theorem3. Indeed, it suffices to verify the following (see Figure 1.4):

431

Stability in ChemicalReactorTheory

(a) The ffmction (t.2(e)/rnin-~¢)¢(x) is a subsolution of problem (,)~ for every A >_ u(t.2(~))/mince. (b) The function Cdp(x) is a supersolution of problem (*)~ if C ~ Aexp[1/e]. (c) There exists at ~nost one positive solution u~(x) C2(~) of problem (*) the order interval [ t~),¢,C¢], m~n~ 0

C~Aexp[1/e].

Therefore Theorem 4 follows from an application ~(x), ¢(x) (t~ (e)/min~¢)¢(x) and~(x) := C¢ (x) 3

CONCLUDING

of Theorem 2.1 with ~(x)

REMARKS

In this section we estimate the constant fl in part (iv) of Theorem1 in terms of the function ¢(x). First we recall the precise definition of the constant fl in Theorem1. For a relatively compact subdomain 12 of D with smooth boundary, we consider the following linear boundary value problem:

{

Aw= xa in D, Bw = 0 on OD.

(3.1)

Here Xa(x) is the characteristic function of f~ in D. It is well known(see [8], [16], [18]) that problem (3.1) is uniquely solvable in the frameworkLp spaces. Moreover we can show that the solution w~(x) belongs to CI(~) and is positive everywhere in D. Then the constant/~ is defined as follows: sup inf wfl (x). Under condition (H), we can prove that re_in ¢ _< ~ _< m_ax ¢. D D

(3.2)

Indeed, if ~l(x) is the eigenfunction corresponding to the first eigenvalue A~ problem (1.2) and if maxu~= 1, then [20, Lemma5.1] tells us that A~m_in~o~ < ~ _ min¢. D

(3.3)

To do so, we choose a sequence {~2j) of relatively compact subdomains of D, with smooth boundary, such that fl~ ~ D as j -~ cx~. If wfl~ (x) is a unique solution problem (3.1) with fl := f)~, then it follows from an application of ~ t heory of linear elliptic boundary value problems that wa~(x)

~¢(x)

inC(D)

asj--~c~.

432

Taira and Umezu

Now,since wnj (x) and ¢(x) are both strictly illf~

positive on ~, we obtain

wD~ = sup~ b ~ : SUPD --, (3.4)

1 mince = inf D ¢ = suPD ~. Howeverit is easy to verify that

1

(x) so that

sup Xfb __~ sup 1 D W~j

as 3 --~ oc.

D 7

In view of formulas (3.4), this implies that infwf b ~min¢

(3.5)

asj--~

Therefore the desired inequality (3.3) follows from assertion (3.5), since we inf flj

wftj

< sup infwn = ft. _ ~D ~2

[]

EXAMPLE 3.1 If c(x) is a positive constant c and b(x’) = 0 on OD, then we find from assertion (3.2) that ~ 1/c, si nce ¢( x) = 1/ c in D (see [20 , Theorem 5.4]).

REFERENCES 1. 2.

3.

4. 5. 6. 7.

H. Amann,Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAMRev. 18 (1976), 620-709. H. Amann,Periodic solutions of semi-linear parabolic equations, In: Nonlinear Analysis, L. Cesari, R. Kannanand H. F. Weinberger (eds.), Academic Press, NewYork San Francisco London, 1978, pp. 1-29. T. Boddington, P. Gray, and G. C. Wake, Criteria for thermal explosions with and without reactant consumption, Proc. R. Soc. London A. 357 (1977), 403-422. K: J. Brown, M. M. A. Ibrahim, and R. Shivaji, S-shaped bifurcation curves problems, Nonlinear Anal. 5 (1981), 475-486. D.S. Cohen, Multiple stable solutions of nonlinqar boundary value problems arising in chemical reactor theory, SIAMJ. Appl. Math. 20 (1971), 1--13. D.S. Cohen and T. W. Laetsch, Nonlinear boundary value problems suggested by chemical reactor theory, J. Differential Equations 7 (1970), 217-226. E.N. Dancer, On the number of positive solutions of weakly non-linear elliptic equations when a parameter is large, Proc. London Math. Soc. 53 (198(;), 429-452.

Stability in ChemicalReactorTheory 8. 9. 10. 11. 12. 13. 14. 15.

16.

17. 18. 19. 20.

433

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, NewYork Berlin Heidelberg Tokyo, 1983. R. WLegget and L. R. Williams, Multiple fixed point theorems for problems in chemical reactor theory, J. Math. Anal. Appl. 69 (1979), 180-193. C.V. Pao, Nonlinear parabolic and elliptic equations, Plenum, NewYork London, 1992. S.V. Patter, Solutions of a differential equation in chemical reactor processes, SIAMJ. Appl. Math. 26 (1974), 687-715. M.H. Protter and H. F. Weinberger, Maximumprinciples in differential equations, Prentice-Hall, EnglewoodCliffs, NewJersey, 1967. D.H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems,. Indiana Univ. Math. J. 21 (1972), 979-1000. D.H. Sattinger, Topics in stability and bifurcation theory, Lecture Notes in Mathematics, No. 309, Springer-Verlag, NewYork Heidelberg Berlin, 1973. J. Smoller, Shock waves and reaction-diffusion equations, Springer-Verlag, New York Berlin Heidelberg London Paris Tokyo Hon Kong Barcelona Budapest, 1994. K. Taira, Analytic semigroups and semilinear initial boundary value problems, London Mathematical Society Lecture Note Series, No. 223, Cambridge University Press, London NewYork, 1995. K. Taira and K. Umezu,Semilinear elliptic boundary value problems in chemical reactor theory, J. Differential Equations 142 (1998), 434-454. M. Taylor, Pseudodifferential operators, Princeton Univ. Press, Princeton, 1981. H. Wiebers, S-shaped bifurcation curves of nonlinear elliptic boundary value problems, Math. Ann. 270 (1985), 555-570. H. Wiebers, Critical behaviour of nonlinear elliptic boundary value problems suggested by exothermic reactions, Proc. Roy. Soc. Edinburgh 102A (1986), 19-.36.

Banach Space Valued Ornstein-Uhlenbeck Processes Indexed by the Circle 1 DepartInent of Mathematics, The University ZDZIStAWBRZE~NIAK Hull HU6 7RX, England, [email protected]

of Hull,

JAN VANNEERVEN Department of Mathematics, Delft University of Technology, PO Box 5031, 2600 GADelft, The Netherlands, [email protected]

ABSTRACT In this paper we study the periodic stochastic abstract lem dX(t) = AX(t)dt + B dWH(t), [0 , T],

Cauchy prob-

X(O) = X(T), where A is the generator of a Co-semigroup {S(t)}t>~o on a separable real Banach space, {WH(t)}t>~ois a suitable cylindrical Wiener process with reproducing kernel Hilbert space H, and B : H -~ E is a bounded linear operator. Weobtain sufficient conditions for existence of Gaussian mild solutions and show that solutions and compute the covariance of these solutions. Wealso obtain sufficient conditions which guaratee that the mild solution is law-equivalent with the mild solution at time T of the corresponding stochastic abstract Cauchy problem with zero initial condition. 1991 AMSSubject Classification:

0

60G15, 60H15, 60B05, 47D03

INTRODUCTION

Periodic stochastic processes do appear in manybranches of mathematics and mathematical physics. Their importance stems from the fact that traces of certain semigroups are equal to some integrals over periodic maps. For example Bismut [Bi] used such representation to provide a probabilistic prrof of the Atiyah-Singer Theorem. This work was partially

supported by EPSRCgrant GR/L 60876. 435

436

Brze~niakand van Neerven

In [ABB]the authors have found that a trace of tile Schrbdinger group can be represented as a Feynmantype integral over a certain Hilbert space of periodic maps. Finally, there is muchresearch, see e.g. [JL], towards finding stochastic processes with values in manifold valued periodic maps. This is believed to be an essential step towards proving certain index theorems for Dirac operators, as conjectured by Witten. In this paper we will be concerned with periodic Ornstein-Uhlenbeck processes (briefly, periodic O-Uprocesses). Werefer to [KL] and [No] for various characterizations of this class of processes. They arise, for instance, in tile stndy of quantum systems in the positive temperature regime are periodic, with their period being inversely proportional to the temperature; see [KL], [AH]and references therein. A standard example of a periodic Ornstein-Uhlenbeck process is the process whose Cameron-Martin space is the Hilbert space H~,2(S~,/~), where a > 0 is a real nmnber and S~ is the unit circle, with norm

Ibll = (l (s)l2 + a2l’~(s)]

2) ds.

It can be shownthat this process can be seen, in an appropriate sense, as a solution to the following periodic It6 equation: d~(t) = -a~(t) dt + dw(t), [0, 2~], ~(0) = ~(2~r), where w(t) is the standard ll~d-valued Wiener process. Klein and Landau in [KL] have also discussed some infinite-dimensional examples; we also mention the papers [DTb] and [DTd]. It turns out that even such questions as existence and uniqueness are nontrivial and their analysis provides deep insight into other phenomena. In particular, despite its importance, the problem of existence of infinite-dimensional periodic O-Uprocesses is not well understood. In the present paper we study the following periodic It6 equation in infinite mensions:

di-

dX(t) = AX(t) dt + B dWg(t), [0 , T], X(O) = X(T). Here A is the generator of a Co-semigroup {S(t)}t>~o on a separable real Banach space E, {WH(t)}t>~o is a suitable cylindrical Wiener process with reproducing kernel Hilbert space H, and B : H -+ E is bounded and linear. In fact, the periodic boundary condition is just a special case of more general one of the form X(0) cOX for a suitable linear map0 : E[°’T] -~ E, where E is some Banach space, and sc~me of our results can be formulated for this .more general setting. Wewill describe next our results in the periodic case, i.e. when cOX = X(T). Under the assumption I - S(T) is invertible in E we prove that (0.1) has a unique solution whenever the corresponding stochastic Cauchy problem dX(t) = AX(t) dt + B dWu(t),

x(o) =

t ~ [0, T],

(o.2)

BanachSpaceValuedOrnstein-Uhlenbeck Processes

437

has a unique solution. In fact, we give an explicit formula that relates the solutions of (0.1) and (0.2). Wemention that necessary and sufficient conditions for unique solvability of (0.2) have been obtained recently in [BN]. Wealso show that this solution of (0.1) is a centred Gaussian process and we derive an explicit formula for its covariance. In order to derivc the formula relating the solutions of (0.1) and (0.2), in Section 2 we first study the deterministic case. More precisely, we consider the periodic inhomogeneous Cauchy problem v’(t)

: Av(t) + f(t),

[O,T],

v(0) As it is well known,under assumption (0.1) this problem has a unique mild solution for every f E Lt ([0, T]; E) which is given explicitly t

f0

u(t) = (I - S(T)) -1 S(t - s rood T)f(s)

t e [0,r].

Weshow that this expression is in fact a special case of a general formula for mild solutions of problems with initial value conditions of the type v(O) = Ov in terms of the mild solutions of the corresponding Cauchy proble~n with initial condition v(0) = 0. As such, this approach is in the spirit of (and to some extent motivated by) the boundary perturbation theory of Greiner [Gr].

1

A MOTIVATING

EXAMPLE

In this section we will perform some computations for the following periodic It6 equation: d~(t) = -a~(t) dt.+ dw(t), t ~ [0, 27r], (1.1) ~(0) = ~(27r), where a > 0 is a given real number and {w(t)}te[o,.~,] is the standard II~d-valued Wiener process. The results of these computations are well-known and will be generalized to infinite dimenstions in Section 3. Equation (1.1) has a unique weaksolution {(t}te[o,’~,]; this solution is Gaussian and its Camer0n-Martinspace is the Hilbert space H~,~(S~; I~~) of all absolutely continuous functions 3’ : S~ -~ lRd such that 2~r

f0

I111:=

e +aZl( )l 2) ds<

Starting from the space H~,~, we will give the process {~(t)}te[o,2,l. For simplicity t ~ [0,2~r] the linear mapHal,~(S~;ll(a) 9 Riesz Representation Theoremthere exists

a direct calculation of the covariance of of exposition we assume that d = 1. Given 3’ ~+ 3’(t) e lt~ is bouuded, hence by a unique ut ~ Hla’2(S1;II~ d) such that

[7, ut]~ = 3"(t), 3’ d) H~’~(S~;ll{

Brze~.niakandvan Neerven

438 Since E(~(t){(s)) = [u,,u,],

’ut(s)

our task reduces to finding ut. Assume now that to E (0, 2~r) is fixed and write u = uto. Then for every 3’ E C~°([0,2rr] \ {to}),

Therefore, u is a weaksolution to -ii + a=u= 0, on [0, 2~r] \ {to}. By the elliptic regularity we have u e H~’=([0, 27r] \ {to}) and so, in particular, limits u(tff), u(t+o), i,(tff), i,(t~) do all exist. Byan integration by parts as aboveand 2,2 1 recalling that u(0) = u(2zr) we then infer the following. For any 7 ~ H, (S ;IRd),

fo

~ u(s) (-5(s) + a~-7(s)) ds + (it(t~) - 7(to) =7(t

~,2/,q~./~a) 9 7 ~ 7(to) ~ IR is onto, we arrive at the following Since the map ... ,~ , characterization of uto. An element u E H~’=([0, 2rr] \ {to}) equals Uto if and only it is a weak solution to the boundary value problem -i~+a~u=O, on [0,2~r]\{to}, ~(t~-) - ~(to+) = 1,

(1.2)

- = o. For to = 0 or to = 27r, the problem (1.2) takes the form -/i+a~u=0,

on (0,27r),

(1.3)

i,(2rr) - = u(2r) - u(0)

Standard but tedious calculations give the following formula for the unique solution Uoto (1.3): cosh(~ra)’ cosh(as) uo(s) = 2~zsinh(as) + sinh(~ra) 1 [sinh(as) + sinh(a(27r - s))], = 4a sinh~ (Tra) /qrl,2(.q,1. ]~d) implies that The rotation invariance of the scalar product in ..~ ~ , ut(s) = uo(s rood 2r r), s

e [0,2zr].

BanachSpaceValuedOrnstein-Uhlenbeck Processes

439

and thus 1 E(~(t)~(s)) = aaslnh2~ra~ [sinh(a(s - t)) si nh(a(2~r - (s- t )) )], s,

t e [0, (1.4)

In Section 4 we will give a far-reaching generalization of this formula. 2

THE ABSTRACT CAUCHY PROBLEM WITH AN INITIAL VALUE OPERATOR

Throughout this section, S = {S(t)}t.~>o will be a fixed Co-semigroup with generator A on a (real or complex) Banach space E. Wewill study the deterministic problem u’(t) = du(t) f( t), t e [0, T], (AOPo)

u(O)= Ou,

where f E L1 ([0, T]; E), and 0 : C([0, T]; E) -~ E is a boundedlinear operator. aim is to give a general formula for the mild solution of this problem in terms of the mild solution of the related inhomogeneous Cauchy problem u’(t) = Au(t) + f(t),

t e [0,T],

(ACP)

=o. Westart

by defining an operator L:E -~ C([0,T]; E) (Lx)(t) -= S(t)x,

[0, T].

Welet Co([O,T];E) := {f e C([O,T];E) : f(O) and for a bounded linear operator 0 : C([0, T]; E) -~ E we define Co([O,TI;E)=

(f G.C([O,TI;E):

f(O):Of}.

PROPOSITION 2.1 For an f ~ C([0, T]; E) the following assertions are equivalent:

(s) ] e Co([O,T];Z); (ii) (I - LOft C0([0, T] ; E) IfI- LOis iuvertible i~ C([0, T]; E), ~he~ I - LOrestricts

co([o,T];E)o.to6 o([O,T]; PROOF(i)=v(ii):

f ~ Co([ O,T];E), so f (O)= Of. Then

(I- LO)f(O) S(O) - L OI(O) & y (o) - S (O)(Oy) = O (ii)~(i):

g :=(I - L D)fCo(J0, T]; E), s o g( 0) - - 0 From f = (I - LO)I + LOf = g

we obtain f(O) = g(O) + Lc~f(O) = LOI(O) = S(O)cOf The final assertion is obvious.

to au isomorphism from

440

Brze~niakand van Neerven

A mild solution of (ACPo)is a function u E Co(f0, T]; E) satisfying u(t)

= S(t)Ou ÷ S(t - s)f(s)

[0, T].

For 0 = 0 the problem (ACP) := (ACPo) has a unique mild solution given by u(t)

= S(t - s)f(s)

u, which

ds, [0 , T];

cf. [Pa, Chapter 4]. PROPOSITION 2.2 Let f 6 Lt ([0,T]; E) be fixed. Suppose u is a mild solution of (ACP). IfI- LO is invertible in C([O,T];E), then v = (I - LO)-tu is solution of (ACPo). PROOFBy assumption we have u e Co(J0, T]; E) and u(t) = S(t - s) J’ (s)ds for all t ~ [0, rl. Hence v = (I- LO)-~u Co([0, T] ; ~) by Proposition 2.1 , and for all t E [0, T] we have v(t) = LOv(t) + (I - LO)v(t) = LOv(t) + u(t) = S(t)Ov + S(t ¯

Let ¢ : LI([0,T]; E) -~ Co([0,T]; E) denote the convolution operator defined Of(t)

S(t - s) f(s)ds, t

[0,T].

If we interpret this operator as the solution operator corresponding to the problem (ACP), then we may regard (I - LO)-~Oas the solution operator corresponding to (ACPo). Let U be a bounded linear operator on E commuting with S(t) EXAMPLE 2.3 for all t ~ [0, T]. In this example we consider the abstract Cauchy problem with holonomy U: x’(t) = Ax(t) + f(t), t ~ [0,T], (ACPv) x(O) = Ux(T). In order to analyze this problem we consider the operator Ov : C([0, T]; E) -* defined by Our = U f(T). Weclaim: if I - US(T) is invertible then I - LOuis invertible in C([0,T]; E) and for all f e L~([0,T]; E) we (I-

LOu)-lof(t) (I-US(T))

-~

(/o

S(t-s)f(s)ds+

t US(T+t-s)f(s)ds

)

BanachSpace ValuedOrnstein-Uhlenbeck Processes

441

Oncewe knowthis, Proposition 2.2 shows that the right hand side is a mild solution of (ACPu). First we show that I - LOgis invertible and that its inverse (I - LOu)-I is given by (I-LOu)-~g(t)

= g(t)+US(t)(I-US(T))-Ig(T),

[O,T ], g ~ C([O,T];E).

Indeed, the right hand side of this identity C([0, T]; E), and for all g e C([0, T]; E) we

defines a bounded operator J on

(I-LOv)Jg(t) = (I - Lcgv) (g(.) + US(.)(I - US(T))-lg(T)) = (g(t)

+ US(t)(I - US(T))-lg(T)) - US(t) (g(T) + US(T)(I - US(T))-lg(T))

=g(t) and J((I - LOu)g)(t) = (I - LOu)g(t) + US(t)(I - US(T))-1(I = .q(t) - US(t)g(T) + US(t)(I -~(g(T) - U S(T)g(

=g(t). This shows that J is a two-sided inverse of I - LOv. Next we compute: (I - LOu)-l~f(t)

= ~f(t)

:[.I.

S(t - s)f(s)

=

S(t - s)f(s)

+ VS(t)(I

ds + US(t)(I

- -~ S(T - s)f( s)

ds + US(T)(I - -~ [,

+(I-US(T))

- US(T))-I~f(T)

-1./,

S(t - s)f(s)

T US(T +t-s)f(s)ds

= (~r _ us(r/)-~ s(t - s/I(s) as + ~s(r + t - s)I(s/as The case U = I corresponds to the periodic inhomogeneous Cauchy proble~n; the above formula theu simplifies to

(I - ~0)-~ ~I(t) = (I - S(T))-~ S(t -

s mod r)l(s)as;

here OI = f(T) and t_smodT={t-s

T+t-s

ift-s)O, ift-s
This expression for the mild solution of the periodic Cauchyproblem is well known;

442 3

Brze~niakand van Neerven THE PERIODIC STOCHASTIC CAUCHY PROBLEM

ABSTRACT

Wewill use the framework of Section 2 to set up a theory for E-valued OrnsteinUhlenbeck processes with holonomy; for this we use the analogy between the linear stochastic Cauchy problem driven by cylindrical noise and the deterministic inhomogenousCauchy problem. Our terminology is standard; we refer to [DZ], [V’rC], It is a well known phenomenonthat the paths of an Ornstein-Uhlenbeck process generally fail to be continuous. For this reason the space C([0, T]; E) may not [°’T] appropriate for our discussion, and we have to consider instead the path space E consisting of the totality of all E-valued functions on [0, T]. With respect to the product topology, E[°’T] is a quasi-complete locally convex topological vector space. Welet E~°’Tt := (f ~ E[°’T1 : f(0) and for a continuous linear operator 0 : E[°’T] ~ E we let

°’rl :={: e EIO,rl Eg : 1(0)=COy}. PROPOSITION 3.1 Let cO : El°,T] -~ E be a continuous linear operator. (i) The operator I - LOmaps(7([0, T]; E) into itsel£ (ii) If I- LOis invertible as an operator on E[°’T], then its restriction to C([0, T]; E) is invertible in C([0, T]; E). PROOF(i): For .f E C([O,T];E) we have (I - LO)f = f - S(.)COf, which is a contiuuous function. (ii): Suppose that I LOis inv ertible in E[° ’r]. Fixf E C([0, T];E ). Let [°’Tl g e E be such that f = (I - LO)g. Then g = (I - LO)g + LCOg= f + S(.)Og, which is a continuous function. This shows that I-LCOmapsC([0, T]; E) onto itself, and being injective, the restriction of I - LOto C([0, T]; E) is therefore invertible. ¯

Wehave the following analogue of Proposition 2.1: PROPOSITION 3.2 Let f ~ E[°’rl.

The following assertions

are equivalent:

(i) f e Eg°’T] Oi) (I - LO)f e °’T]. If I - LCOis invertible as an operator on Ei°’rl, from Eg°’r] onto °’rl. E~

then it restricts to an isomorphism

443

BanachSpace ValuedOrnstein-Uhlenbeck Processes

Let H be a separable real Hilbert space. A cylindrical Wiener process with reproducing kernel Hilbert space (RKHS) over a filtered probability space (ft, r, {grt}te[O,T], P) is a f mnily of bounded linear operators {W~(t)}te[O,T] from H into L2 (P) with the following properties: (i) For all t e [0, T] and h ¯ H, Wg(t)h is an 5rt-measurable centred Gaussian random variable; (ii) For all h ¯ H, the process {Wg(t)h}te[O,T] is stationary; (iii) For all t, s ¯ [0, T] and h, g ¯ H we have E(WH(t)h.

Wg(s)g) = (t A s)[h,

Instead of WH(t)h we will usually write [WH(t), hi. Note that for all h ¯ H the real-valued process {[Wg(t),h]}te[O,T] is a Brownian motion; in particular it admits a continuous modification. Throughout the rest of this section E is a separable real Banach space, H is a separable real Hilbert space and B ¯ /:(H,E) is a bounded linear operator. Let 0 : E[°’T] -~ E be a continuous linear operator. Weconsider the stochastic abstract Cauchy problem dXt = AXt dt + B dWH(t),

t ¯ [0,T],

Xo = OX,

(sACPo)

where {Wu(t)}te[O,Tl is a cylindical Wiener process with RKHSH. A weak solution is an E-valued stochastic process {Xt}te[O,T] such that for all x* ¯ D(A*), t ~-~ (A’x*, Xtl is integrable almost surely and t

{Xt,x*)

j~0

= (OX, x*) + .(Xs’A*x*) ds + [WH(t),B*

(3.1)

almost surely. Here the random variable OX is defined by

= a(x@)), where X(a~) ¯ [0,T] i s g iven b y ( X(co))(t) = X~(w). Note that if {Xt}~e[o,~’l is a weaksolution, then by (3.1), (OX, x*) is a measurable function for each x* ¯ D(A*), and hence, by an approximation argument, for each x* ¯ E*. Since OX is also separably valued, it follows that OXis strongly measurable. Unless 0 = 0, because of ~he initial condition it makes no sense to imposepredictability of {Xt}te[O,Tl. If 0 = 0 we refer to the avove problem as (sACP). Necesary and sufficent conditions for the existence of a predictable weak solution of (sACP)are given in [BN]; for the convenience of the reader we state the result. THEOREM 3.3 The following

assertions

are equivalent:

(i) The problem (ACP) has a predictable E-valued weak solution {Xt}te[O,Tl on

[0,T];

444

Brze~niakand van Neerven

(ii) The operator QT ~:(E*, E)defined by QTX* := S(t)BB*S*(t)x*

dr,

x* ~

is the covariance operator of a centred Gaussian Borel measure on E. In this situation, the solution { Xt } tE[O,T] is unique, and satisfies f .!

(X,,x*)

: ]o (S(t - s) o B dWn(s),x*),

t e [0, T], x* e E*.

The process {Xt~}~e[o,T} is an Ornstein-Uhlenbeck process with drift A and forcing operator B, i.e. a centred Gaussian process with covariance ~As

E((Xt,x*)(X.~,y*))=

f [B*S*(t-u)x*,B*S*(s-u)y*]Hdu

(3.2)

JO

for all x*, y* E E* and 0 <~ t, s <~ T. It has a version with a/most surely sq~are integrable trajectories. If the semigroupgeneratedby A is analytic, then { X~} t.e [o,T] has a version ~vith continuous trajectories. Parallel to Proposition 2.2 we have the following result: THEOREM 3.4 Let {X~}~e[0,T] be a weak solution dX~ : AX~ dt + B dWH(t),

of the problem t [0 , T] ,

Xo=O. Let 0 : E[°,T] --~ E be a continuous linear operator such that I - LOis invertible in E[°’T]. Then the Gaussianprocess {]’~}~[0,T], where 1"), = ((I - L~)-Z X)~, weak solution of the problem dYt=AVtdt+BdWH(t), Yo = OY.

t ~ [0,

PROOFAs was stated in Theorem 3.3, the weak solution are almost surely square integrable:

b~ IJX~(~)[I < ~ fo

for almost

T],

{X~}~e[O,T] Of (sACP)

all ~ ~

It follows that also the process {l’~}te[o(r I is ahnost surely square integrable. Hence if x* ~ D(A*) is given, the map t ~ {l~,A*x*} is integrable almost surely. For almost all w ~ f~ we have ((1 - LO)(Y(w))(t) = X,(w) = (X(~))(t)

= (I - no)(X(w))(t)

Multiplying both sides with (I - LO)-~, we obtain

~ = x~ + LO(Y(~))= X~.

+ LO(X(w))(t).

445

BanachSpace ValuedOrnstein-Uhlenbeck Processes Using this,

we compute

f

0(Ys, A’x*) ds (Xs, A’ x*) ds + { S( s)OY, A’x*) ds = (Xs, A’z*) ds + (S(t)OY

Hence,

Define the continuous linear mapOH : L~([O, T]; H) -~ [°’Tl by

OHI(t) := S(t - s)BI(s) ds. Wewill also interpret L as an operator from E into E[°’TI. In order to stress that the range space is the locally convex space EI0,rl, the continuous adjoints of ~ and L will be denoted by O~ and L~, respectively. LEMMA3.5

We have

¢~(~®z*) = ~io,~l(.)B*S’(t,PROOFFor all

f ~ LZ([O,T];H) we have

= [l(s)., B’S*(t- s)]H = [f, X[o,q(’)B*S*(t -

")X*]L2(tO,TI;H).

Thanks to this lemma, the covariance of an Ornstein-Uhlenbeck process can be rewritten in terms of ~g as follows:

~((x~, ~*)(x~,v*)) = [~(~ ~*), ~(® THEOREM 3.6 Let X = {X~}t,e[0,T]

be a weak solution

Y = (I-

LO)-zX

of (sACP) and let

446

Brze~niakand van Neerven

be the corresponding we3k solution of (sACPo). Then for all x*, ~* ~ E* and all 0 <~ t, s <~T we have

E(0"i,z*)0;,y*)) [(I,~(I - O’L’)-’ (St x*), +~ (I - O’L’)-1(Ss PROOFWe can reformulate

(3.3)

(3.3) by saying that

[(X(.),6~ ®z*) (X(.),6s ® V*)]~(r~) = [¢I:’~-(6, ®x ),+t,(6~ ®V)]g~([O,T];t~>. ’It is a routine exercise to check that every element in the topological dual (E[°,TI) of E[°,r] can be represented as a finite linear combinationof functionals of the form gt ® z* with t ~ [0,T] and x* ~ E*. By bilinearity, it follows from (3.4) that

[<x(.),~>,<x(.),¢>]r~(~ re, / = [+;/(¢),+},(¢)]~
we compute

An E-valued centred Gaussian process whose covariance is given by (3.3) will be called a 0-Ornstein-Uhlenbeck process with drift A arid forcing operator B ~

EXAMPLE 3.7 We will work out the formula for an Ornstein-Uhlenbeck process with holonomy U, i.e. an 0-Ornstein-Uhlenbeck process with 0 = Ov given by Our = Uf(T) as in Example 2.3. As before, we assume that U is a bounded operator on E commuting with each S(t). Arguing as in Example2.3, we see that if I-US(T) is invertible, then the operator 1 - LOuis invertible in EIO,TI and its inverse is given by (I - LOu)-’g(t)

= g(t)

+ US(t)(I

- US(T))-’g(T),

E[ °’ T].

For 9 ~ E[°’TI we compute: ((I - LOv)-~ g, 5t ® x*) = (g(.) + US(.)(I - US(T))-’ g(T), 5t = (g(t) + VS(t)(I - US(T))-~g(T), = (9, (~, ® z* + ~T ® U*S*(t)(I - U*S*(T))-~x*)). From this we see that (I - 0’v L’~-t+~ ~ t t®x*)=~®

*+~rN(U*S*(t)(I x*

U*S*(T))-~x

BanachSpaceValuedOrnstein-U’hlenbeck Processes

447

and hence,

+’~+(I-O’oL’)-I (~, : +~(~, ®x* + ~ + *) (u*s*(t)(~- U’S*(T))-~ = ~to,~](.)B*S*(t - .)z* *B*U*S*(T -.) S*(O(I - U*S*(T))-~z -~ (X[o,,,](.)S*(*- .)x* + Xi~,TI(’)U*S*(T = B*(I- U*S*(T)) + t -(3.5) Note the similarity with the formula obtained in Example 2.3. In combination with (3.3), this gives the covariance of an Ornstein-Uhlenbeck process with holonomy Let us nowtake a look at the periodic case U = I. For the semigroupS(t) = e-~*I, with a > 0 a given real number, the problem (sACPo) reduces to (1.1). We compute the covariance of its weak solution {Y~.}~[0,T] explicity: from (3.3) and (3.5) we obtain after an elementary computation that 1

~B*x* B* *~

This is exactly the formula (1.4) derived in Section 1, where we took T = 2u. The case U = -I in this example corresponds to the ’twisted’ discuss briefly next.

case, which we

EXAMPLE 3.8 For U = -I and S(t) = e-~*I with a > 0, a comutation to the one above gives the following formula for the covariance:

similar

~((Y~,~’)(v~, 1 - ~. ~°~ ~(~)(-~i~(~(~- ~)) + ~i~(~(T- ~)))) [~’~ In the present case, we may take any. a ~ ~; witl~ a = 0 the resulting for the resulting ’twisted Brownianbridge’ becomes

covariance

E((Yt,x*)(Y,,y*))

4

THE PERIODIC PROBLEM VERSUS LAW EQUIVALENCE

THE CAUCHY PROBLEM:

In this final section we suppose again that E is a separable real Banach space and that A is a generator of a Co-semigroup.(S(t)}~>~o on E with generator A. Our aim in this section is to comparethe laws of the periodic O-Uprocess (]~}te[0,v] with those of the O-U process (Xt}~e[O,T] corresponding to the Cauchy problem with initial value operator 0 = 0 (cf. the notation used in Theore~n 3.3). Throughout this section we assume that I - S(T) is invertible. The laws of the random variables Y~ en Xt will be denoted by vt and p~, respectively; these are centred Gaussian Borel ~measures on E. By Ht and Gt we denote thd reproducing kernel Hilbert spaces of v~ and #~, respectively, and by R~ and Qt their covariance operators.

448

Brze~niakand van Neerven

LEMMA 4.1 For all t ¢ [0, T] we have Rt = RT = (I - S(T))-IQT(I PROOF For allt¢[0,

T]andx*

S*(T))-’

~E* we have

II¢ (x-- O’L’)-~(~T ~ Z )[IL2(fO,T];If) ¯ = x*>. =

From this we infer that Rt = RT. Moreover by (3.2),

(RTX*,X*) = .]a IIB*(I - S*(T))-~S’(s)x*ll~ds -- IIB*S*(s)((I

S*(T))-lz*)II~

ds

= (QT(I - S*(T))-~x *, (I - S*(T))-lx*). This shows that -~. RT = (I - S(T))-~QT(I - S*(T)) Because a centred Gaussian measure is determined by its covariance operator, the first equality in Lemma4.1 implies: COROLLARY 4.2 For MI t, s e [0, T], the laws of Y~ and of Yt are the same. Next we compare the laws of ]"~r and XT. V~’e start by presenting two sets of sufficient conditious for equality of the associated reproducing kernels. PROPOSITION 4.3 on H such that

Suppose that there exists a bounded linear operator SH(T) B o SH(T) = S(T) o B

and assume that I - SH(T) is invertible GT = HT. PROOFThe assumptions

in H. Then, as subsets of E, we have

imply that

B* o (I- S*(T))-’

= (I-

S~(T))-’

449

BanachSpaceValuedOrnstein-Uhlenbeck Processes Hence, for all x* E E* we have

(RTX*,X*)=- IlB*S*(s)((I - S*(T))-lx*)ll~Hds P

T

ds - Z liB*((/- S*(T))--1S*(s)z*)I[~H

= II((I S*~(T)I-IB*S*(s)z*II5 ds ~
. : I1(~- S~(T))-~II:(Q:*,x*) This implies that HTC GT. Similarly, for all x* E E* we have P

(QTx*,x*)

T

: Z [IB*S*(s)x*[~ds P

T

= Z ]l(I ~ III-

- S~(T))B*S*(s)(I S~(T)[[ ~ IIB*S*(s)(I

- S*(T))-~x*[[~ds S*(T))-lz*[I~

ds

from which we infer ~he opposite inclusion GT C PROPOSITION 4.4 Suppose that S(T) maps GT into itself and assume that the restriction of I - S(T) to GTis invertiSle in GT. Then, as subsets of E, we have GT = HT. PROOFDenote by Sa~(T) the restriction of the operator S(T) to GT. The assumptions imply that (I - S(T)) -~ .maps G~ into itself, and that the restriction to GTcoincides with the inverse of I - Sc,’r (T). Let iT denote the inclusion map from GTinto E. By general facts about reproducing kernels (cf. [DZ, AppendixB] or [Ne, Section 1]), we have the factorization QT= iTiT. Denoting by Sat(T) the restriction of S(T) to GT, it follows from the identity ~T o S*(T)x* S~r(T ) o ~Tx th at *. (I- ¢* :T~-~i* x* = i~(I S*(T))-~x Hence, for all x* ~ E* we have

*, (~ - S*(T))-~ *) (n:*, x*) : (q~(~- S*(T))-~x = IIi~’(/-S (T)) II~.~* -~ * ~ -1., ~ : II(Z-SaT(T)) * ~T~,I1~ ¯

T

--1

2 ’*

* 2

* * * T--1 II"(Q~x,:I: = I1(I - S~,,~()) ).

This gives the inclusion HTC GT. Similarly, for all x* E E* we have

~(n:rz *,x *>. (QTX*,X*) <. t1~’ * - S~(T)II from which we infer the opposite inclusion GT C HT.

¯

450

Brze~niakand van Neerven

The assumptions of Proposition 4.4 are satisfied if GTis an interpolation space between D(A) and E. In particular, they are satisfied in tim situation when H = E and {S(t)}t~>o is an analytic semigroup on E: in this case we have GT= DA(½, 2) [DZ, Appendix B]. Wealso mention the following sufficient condition for S(T)-invariance of GT [Ne, Lemma1.10]: S(T) mapsGTinto itself if there for all t e [0, T] there exists a bounded linear operator SH(t) such that B o SH(t) = S(t) and T

~

o IISg(t)lldt

< co.

This gives a link between the conditions of Propositions 4.3 and 4.4. Next we investigate under what conditions the laws of YT and X:r are equivalent. THEOREM 4.5 Suppose that S(T) maps GT into itself and that the restriction S(T) to GT is Hilbert-Schmidt. Assume moreover that the restriction of I ~ (T) to GT is invertible in GT. Then the laws of YT and XT are equivalent. PROOFDenote by VT : HT --~ GT the map defined VT(RT.T*)

----

QTX*,

by

x* ~ E*.

By Proposition 4.4 we have GT= HTas subsets of E, and therefore, by general facts concerning reproducing kernel spaces, the map Vr is well-defined and extends to an isomorphism of HT onto GT. Then by the closed graph theorem, as an operator on GT, VT is an isomorphism as well. Choose a sequence (x.~) in E* such that (h,,), with ha = RTX*n, is an orthonormal basis for GT. Then, E [l(I

- VT)ha[I~.~ = E [[(I - "T)RTXnIIG~ *

2

~*"2

....Z II(I S(T))-~QT(I Q:r. S*(T))-tx~ ,~ll~’~

= ~ ~1(~- (~ - S~(T))(~

*

n

This shows that I - VT is an Hilbert-SChmidt operator on GT if and only if the operator

* * ~ -(~ -Sa~(T))(I- S~(T)) = Sa~(T) ~,,(T) Sa~(T)S~,~(T) is Hilbert-Schmidt on GT. Nowby assumption, SGr (T) is Hilbert-Schmidt on GT, and therefore we conclude that I - VT is Hilbert-Schmidt on GT. The equivalence of the measures PT and no~v follows from the Feldman-Haj~k theorem.

BanachSpaceValuedOrnstein-Uhlenbeck Processes As an illustration

we consider the case of a self-adjoint

451 diagonal generator A:

EXAMPLE 4.6 Let E be a separable real Hilbert space. Suppose that selfadjoint operator ill E such that for some orthonormal basis {ej}jE N some sequence of real numbers Aj one has

in

A is a E and

Aej = Ajej, j E N. We further let H = E. From Proposition 4.3 (or from Proposition 4.4 and the remarks following it) it is immediately clear that GT = HT. A simple computation gives, for x = ~ xjej,

and

Hence by the Feldman-Hajdk theorem, the measure #T and vT are equivalent,

J

J

iff

J

This holds if and only if S(T) is a Hilbert-Schmidt operator on E. Since~1/2 ~T is an unitary isomorphism of E onto GT commuting with S(T), this is equivalent to S~r (T) being Hilbert-Schmidt in GT.

ACKNOWLEDGEMENT Part of this research was carried out while the first named author visited the Department of Mathematics at the University of Delft. He would like to thank the membersof the department for their hospitality.

REFERENCES 1.

2. 3.

4.

S. Albeverio, A.M. Boutet de Monvel-Berthier, and Z. Brzelniak, The trace formula for SchrSdinger operators from infinite dimensional oscillatory integrals, Math. Nachr. 182 (1996), 21~65. S. Albeverio and R. HSegh-Krohn, Homogeneousrandom fields and statistical mechanics, J. Funct. Anal. 19 (1975), 242-272. J.M. Bismut, The Atiyah-Singer theorems: a probabilistic approach. I. The index theorem, J. Funct. Anal. 57 (1984), 56-99; II. The Lefschetz fixed point formulas, J. Funct. Anal. 57 (1984), 329-348. Z. Brzeiniak and J.M.A.M. van Neerven, Stochastic convolution in separable Banach spaces and the stochastic linear Cauchy problem, submitted for publication.

452 5.

6. 7.

8. 9.

10.

11.

12. 13. 14. 16.

Brze~niakand wanNeerven G. Da Prato and L. Tubaro, Someresults on periodic measures for d[iffcrentia.l stochastic equations with additive noise, Dynamic Systems Appl. 1 (1992), 131-139. G. Da Prato and C. Tudor, Periodic and almost periodic solutions for semilinear stochastic equations, Stochastic Anal. Appl. 13 (1995), 13-33. G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensic~ns", Encyclopedia of Mathematics and Its Applications, Vol. 44, Cambridge University Press, Cambridge, 1992. G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math. 13 (1987), 213-229. J.D.S. Jones and R. L~a.ndre, A stochastic approach to the Dirac ope~ator over the fl’ee loop space, Tr. Mat. Inst. Steklova 217 (1997), Prostran. Petel i Gruppy Diffeomorf., 258-287. A. Klein and L.J. Landau, Periodic Gaussian Osterwalder-Schrader positive processes and the two-sided Markovproperty on the circle, Pac. J. Math. 94 (1981), 341-367. A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems", Progress in Nonlinear Differential Equations and their Applications Vol. 16, Birkh/iuser Verlag, Basel, 1995. J.M.A.M. van Neerven, Nonsymmetric Ornstein-Uhlenbeck semigroups in Banach spaces, J. Funct. Anal. 155 (1998), 495-535. J.R. Norris, Ornstein-Uhlenbeck processes indexed by the circle, Ann. Probab. 26 (1998), 465-478. A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, 1983 N.N. Vakhania, V.I. Tarieladze, and S.A. Chobanyan, "Probability Distributions on Banach Spaces", D. Reidel Publishing Company, Dordrecht-BostonLancaster-Tokyo, 1987.

Some Properties

of the KMS-Function

JAN A. VANCASTEREN Department of Mathematics & Computer Science, versity of Antwerp (UIA), Universiteitsplein 1, 2610 Antwerp, Belgium Telephone: +32 3 820 24 02 (department), +32 3 239 77 33 (home), Fax: +32 3 820 24 21 (department), E-mail address: vcaster©uia.ua.ac.be

Uni-

ABSTRACT Let, for j = 0, 1, Hj = Hi >_ -wj > -oc, be a self-adjoint operator in the Hilbert spaces ~j. Let T : 7-/1 -~ "H0 be a linear operator with domain in 7-/1 and range in ~/0. Let l~(t) = exp(-tHj), t >_ 0, be the strongly continuous semigroup generated by Hy, j = 0, 1. If the operators (aI ÷ Ho) Vo (to) T (aI -1 and (aI ÷ Ho)-1 T (aI ÷ Hi) 1~(to) are compact, (Hilbert-Schmidt, Trace class), then so is the operator

~o

t° Vo(u)TI~](to - u)

The result is applicable if T = ,TH~ - Ho~, where fl : "H~ ~ 7~o is a boundedlinear (identification) operator. In this case f~o Vo (u)TV~(to - u) du = Vo (to) ~ - ~VI i.e. the difference of the semigroups. Someconvergence and approximation results are presented as well. For example the operator f~o Vo(u)TV~ (to- u)du is expressed in terms of the operator toV0 (to/2) TI/’~ (to/2).

1

INTRODUCTION,

MOTIVATION

In [6] the authors proved a result of the following sort. The author is obliged to the University of Antwerp (UIA) for its material suppo[t. The author also sincerely thanks two anonymous referees for their remarks and suggestions to improve and correct the paper.

453

454

van Casteren

THEOREM Let V and W be Kato-Feller potentials on ]~" for which W- V belongs to L1 (I~). Then the scattering system (HoJrV, Ho~-W)is complete, in the sense that, for f E Pac (Ho~-W)2 ( R~), t he l imits Q:e (Ho-~V, Ho-i-W) f :: lim ei~(H°~-V)e-it(H°$W) f, f 2 (~) exist,

and that the operators ~+ (Ho-i-V, Ho-i-W) are unitary fwm

(Ho4-W)

o,

to

L:

~ ~ in L(ll~"), In order to prove this we write, with Ho = -~A Vo (to) : exp (-to (Ho~-V)), and V1 (to) : exp (-to (Ho~-W)). From our assumption that W - V is an LLfunction it follows that the operator D (to) := Vo (to) - V~ (to) is compact (in fact it is Hilbert-Schmidt, but need its compactness). In addition, W- V being an Ll-function implies that the operator toVo (t0/2) (W- V) V~(t0/2) is of trace The latter property is even more important than the compactness of the operator D (to). The problem left out by this approach is establishing a direct relationship between toVo (to) (W- V) t~ (to) and D (to). Such a link exists. To be specific,

whereU1 is a randomvariable, whichis logistically distributed: pIogistlc [Ul ~ B] --~

(coshTrT)2dr. Then toVo (to/2) (W- V) V1 (to/2) = Q (to) D (to), and, more significantly, D(to)=

lim

~

’~-*~--o J+ (-1)~(t°)~(t°V°(t°/2)(W-V)Vl(t°/2))" For this, see e.g. Proposition 4 and Theorem 11 below: the operator Ho in §2 rnay be any self-adjoint operator, which is bounded below. Duhamel’s formula shows that 7) (to)(W-V) = D(to) = Vo(to)- V1(to). 7)(t) (W- V) =

Vo(t - - V)V 2

MAIN RESULTS

Like in the abstract, let l,b(t) = exp(-tHo) and lq(t) = exp(-tH~), t self-adjoint semigroups, generated by Ho respectively H~. Here Ho and H1 are self-adjoint linear operators, which are bounded below in the respective HiIbert spaces 7/o and 7/1. Let T be a linear operator with domain D(T) in 7/~ and

Properties of the KMS.Function

455

range R(T) in 7{0. Suppose that there exists a dense subspace 7{~ of 7{-/0 together with a dense subspace 7{~ of 7{~, such that for every t > 0 the inner-products {TVI(t- s)f, Vo(s)gln o, 0 < s < t, f E 7{’1, g ~ 7{’0 can be given a meaning and that there exists a constant C(t) such that

Then there exists a continuous linear operator, denoted by 1)(QT, for which (1)(t)Tf,

g)uo= (TVl(t-s)f,

Vo(s)g)~todS,

re7{1,

If, e.g. J : 7{~ -4 7{0 is a continuous linear operator, and if T = JH1 - HoJ, where, as above, H0 and H1 generate the semigroups V0(t) and t~ (t) respectively, then 1)(t)T = Vo(t) - t’](t), i.e. 1)(t)(JH~ -HoJ) = V0(t) - Vl(t). The operator I)(to)T is defined by I)(to)T = f~o Vo(u)TVI (to - In the fol lowing theorem we present the KMS-function, and some of its properties. Wesay that the operator 1)(t)T is defined in form sense. In Theorem1 below a similar notion is in vogue. It is related to the KMS-condition in the theory of von Neumann-algebras: e.g. see Sunder [9, page 63 and following] and the remark prior to Theorem2. 1. THEOREM (a) Suppose (7, s) belongs to the strip ll~ x (-½,½), and let 7{1 -~ 7{o be a linear operator with the property that the operators Vo(to)T and V~ (to)T* are densely defined. Then, in form sense, the following identity is true: toVo (-iTto) Vo ( (~ + s) to) TVl ( (~ - s) to) to foo

cos

7r8

= ~-~_~ cosh ~r(~: = ~-~ - sin zcs Vo (to)Vo (-into)TV~ (into)da cos ~rs to Vo (-into) TV1(into) V~ (to) + ~-]~ ~ coshTr(T----- ~-~-+ sin ~rs (b) The following .identity is valid as "well: to f °~ log (cosh~r~-+l) \~ - 1 1/~ (--iTto) 1)(to)T = ~ to

log

(1)

{Vo (to) T + (to) } Vt (iTt o) dT

.z~rr 1/o (-irto) {Vo (to) T + TI/~ (to)} V~ (irto) dr.

The identity in (2) will be refined and improvedin Theorem2. It is useful, because it expresses the operator 1)(t) in terms Vo(t)T + TVI(t). SeeCorollary 3 as well. Proof. Weoutline a proof. (a) This formula follows by virtue of the following observation. The function at the right hand side of formula (1) is harmonicon the strip ll~x (-1~, ½) and it possesses boundary values toVo(to) 1/o (-i~-to) TV1(iTto),

for s = ½, and

toVo(-i’rto) TI,q (ivto) Vl

for s - 1

(4)

456

van Casteren

The left hand side is harmonic on the same strip (in fact it is holomorphic in the variable r + is there), and has the same boundary values. The holomorphyfollows from the identity Vo(-irto)tb ((s + 1/2) to) = Vo (to/2) ~ (-i (r + is)). A similar equality is valid for the semigroup I.~ (t). The uniqueness part on the existence solutions to the classical Dirichlet problemon a strip, yields the formula in (1).. (b) The identity in (2) follows upon integrating the identity in (1) with respect s (between -} and }). The equality in (3) is a consequence of a very classical identity: cosh rr + 1 = coth ~ ~rr. cosh nT -- 1 Remark1.

~Veconsiderthespace

~ox~l

togetherwith

V(ir)=(Vo~T)

~i(i~)

Define the flow A~ on B(Ho x Hl,~O X ~1) by A~(T) = V(--ivto)TV(ivto). Define for (f,g) ~ Ho x ~, and T e Y(Ho x H~,Ho ~ ~) the function Ff,9(7 + is), ~ <s< 1 by

The function FI,~ is K(ubo)-M(artin)-S(chwinger)-admissible fbr the operators V(t0) and T in the sense that it is continuous on the closed strip ~ < s <_ } ,} and holomorphic on its interior. Moreover, {r+is:r~R, -~_

Ff, a (r-~i)=to

(A~(T)V(to)(:),

If T = T’, then Ff,g (v - ½i) = FI, ~ (r + ½i). Remark 2, The family of operators t ~~ (Vo(t)V(t)T) 0 V~ (t) is a strongly

continuous

semigroupon the space 7go × 7~. hnplicitly, this fact is used in the proof of equality (2~). The usefulness of Theorem2 is to be found in the Corollaries 3 and 14. 2. THEOREM Let the hypotheses be as in Theorem 1. Suppose that a > 0 belongs to the intersection of the resolvent sets of -Ho and -Hi. The following equalities are valid:

Properties of the KMS-Function

457 e -irt°(al+H~)

- i _’m (e~rt°(al+H°) -1) -~-taIlh

(~to(aI~-.o))

- I)

(aI

+ Ho)-~TVl

((aI-~-So)-lTUi

sinh

7rT dr

- I

(to)-Uo(to)T(aI-~"l)

log cosh ~v + ~ei~o~o (aI + Ho) Vo (to) T (aI + -~ e-i ~t°~dv ~ cosh nr 1 ,i~toHo(ai+Ho)_~T(ai+H~)V~(to)e_irto~dr to f~ log coshur+l coshnr--~e + ~ to 2u

~ ~

+i -i

eirto(aITHo)

sinh ~r - [Vo (to) T (aI -1 H~) e i~t°(aI+H°) (aI + Ho)-~ TV~ (to)

~-irto(aI

+ H~

sinhnv

- IdT

+ tanh (~to (aI + Ho)) (aI + Ho)-l TV[

Proo.f. As above we write ~ (to) T = f~o ~(u)TV~ (to - u) du, Ao = aI + AI : aI + H~. Then, from the KMS-formula (see Theorem 1 (b)) it tbllows ~ (to)T

=

to ]ff cosh~ + 1 e~to~ (to)) -~t°a~dv o (Vo(to) T+ TV~ log cosh rT -- 1

1 f~, coshrr+l = -~ ~ ,og e~t°AOVo(to)TA~ 2r J_~ cosh ~r - 1 + ~1 f~

~ 0 (e_~toA~ ~

_I)

log coshur + lcosh ur-- 1 0(i~)0 (ei~toAo_i) A~TV~(to)e_~toA~dT

(integration by parts) log coshrT -- e ~t°~° Vo (to) TA~~ (e -i~t°~ - I) dr = -2~ ~ ~(iT) " (to) 0 lo (g cocosh~+l 1 (e ~*t°A° - I) A~ITI6 shnr-- 1e -i*t°A’ d~ f; ) = 2~t° j_~f~ log coshC°Sh~r+l#*t°A°A°~"b(t°)TA~i~r - 1 (e-~*t°A’ -I) 1 -i~t°A~ 1 eirt°A°Vo (to) TA~ (e - I) dT + i ~ sinh r~

f~

to f~ cosh~T + 1 (e~oAo I) A~tTA~Vi (to) + ~ log -. ~ coshnr 1 ~ sinh nr

e-i~t°A~dT

458

w~nCasteren cosh~n- + ~ -i~’t°A’dr = --~ log 2zr cosh ~rr -- ei~t°A°AoVo(’to) TA-[~e to/_ ~ to ~_: log cosh~r~-+ leiTtoAodrAoVo(to)TA[ , 2~r

cosh 7rr -- 1 1 (e iTt°A° -- I) Vo (to) 1 (e- i~t°A~ - I ) dT o~ sinh rrr

+i

+ Vo (to) TA-( 1 i

1 (e -irt°A’ -- I) dr ~ sinh ~rT to faz coshzr~- + 1 + ~ log -- eiTt°A°A~ITAxV~ (to) e-i~t°Aldr c~ cosh r’r 1 ~ l_ - ,4~-1TA1l,q (to) tO ~-~. log coshTrT cosh ~rr -+ le_irtoA~dT

- i

1 (e ~*°A° - I) AgITVI (to)(e -i~t°a’ ~ sinh 7rr

/;

- i oo sinh irt°a° ZrT ? 1 (e

- I)

- I) dr

(7)

dTA~’TVi (to).

From(7) together with the equalities:

we obtain

v_-(to) to

log cosh 7rr + leirtoA ° (Aol/b (to)TA[~ + A~-XTA,V~ (to)) e-irt°A’dr

2~r c~

cosh rrr - 1 1 (e irt°A° - I) (Vo (to) TA~x - A~xTVx(to)) (e -i’t°A~ - I) dr + i ~_~ ~ sinh rrr + tanh ( ~to (al + Ho)) (A~lTV~(t.o)

- Vo (to) ~)

-(A~lZVl(tO)-Vo(to)rA-~l)tanh(~to(a[+H1))..

(9)

This proves equality (5) in Theorem2. Another appeal to (8) yields equality 3. COROLLARY If the operators (aI + Ho) Vo (to) T (aI + -~ and(aI + Ho-x T (aI + H,) V,(to) are compact(Hilbert-Schmidt, trace class), then so is the operator 79 (to) 7[’. More-

Properties of the KMS-Function

459

over, II fot° Vo(u)TVl(to - u) I to

Here 111 stands respectively for the usual operator norm (in the compact operator case), Hilbert-Schmidt norm (in the Hilbert-Schmidt case), or the trace norm the trace class situation). Pwof. This corollary is obtained from Theorem2 (equality (6)) via the following observations: ~ ~ ~

log cosh~r + 1 cosh ~r -

=2

~dr = 1, ~ sinh ~r

in conjunction with the operator equalities eirt°A°

--

I = irtoAo

e~’~rt°A°ds, and I - e-~rt°A~ = iTtoA~

e-i~rt°A~ds,

and the inequalities: ~tanh(~toAo)~ I~land This concludes the proof of Corollary 3. The result is applicable if T = ~H~. - Ho~, where ~ : ~ ~ ~o is a bounded linear (identification) operator. In this case f~o Vo(u)TV~(to - u)du = (t0 ) ~ ~V~(to); i.e. the difference of the semigroups. Amongother things, in the sequel identity (11) below is used substantially. 4. PROPOSITION identities are true:

= g ~ (cosh.~)

N the presence of the hypotheses of Theorem 1, the ]ollowing

"

Yo(u)TV~(tou)duY~(-irto)dr.

Here T is a linear operator from "]~1 to no, and U1 is a logistically random variable: ~|ogistic [U 1 (~ B] ~-~ ~ (cosh 7rT)e ~ dr = 7r (coshTrr)3 ~

(11) distributed

1B(a)da dr.

460

wanCasteren

Remark. A not necessarily rigorous proof of Proposition 4 will be given in Proposition 10 equality (22). The method used there is based in double Stieltjes operator integrals: [1, 2, 3, 4]. Someinformation on this topic can be found in Yafaev [14], page 225-228 as well. It is not very clear under what circumstances these double Stielt.jes operator integrals are well defined. 1 Put ~(a) - 2 (cosh 2’ Another rel evant for mula is thefoll owing one. The identity in the following proposition expresses, up to bounded operators, the operator 7) (to) T - toV0 (t0/2) TV~ (to/2) in terms of operators of the form (aI + Ho)-I Tt’](to) (aI H1) an d V0(to) (a I + Ho) T (aI + -~. It wo uld be nice, if some applications can be found for this representation. It indicates that some kind of anti-commutator estimates are relevant. 5. PROPOSITION Under the assumptions of Theorem 1, the following true:

identity

7) (to) T - toVo (t0/2) TVi (to/2) =

(12)

So’S_

exp (-isato (aI + Ho)) (aI -~TV~ (t o)

L1S/ 0 (e~(~)exp(iseto(aI+Ho)-I)) 15 (to) T (aI + -~ exp(-is eto (aI + H~)) d~ + I - ~ (aI

+ Vo(to)

+ Ho) sinh

(aI

+ Ho)

to. T (aI + -1 I- ~

(aI + g0)-I TV1 (to)

(aI + Hi) sinh (aI + Hi)

Proof. From equalities (23) (with n = l) and (24) in Proposition 10 (or from in the proof of Theorem9) below we obtain

= itoE’°gisti~ [u, ~’ Vo (isU~to) (TV,(to) - Vo(to)T) V, (-isU,

ex~ (iseto (aI + H~) ) de -

~(e)ito exp (-iseto

(~I + Ho)) ~5 (to)

exp (is~o (aI + H~)) d~ = ito

is

~(~) exp (-is~to

(~I + Ho))

461

Properties of the KMS.Function

(integration by parts) =

exp (-isato (aI + Ho)) (aI -1 TV~ (to)

-/o’/:° ~ (to) T (aI + -~ exp(isa to (aI + H~)) da ds

- itoI4 (to)

e~(e) exp (is~to (M + H~)) de

In the latter expression the last 2 integrals can be calculated, and in the second integral -e replaces e to yield (12). This concludes the proof of Proposition In what follows the sequence U~,... , U,~,... denotes a sequence of independent identically distributed randomvariables, each distributed according to the logistic law, and U0 = 0. The operator ~(t0)(T)’is defined ~ (tO) (T) I°gistic [ei t°U~H°Te-it°V~H~].

462

van Ca~teren

From Proposition 4 it follows that toVo(to/2)TVl (to/2) = Q (to)[~ (to) 6. THEOREM The following (I-

identity

Q(to))’~+l [/o~VO(s)Tl~(a-

is true: ]

(la)

exp (-it~ (Uo + U~ +... + U~) g~)].

Pro@~or n = 0 the above equality is a consequence of (24) in Proposition 10. For general n employ induction. 7. COROLLARY Let all operators 4 the form Vo(t)TVl(t), (Hilbert-Schmidt, trace class). Suppose that, for t > lim

t2 >tl )O, t2¢ 0

kb(t)T

e-~’~l,’~ (s)ds =t2lira >tl

>O,

t2¢ 0

t > O, be compact

e-’~t’b(s)dsTl~i(t) 1

Then the operators ~(t)T, t > O, are compact. A similar statement is valid in the Hilbert-Schmidt (trace class) situation: Hilbert-Schmidt (trace class) norm replaces the operator norm. 8. COROLLARY Suppose that S(to) := ito l°gis~ic = it0~ 2= ito

re(r)

the

the operator ei~t°~° (TV~(to) - Vo (to) T) e-i~t°~t1~ds

e~s~°H~(TV~(to) - (~o) T) -i~t°H~ ds (sgn(r) -- tanh ~r) it°rH° (TV; (to) - Vo(to

e-itov H~dT

is Hilbert-Schmidt. Then, in Hilbert-Schmidt norm, the difference ~t° ~,~(s)TP~ (to - s)ds- ~ (~ + ~)(-l)JQ(to)J j=o +

(to,~b

(~[) )(14

converges to zero if n tends to ~. Proof. From Theorem 6 and Proposition 4 we get j=o kJ + (-1)JQ (t°)j

[toVo (to/2) TV~(to/2)]

Properties of the KMS-Function

463

= (z - Q(to))"s (to).

(15)

An argument as in the proof of Theorem2.5 in [13] yields the conclusion in Corollary 8. More precisely, let S : 7-Lt --+ 7-/0 be a Hilbert-Schmidt operator. First approximate S in Hilbert-Schmidt norm by an operator-valued integral of the form S~ := f £o(a)Vo(iato)SVt (-iato)da, where ~ is a rapidly decreasing function on I~. 0, and since II ( , Q(to)I]lgSllHS--< IlSll ~s the desired conclusion follows. 9. THEOREM Suppose that the operators (aI + Ho)-~ TV~ (to) + Vo (to) T (aI -~and

(16)

(aI + Ho)-: T (aI Hi) V:(to ) - ( aI+ Ho) Vo ( to) T (aI -1 (17) are compact (Hilbert-Schmidt, trace class). Then the operator f~o Vo(u)TVi (to - u) du is compact(Hilbert-Schmidt, trace class) if and only ifVo (½to)TV1 (½to) is so. Moreover, i] (16) v are sa tisf ied, then the Riemann approximation {2j+l,~ to~-t~ Vo < ~o) rv,(2n-2j-1) j=0

to

converges with respect to the operator norm (in the compact case), in the HibertSchmidt norm (in the Hilbert-Schmidt case), and in the trace norm (in the trace situation). Remark. From the proof it will follow that (16) and (17) may be relaxed to following assumption. For T 6 N the operator i ei~n° (TV~ (t0) - 1,~ (t0) -wn~da (int egration by p art s)

=~,~o ((a~+Ho)-’ rV,(to)+ ~4(to)r(~Z

~-i~H~

- ((aI + Ho)-’ TV, (to)+ Vo (to)T (aI + H,)-’)

+~f~~,.-o((.z+~o)-lT (aZ+m)V, JO

- (al+no)Vo(to)r(~Z+~,)-~)~-~ma~

(~8)

is compact (Hilbert-Schmidt, trace class), and converges to 0 in the operator norm (or in Hilbert-Schmidt norm, or in trace dlass norm, as the case maybe), if r tends to 0. For the proof of Theorem9 a nmnberof identities are required. They are collected in the following proposition. 10. PROPOSITION Let the hypotheses :D (to)r =

be as in Theorem 9. Put

Vo(u)rkl (to - u)

(19)

464

van Casteren

vn(to)r= t_o }2~: to ° t,--~-~~°Jrv, n-1 /’2j+l,’~ (2n-2j-1) n

(20)

"

-2-~

j=0

The following equalities are valid:

Proof of Proposition 10. A proof of (21) reads as follows (see Remark 2 prior Theorem2 as well):

An informal proof of (22) based on double Stieltjes follows:

operator integrals

reads

A proof based on complex function theory is to found in [13] and in [11]. A proof of (23) proceeds as follows:

(to)T= to ~ I’b (-~to)Vo (~ntO)Tiff 12 j=O

(2--~to)V]

(n-

~"

Properties of the KMS-Function

465

(employ (22) with to/n replacing to)

(In the ultimate equality we employedequality (21).) This shows equality (23). order to prove (24) we proceed as follows. The reasoning can be made rigorous upon replacing T by (aI + Ho)-1 T (aI + HI)-~. Wewrite e~H°~ (to) -~z~ - ~ (t o) T = i eiztl°D

(to) (HoT - TH1) -i~rz~ da t°

~iaHo

~0

Yo (u) (HoT - TH1) VI (to - u) -i¢ H~ da

ei°Z° fot° {-~u (Vo(u)TV~ (to - u))} due-~Z’da ei~z° (TV~ (to) - l~ (to) T) e-i~Sda. Proof of Theorem 9. The assertion in Theorem9 follows from the identity: Dn (to) T - D (to) = i n

(25) eiaH° (TV~ (to) - Vo (to) -i~rH~ da dr (seeformula (18))

¢(nr) ~x~

J0

-ir :n t°H’ ¢(nT){eirt°H°((aI+Ho)-l’TVl(to)+Vo(to)T(aI+H1)-l)e --((at

+ in

+ Ho)-ITV1

(to)+

Po(to)T(aI

H1 ) -1)

)d

f. f,,o (

ei~H° (aI + Ho)-~ T (aI + H~) Vi (to)

¢(nr)

~ -~

dO

7. ei"t°n° ((aI + Ho)-~ TV~ (to)+ Vo (to) T (aI + -i¢t°z’ is continuous for the operator norm, and that the same is true for the function e~aH° (aI + Ho)-~ T (aI + H~) V~

7 ~ ~0

- (aI + Ho) Vo (to) T (aI + H~)-~ e-~H~da.

466

van Casteren

Consequently equality (25) yields lim,,~ 79,, (to)T = 79(to)T for the operator norm. Another appeal to (25) for n = 1 yields the compactness of 79 (to) T. follows because 791 (to) T ~- toVo (to/2) TV1 (to/2). A similar argumentworks in the Hilbert-Schmidt and trace class context. The equality in (25) is a consequence the equalities in Proposition 10: 79,~ (to) T - ~ (to) T (see = n

~(nr)ei~°H°~ (to) -irt°H~ dr- ~ (t o)

=~

¢(n~)(~,~to-o~ (to) -~°~’- ~ (to) ~)d~(~ (~4

= in

¢(nr)

e i~t°

(Tl/i

(to)

1"? (t o) T) e-i~mdadr.

(26)

J 0

The equality in (25) then follows from (18) and (26). In the proof of Theorem 12 the following lemmawill be used. The symbol ~(~) denotes the first derivative o~ with respect to ~. A similar convention is employed for the higher derivatives of 11.

LEMMAPut

¢(a)

2 (coshna) ~" Then its Fourier" transform V) is given

by ~(~) si nh2~ . In addition

seta~(~)

=~

1-~(~)

Then the following assertions are valid. (a) The derivative of C,(~) is given

+ (n + 1)(1- ~(~))~

~(n+

(b) Moreover, ~here ezists

116 a constant C ~ ~ such ~hat, for n ~ 2,

.

Properties of the KMS-Function

467

(c) Consequently, for "r > O, and n >_2

/o

(

2~d~ (r~" +1)

T~ + 1 -

(d) For -r > 0 the following inequalities are valid: C

sin T~ d

oo

oo .

Proof. (a) The expression for C~(~) is obtained after a tedious, but straight-forward calculation. (b) Wenote that the functions ~ and its first, second and third derivative obey the following inequalities (~ > 0):

o < -~’(¢)

;

20

I;’(¢) ~ ~’(¢) + ¢, and 051-~(¢)~min ~¢,~ The validity of these inequalities 1 1 1 f(~) : -~ + ~ coth

is guaranteed by the following arguments. Put

Then 0 _< f(~) _< rain , ~ , and -~-~ _4 ~2 2f(~)~ -< 0. We also have

and, finally,

Moreover, we have sinh ~ - ~ 1 0_< ¢(cosh¢- 1) <- ~’ and

468

wanCasteren ½(cosh2~+l)+4cosh~-5-3~sinh~

<

o_<½~-si-~-::-: ~sFn~-(-~-~s~=;~o--o~.:= 3So we obtain

1

I +:(n+:)n(~ :)

(I ~

(1 ~(~))’~-~

~’(~)

~-~

: :n + :) (l - :(<)) }~’:<) +~(n+l) +:(n3

+l)n(’-:(<))~-’:(<)

:’(<) +~(n+l)(l-~(<))n

1 (i ~ "~-2

+~

1 2 (i_~(:)) + I

+ ~(~ - ~.):(:)~ + ~

~ : 16,0 < x < i) (applythe inequalities n(l-x)~’-~x ~ 2 and(n+2)~(l-x) ~"-~x

Properties of the KMS-Function

469

This proves the statement in (b), where the constant C satisfies

116 C _< -~-

(c) This inequality follows from (b) in conjunction with integration by parts. precisely the term which contains C yields the term in (c) with C in it. The second term is obtained as follows:

From the inequality

~ 4 ( ^ \ ½"+~ 1 ~ 1 - ¢(r~)) d~ _< ~ the result

/i

in (c) follows.

(d) The first inequality in (d) is obtained upon integration one is a consequence of d~ <_

3+2

l+p’

by parts. The second

p>0,

which in turn can also be proved upon employing integration

by parts.

12. THEOREM Suppose that the operator TV~ (to) - Vo (to) T is bounded (HilbertSchmidt, trace class). Then, in operator norm ’Hilbert-Schmidt norm, trace class norm) sense, the difference

j=o converges to zero if n tends to ~c. 1 Proof. Put X(r) = ~ (sgn(T) -- tanhnr). Then its Fourier transform is given

=

¯

(27)

First we notice (see Corollary 8):

(apply Fubini’s theorem) =~

(sgn(r) -- tanh ~r~-) eit°~H° (TV~ (to) - Vo (to) -it °~H’dT.

(28/

470

w~nCasteren

Put F(~-) = exp (itovHo) (TV~ (to) - Vo (to) exp(-it ovHl). We also write me.(B) = J~ V(T)dT : plogisti(’ [/l ~ B], if B is a Borel subset of R. With 5 we denote the Dirac measure at 0, and with f * #(v) = f f(7 - a)d#(a) the convolution product of the function f and the Borel measm’e p. From (27) and (23) cock, unction with (15) we infer It° Vo(s)TV~(to - s) ds -~

(~ ++~)(-i)~Q(to)~[toVo(to/~)TV~ (to/2)]

=ito (I - Q (to))" ~°gistic I° = gist it0 ’c ~ (~.) (-i)JE j=0

exp (ito

(U1 +...

+ Uj)

/

Ho)

o~+’ F(r)dr exp (-ito

= ito IX * ((f

to

~

l//(

(Uj +... + Ui) H1)]

- m~)*’] (~-)F(r)dr

1 - ~(() n+~ 1 sin(v~)d(F(r)dr

(integration by parts)

_-__to

c,,(~)

)

d~ ~(F(~)

-F(-~))d~

(integration by parts)

= -where,

c,’~(~)

)1

(29)

d, d~ ~ (r(~) F(-~)) ~,

as in Lemma 11, Cn(~) = ~ 1-

n+~ 1-5(~))

. Then by

Lemma 11 II~ t° Vo(s)TF~(to - s)ds -~=o ~

(~+l)(-i)’~Q(t°)J[t°V°(t°/2)TVl(t~/2)] 1+1

Properties of the KMS-Function

471

(30) where

Co = ~ + 2 C +

< 86. By Lebesgue’s dominated convergence the-

orem, the final expression in (30) tends to zero, as n tends to ~. This proves Theorem 12. Upon specializing following corollary.

Theorem 9 to th case where T = flH~ - Hoff we get the

14. COROLLARY The semigroup difference operators Vo(t)ff - ffVl(t), t -~ -1, are compactif and only if the resolvent differences (aI + Ho) ff - ff (aI + H1) -a E res (Ho) V~res (H1), are compact. Proof. If the operators V0(t)ff - ffl,~(t), since for a > 0 large enough,

t > 0, are compact, then the operator,

(aI + Ho)-’ ff - ff (aI + -~ = e -a S(Vo(s)ff -,T

Vl(s)) ds,

(the integral are taken in normsense) is compact. For other a with -a in res (H0) res (H~), one may employthe resolvent identity. Conversely, if the resolvent differences are compact, i.e. if the operators (aI+Ho)-~(ffH~-Hoff)(aI+H~)

-~,

-aeres(Ho)~res(H~)

are compact, then the operators (aI + Ho) Vo (to) T (aI -~ = (aI + Ho)~ Vo (to) (aI + -~ (,T H~ - Hof f) (aI + H~)- and (aI + Ho)-~ (ffH~ - Hofl) (aI + -~ (aI + H~) 2~ (to are compact. An application of Corollary 3 yields the desired result.

REFERENCES 1. 2. 3.

M.Sh. Birman and M.Z. Solomyak, Double Stieltjes operator integrals I, Problemy Mat. Fyz. 1 (1966), 33-67. M.Sh. Birman and M.Z. Solomyak, Double Stieltjes operator integrals II, Problemy Mat. Fyz. 2 (1967), 26~60. M.Sh. Birman and M.Z. Solomyak, Double Stieltjes operator integrals III, Problemy Mat. Fyz. 6 (1973), 27-53.

472 4.

5. 6.

7. 8. 9. 10. 11. 12.

13. 14.

van Casteren M.Sh. Birman and M.Z. Solomyak, Operator integration, perturbations, and commutators, Zap. Nauchn. sere. Leningrad, Otdel. Inst. Steklov (LOMI)170 (1989), 34-66. M. Demuthand S. Eder, A trace class estimate for two-space wave operators, Preprint (1999), Universit~t Clausthal, Clausthal. M. Demuth and J.A. van Casteren, Completeness of scattering systems with obstacles o]finite capacity, Operator Theory: Advancesand Applications, Proceedings of the OWOTA conference, 1995, Regensburg, vol. 102, Birkh~iuser, Basel, 1998, pp. 39-50. M. Demuthand J.A. van Casteren, Stochastic spectral theory of Feller operatots, book in preparation, Birkh~iuser Verlag, Basel, 1999 (to appear). M. Evans, N. Hastings and B. Peacock, Statistical distributions, A Wile), terscience Publication, second edition, John Wiley and Sons, NewYork, 1993. V.S. Sunder, An invitation to yon Neumannalgebras, Universitext, SpringerVerlag, NewYork, Berlin, 1987. J.A. van Casteren, Cauchy semigroups and wave operators, WarwickPreprints 64/1995 (1995). J.A. van Casteren, On differences o] sel]-adjoint semigroups, Annales Math6matiques Blaise Pascal, no. 3, vol. 3, 1996, pp. 165-188. J.A. van Casteren, On differences o] sel]-adjoint semigroups: a compactness property, Proceedings Conference Internationale de Math6matiquesAppliql~6es et Sciences de l’Ing6nicur (CIMASI), 14-15-16 November1996, pp. (19 supplementary pages). J.A. van Casteren, Compactnessproperties o] di~erences o[ sel]-adjoint semigroups, Preprint UIA(1998), University of Antwerp (UIA), Antwerp. D.R. Yafaev, Mathematical Scattering Theory: general theory, Translations of Mathematical Monographs, vol. 105, Amer. Math. Soc., 1992.

A Generalization

of the Bismut-Elworthy

Formula

SANDRA CERRAI Dipartimento di Matematica per le Decisioni, University of Florence Dimadefas, Via Lombroso6/17, 50134 Firenze, Italy

ABSTRACT We give a generalization of the Bismut-Elworthy formula, which allows us to cover a larger numberof situations than the classical one. It applies to the case of stochastic equations having the diffusion term possibly degenerate, so that the corresponding semigroup cannot have a regularizing effect. It also applies to the case of somestochastic equations both in finite and in infinite dimension whose solutions are not mean-squaredifferentiable with respect to the initial datum.

1

INTRODUCTION

Consider a general stochastic problem in a general separable Hilbert space H d~(t) = b(~(t)) dt + a(~(t)) dw(t), ~(0)

(1)

Here w(t) is a cylindrical Wienerprocess defined in a probability space (gt, $’, ~-tlP) with values in a Hilbert space U such that the injection H C U is Hilbert-Schmidt and b: D(b) C_ H ~ H and a: D(a) C_ H -~ ~.(U,H). If the coefficients b and a are such that the problem (1) adnfits a unique H-valued adapted solution ~(t;x), we can define for any 99 Bb(H) (~) and x e H Ptq~(x) = IE~(~(t;x)),

> 0.

Pt is the transition semigroup corresponding to the problem (1). Now,assume that the process ~(t; x) is k-times mean-square differentiable respect to the initial datum x ~ H, that is, for any t > 0 the mapping

with

H ~ L~(f~; H), x ~ ~(t; 1If x is a Banachspace, we shall denote by Bb(X) the Banachspace of all boundedand Borelianfunctions~o : X---r IR andby C~(X)the subspaceof all k-timesdifferentiablefunctions whichare uniformlycontinuoustogetherwith their derivatives, up to the k-th order. 473

474

Cerrai

is k-times Fr6chet differentiable. Moreover, assume that the diffusion term a is strictly non degenerate, that is a(x) is right invertible with a boundedinverse for any x ~ H and sup < ~H

Under ~hese conditions it is possible ~o prove tha~ ~he semigroup Pt has a regularizing effect, ~h~is for any t > 0 ~ e B&(H) ~ Pt~ e C~(H). Furthermore, for any ~ ~ C&(H)the Bismut-Elworthy formula holds
x)) -~ (( (s;

x) )D((s; x)

h, dw (s))H" (2

Such a formula is quite important, because it provides an explicit expression for the first derivative of Pt~ in terms of ~ and not of its derivative. For the proof in the finite dimensional case we refer to the papers by Bismut and Elworthy [1] and [6]; for the generalization to the infinite dimensional case we refer to the paper by Peszat and Zabczyk [9]. Such a formula is first proved for functions ~ ~ C~(H) and hence it is extended to all fimctions ~ ~ C~(H). Notice that, by using the semigroup law, the formula above implies the differentiability of P~, for any ~ ~ Bb(H). In applications there exist several interesting situations in which the mean-square differentiability of the solution with respect to the initial datumand (or) the strict ellipticity of a fail to be true (see for example[8], [2], [3] and [4]). In spite of that Pt has a regularizing effect in the whole space or, as in the case of degeneracy of the diffusion term a, along suitable directions. With these examples in mind, we want to give here a generalization of the formula (2), which allows us to treat a wider class of situations. Namely, we will show that if for some ~ ~ C~ (H) and x, h ~ we have

h)>, t > 0,

h>. =

for a suitable adapted process v(t; x, h) L2(0,T; L2(D)), th en th e fo llowing fo mula for the derivative of P~ holds

1.1

Notations

In the sequel we shall denote by H a separable Hilbert space, endowed with the scalar product (., ’)H and the associated normI" IH. If let.} is a complete orthonormal basis in H, we define the mappings Tn : IRn -~ H, ~ = (~1, ....

~n) ~ Z ~iei, i=1

and

H,~: H-+ IR~,

x ~ ((X,

el)H ,...,

(X, en)H).

Generalizationof the Bismut-Elworthy Formula

475

We denote by Bb(H) the space of all mappings ~ : H -} IR which are bounded and Borelian. Bb(H), endowed with the norm II~ollo : Sllp [qO(X)l ,

is a Banachspace. Cb(H) is the subspace of all uniformly continuous functions. For any k > 1, C~(H) is the subspace of Cb(H) of all functions qo which are k times Frfichet differentiable, with boundedand uniformly continuous derivatives up to the k-th order. If we define [~]h= suplD~qo(z)[,

h=l,...,k,

a: E H

then C~(H), endowed with the norm k h=l

is a Banach space.

2

THE GENERALIZED

BISMUT-ELWORTHY

FORMULA

For any ~o ~ Bb(H) and x ~ H, let us define Pt~o(x) IE~o(~(t; z) ), t

>_0,

where ((t; x) is the unique solution to the problem (1). Our aim is providing expression for the derivative of Pt~o, along suitable directions, when ~o e C~ (H). Such an expression has to involve only’ ~o and not its derivative. THEOREM 2.1 Assume that .for ~o ~ C~ (H) and t >_ 0 the ]unction P~o is differentiable. Moreover, assume that for t > 0 and ]or some fixed x, h ~ H, it holds
(a)

where v(t; x, h) is a suitable adapted process such that o’ Iv(s;

x, h)lds ~ < oo.

(4)

Then, for the same x, h ~ H we have
71E~(((t;x)) (v(s;x,h),dw(s)),.

PROOFLet us fix ~o 6 C’~(H), such that for any x ~ H Tr [DZ~o(x)]

(5)

47{i

Cerrai

According to the It6’s formula it is possible to prove that ~(((t;x))

Pt H. ~p(x) + (D(Pt-~)(((s;x)),a(¢(s;x))dw(s)}

Thanks to (4) we can multiply each side by

(v(s; x, dw(8)) H and, by ta king

the expectation, we get E~(¢(t;x))

{V(s;x,h),dw(s)) u

Due to (a), this implies that

so that (a) follows. Now,if ~ e C~(H), there exists a sequence {~,~} C C~(H) such that Tr [D~(x)]

sup II~ll~

< ~ and

<~

and such that, for any x, h ~ H, ir holds lim ~,~(x) = ~(x),

(6) lim (D~n(x),h) u = (D~(x), n~+~ Actually we can define for each n ~ ~

where {Pn} is a sequence in C~(~.’*) such that supppn

C {~ ~ ~; ~] ~ l/n}

and

fn.p,~(~)d~=l.

Now, as proved before for each n ~ N we have

and then, due to (4) and (6), by the dominated convergence theorem we n~+~

Generalizationof the Bismut-Elworthy Formula

477

On the other hand, for each n ~ ~N we have (D(Ptc;,~)(x),h)~r

= ~lE~.,,(~(t;x))

.H (v(s;x,h),dw(s))

Thus, by using once more (4) and (6), from the dominated convergence theorem have lim (D(Pt~n)(x),h)~= This implies that (5) holds, for any

3

APPLICATIONS

The formula (5) coincides with the classical Bismut-Elworthy formula, when a strongly elliptic and the solution [(t; x) is mean-squaredifferentiable. As a matter of fact, in this case, since ~(t; x) is mean-squaredifferentiable with respect to x, can differentiate under the sign of integral and we get (D(Pt~)(x), h)t I = IE (D~(~(t; x)), D~(t; x)h)H. Thus, since a is invertible, if we define v(t; x, h) = -~ (~(t; x) ) D~(t; x due to the strong ellipticity (5) coincides with (2).

3.1

It6 Stochastic

of a, the process v(t; x, h) fulfils

Equations

Solvable

by Euler’s

(4) and the formula

Method

In the book by Krylov [8J-chapter V, it is presented a class of finite dimensional stochastic equations which are solvable by Euler’s method and whose solutions ((t; x) are not mean-squaredifferentiable, even if the coefficients b and a are smooth. This means that in order to get the derivative of Pt~, for some ~ ~ C~(IRd), it is not possible to proceed directly by deriving under the sign of integral. Nevertheless, in [8] (see Section 7 of Chapter V) it is proved that if ~v e 1 (I d) the function Pt~ is differentiable and for. any x, h ~ IRd it holds (D(Pt~2)(x),h)~.~

= IE(D~(~(t;x)),~lh(t;x))~x~

> O,

(7)

where r/~(t; x) is the solution of the first variation equation corresponding to the problem (1) d~(t) = Db(~(t; x))~?(t) dt + Da(~(t; x)),~(t)

~(0) =

(8)

Krylov proves that under the general hypotheses given for the coefficients b and a, it holds

fo

This yields

ds

t >_0.

(9)

479

Generalizationof the Bismut-Elworthy Formula

In [4] are given several examples in which the above conditions are verified. Moreover it is prove that under the above conditions the following result holds. PROPOSITION3.2 For any x E IR d, h ~ IR r and t >_ O, there dimensional random variable vh(t; x) such that D((t; x)a(x)h = a(((t; and

t >_ o,

fo~" a suitable continuous increasing function c(t). dimensional randomvariable uh(t; x) such that

(~6)

Moreover, there exists a r-

D:((t; x)(a(x)h, a(x)h) = a(((t; x) )uh(t; sup IEluh(t;x)l

a r-

(15)

x))vh(t; x), IP -

2 < c(t)lhl ~, sup~[vh(t;x)l

and

exists

~ <_ c(t)lhl

]P - a.s.

(17)

~, t >_ 0

(18)

for a continuous increasing function c(t), possibly different from the one introduced in (16). If ~ e C~ (IRd), by deriving under the sign of integral, for any x ~ IFt d, rh ~ IR and t >_ 0 we have (D(Pt~)(x), a(x)h)~e = IE (D~p(~(t; x)), D((t; x)a(x)h)~, = ]E (D~(~(t; x)), a(~(t; x))vh (t; Thus, due to (16), the hypotheses of the theorem 2.1 are satisfied following result.

and we have the

PROPOSITION 3.3 If the coefficients b and a verify the conditions described above, for any ~2 ~ C~ (IR~) the function Pt~o is differentiable along the directions a(x)h, with x, h ~ u, and it holds 1

f

t

(D(Pt~)(x), a(x)h)~ = ~]E~(~(t; x))~o/ (vh( s; x), dw(s))~ This meansthat it is possible to give an explicit expression for the derivative of Pt~o along the directions of a, in terms of ~o and not of its derivative. In particular we have

supI(D(Pt~o)(x), b(x))a.I= supI(D(Pt~o)(x), a(x)~(x))a.l 1)-:/~]1~Ollo.

xE lR,d

x~ ltfl

Wecan proceed similarly for the second derivative and tbr arty 99 i = 1,...,r we get sup I
~

C’~(IPtd) and "

480

Cerrai

Therefore, if we denote by .A0 the diffusion operator corresponding to the equation

due to the ItS’s formula and the above estimates, we have that for any ~ ~ d) C~(]R and x d~ 11~, sup xE ~(d

=

sup

_<

By proceeding as in [4], this yields the analiticity the degenerate operator ~lo in the space Cb(IRd).

3.3

1)-11110.

XE ~,t

Stochastic Reaction-Diffusion Having Polynomial Coefficients

of the semigroup generated by

Systems

The theorem 2.1 can be applied also in the infinite dimensional case, to the study of the transition semigroups corresponding to some stochastic partial differential equations. Weshow how in the following example. Consider a stochastic reaction diffusion system in a bounded regular open set ~D C IRd, with d _< 3,

(19)

For each k = 1,... ,n, Ak(~,D) is a uniformly elliptic coefficients given by

operator with regular real

and B~(~, D) is a linear operator acting on the boundary of 79, which may be taken either equal to the identity operator (Dirichlet boundary condition) or equal to the conormal derivative (conormal boundary condition). The function f = (ft,..., fn) : ~× IRn ’-~ lRn is continuous. For any fixed ~ ~ the function f(~, .) : " -~IR’~ is of cla ss Ca, has polynomial growth toge ther with its derivatives and fulfils suitable dissipativity conditions (see [2] and [3] for all details). B~ are non negative bounded linear operators from ~(79) i nto i tself a nd C02Wk/COt Cq~ are independent space-time white noises defined on a probability space (~, ~:, ~:t, ~’). Whend > 1, the operator B = (B~,...,B,~) cannot be taken invertible with a bounded inverse. Then we have to consider degenerate diffusion terms, in general. Nevertheless, the domain of B-1 has to be not too small. Namely, if we denote by

Generalizationof the Bismut-Elworthy Formula

481

.4 the realization in L~(T~; IR’~) of the operator A = (A1,..., A,~) with the boundary conditions/~ = (/~1,...,/~,~), we assume that there exists e < 1 such that D((-A) ~/2) C D(B-I).

(20)

As proved in [5] (see also [2] and [3] for more details) for any x E H = L2(T~;n) the system (19) admits a unique weak solution u(x) ~ C([0, +~];H), ~-a.s. z ~ E = C(~; ~’), then u(x) is a mild solution and u(x) e C((O, +~); E), ~-a.s. Moreover, if x ~ H and {xn} C E is a sequence which converges to x in H, we have that lim U(Xn) = u(x), in C([O,T];H), ~-a.s., for any fixed T > 0. As proved in [2], for any ~ ~ 0 the mapping E ~ L~(~}; E),

x ~ u(t;x)

is Fr~chet differentiable. This implies that for any x, h ~ E there exists the meansquare derivative of u(x) along the direction h, which we denote by Du(t; x)h. Due to the condition (20), it is possible to showthat for any t > Du(t;x)h

~ D(B-~).

By deriving under the sign of integral, for any ~ ~ C~ (E) it holds D( Pt~)(x) . h = ~ D~(u(t; x) ) . Du(t; Now,in [3] it is shownthat if x, h e H and {x,~} and {h~} are two sequences in E converging respectively to x and h in H, then the sequence {Du(x,~)h,,} converges to a/-/-valued process v(t; x, h) which belongs to D(B-~), ~-a.s., for any t > 0 and such that ~H

Thus, if we fix ~ ~ C~ (H) and z, h ~ H, and if {z~,} and {h,~} are two sequences in E converging in H respectively to x and h, from the dominated convergence theorem we easily have li~n D(P~)(x~.).

h~ = ~D~(u(t;z)) . v(t;x,h).

Since P¢~ ~ C~(H), for any ~ ~ B~(H) and t > 0 (see [g]),

we can conclude that

= IE (D~o(u(t; x)), B(B-iv(t; Therefore, due to (21), we can apply the theorem 2.1 a~ad we can conclude that PROPOSITION 3.4 The transition semigroup P~ associated with the system (19) maps Bb(H) into C~ (H), for t > O. Moreover the following Bismut-Elworthy formula holds (D(Pt~o)(x),

h)H = 1-1E t ~o(u(t;x))~0

(v(s;z,h),dw(s)),.

482

Cerrai

REFERENCES 1.

2.

3.

4.

5. 6. 7. 8. 9.

J.M. Bismut. Martingales, the Malliavin Calculus and Hypoellipticity General HSrmander’s Conditions. Z. Wahrscheinlichkeitstheorie verw. Gebiete 56: 469505, 1981. S. Cerrai. SmoothingProperties of Transitiot~ semigroups relative to SDE’swith Values in Banach Spaces. Probability Theory and Related Fields 113: 85-114, 1999. S. Cerrai. Differentiability of Markov Semigroups for Stochastic ReactionDiffusion Equations and Applications to Control. Stochastic Processes and their Applications 83: 15-37, 1999. S. Cerrai. Analytic Semigroups and Degenerate Elliptic Operators with Unbounded Coefficients: a Probabilistic Approach. Preprint Scuola Normale Superiore of Pisa, 1998. Submitted. G. Da Prato, J. Zabczyk. Stochastic Equations in Infinite Dimensions. Cambridge: Cambridge University Press, 1992. K.D. Elworthy, X.M. Li. Formulae for the Derivatives of Heat Semigroups. J. Functional Analysis 125:252-286, 1994. A. Fried~nan. Stochastic Differential Equations and Applications. NewYork: Academic Press, 1975. N.V. Krylov. Introduction to the Theory of Diffnsions Processes. Providence: American Mathematical Society, 1995. S. Peszat, J.Zabczyk. Strong Feller Property and Irriducibility for Diffu~,~ion Processes on Hilbert Spaces. Annals of Probability 23: 157-172, 1995.

Dirichlet Operators for Dissipative Gradient Systems G. DAPRATO Scuola Normale Superiore di Pisa, piazza dei Cavalieri 7, 56126, Pisa, Italy

1

NOTATION

AND SETTING

OF THE PROBLEM

Weare given a real separable Hilbert space H (norm I" I, inner product (.,.),) selfadjoint strictly negative operator A : D(A) C H -~ such th at A -1 is of tra ce class, and (Ax,x) <_ -wlxl 2, x E H, (1.1) for some w > 0, and a convex proper lower semicontinuous mapping U : H -+ [0, +oo]. For any x E H we denote by E(x) the subdifferential of U : E(x) = {z e H : U(x + y) >_ U(x) + (y,z), and set D(E) = {x 6 H: E(x) ¢ It is well known that D(E) is not empty and that E(x) is convex closed for any x ~ D(E). For any x 6 D(E) we denote by Eo(x) the element from E(x) having minimal norm. Weshall assume that

/He-2U(z)#(dx) > O,

(1.2)

where # is the Gaussian measure Af(0, Q) on H with mea,p 0 and covariance operator -1 Q=-½ . A Weare concerned with the following operator N0%o= L~ - (cgU, D~) where OUis the subdifferential

of U, and L is the Ornstein-Uhlenbeck operator

1 Lv = ~ Tr

[D2~]+ (Ax, 483

484

Da Prato

The main goal of the paper is to show that No (with a properly defined domain) is essentially self-adjoint in the space L2(H, ~) where ~ is the Borel measure on H defined as ~(dx) = p(x)#(dx), and

e-2U(.~.) p(x) = fH e-2V(~)#(dy) := ae-2u(’~:)’ x E

Let us denote by N the closure of No and by Pt = etN, the semigroup generated by N. Weshall showthat the semigroupPt is strong Feller, that is that Pt~ is Lipschitz continuous for any bounded and Borel function ~. Finally we shall characterize the domain of N and, following [11], we shall study asymptotic properties of the semigroup P, = e tN, as ergodicity, strongly mixing and spectral gap. The present paper is a generalization of [9], where U was supposed to belong to W~’P(H, p) for some p > 2. The main novelty is that we do not require here any regularity of U except convexity, and so the subdifferential OUcan be multi-valued. Wenotice that N is naturally associated with the differential stochastic inclusion dX - (dX - OU(X))dt-

dW(t) (1.3)

x(o) that was intensively studied by variational methods, see [14], [2], [1]. In the finite dimensional case see [3] and references therein. However, our assumptions are weaker, and we are not able to solve directly problem (1.3). ~’4 (H, p), the ~Ve notice that when U is not necessarily convex, but belongs to W problem of essential self-adjointness of N0 was studied in [15] and in [12]. Weconclude this section by recalling some well knownresult about the OrnsteinUhlenbeck semigroup, see e.g. [13]. Wedefine (1.4) nt~(x) = ],, V(et~x + y)p,(dy), ~ ~ Cb(H)(~), where p, = ~(0, Q,) and 1 A_~(1 _ e~,m). The infinitesimal generator L of R~ is defined as in [4] through its resolvent R(A,L)~(x)

e-~R~(z)dt,~

e C~(H).

Werecall, see [13] that for all ~ e Cb(H) and for all A > 0 we have R(A,L)~ C~ (H) and the following estimate holds

II 10,z eH.

(1.5)

llf H and K are Hilbert spaces we denote by Cb(H; K) the Banachspace of all continuonsand boundedmappingsfrom H into K, endowed with the sup norm~[ ¯ ][o. Moreover,for any k ~ N, C~(H;K) will represent the Banachspace of all mappingsfrom H into K, that are continnous andboundedtogetherwiththeir Fr~chetderivativesof order less or equal to k endowed with their natural norm[]. ~[. If K = ~ we set Cb(H;K) = Cb(Y)and C~(H;K) = C~(H).

Dirichlet Operatorsfor Dissipative GradientSystems

485

¯ Wewill use the fact that the semigroup Rt also acts on functions with polynomial growth, see [5]. In particular we are interested to consider Rt on the space Cb,I(H) consisting of all continuous functions V : H -+ l~ such that sup

~ < +~.

¯ eH1 +Ixl

We shall denote by D(L~) the doInain of the generator of Rt on Cb,I(H) again defined through its resolvent.

2

CONSTRUCTION

OF THE SELF-ADJOINT

EXTENSION

OF N

Weshall assume, besides (1.1) and (1.2), /’HIEO(x)Ie#(dx)< +eC, u (D----(~)

(2.1>

=

where D(E) is the closure of D(E). Then we consider the operator No on L2(H, v) defined as NoV = LV - (Eo, D~p), V E n(No), where n(Uo) = C~(H) ~ D(L,). This definition is meaningful because

and [j,(1

+ )y])2~(dy) <

Our aim is to show that (i) No is symmetric on (ii) the image of ~ - N0 is dense on L~(H, ~) for all A > 0. By a well knownresult of Lumer-Philips, this will imply that the closure N of No is self-adjoint. To perform this program it is useful to introduce an approximating problem. First we introduce the Yosida approximations U~, ~ > 0, of U : U~(x) = inf U(y) +fx - y[2~,YeH

}¯

It is well knownthat U~ ~ C~ (H) and Ea :’= DU~is Lipschitz continuous. Moreover for any x ~ D(E) we have, see e.g. [7], that lim E~(x) = Eo(x).

(2.2)

486

Da Prato

Weshall assnme that E~ is differentiable. by introducing another approximation

This restriction

E~,~(x) =/H e~AE~(e~Ax + y)#~(dy), Nowwe consider the differential

can be easily removed

~ > 0,8 >

stochastic equation

{

dX~ = (AX~ + Ea(X~))dt

+ dW(t) (2.3)

x~(o)=

Since E~ is Lipschitz continuous problem (2.3) has a unique solution X~(t, x), see [13], moreover X~(t, x) is differentiable and the following estimate holds

IIDX~(t,x)[I <_-~t, t>_ o. Weconsider the transition

(2.4)

semigroup

P~o2(x) = E[o2(X~(t, x))], C~(H), x E H , > O. Wedenote by No. the infinitesimal generator of Pt ~, on C~(H)again defined throngh its resolvent. Wegive now some regularity properties for the solutions of the equation, ~ -

N~ = f,

(2.5)

where f ~ Cb(H) and A > 0. LEMMA 2.1 Let f ~ C~ (H) and A > O. Then equation (2.5) has a unique solut.ion ~ ~ C~ ( H). Moreover the following estimate holds 1 < ~ H/Ill. [1~11~’_

(2.6)

Finally ~o~ ~ D(L1) and )~ - L~ + (E~,Dpa) PROOFSince E~ is differentiable (D~(z), h) =

=

(2.7)

we havc for h ~ H,

e-~tE[(Df(X~(t,

x), x)h)] dt.

(2.8)

Moreover,by (2.8) it follows, taking into ’account (2.4) 1 I(~(x),h)l Il YlI~ Ihl, h e H, that yields (2.6) for the arbitrariness of h. The last statement follows since {E~, D~) has a sublinear growth. I Finally we denote by ~ the Borel measure on H

~(d~.) =~(x)~4d~’),¯

Dirichlet Operatorsfor Dissipative GradientSystems

487

where p~(x)

LEMMA2.2

e-2Uo = fH e-2U~(v)#(dY)

:= a~e-2U°(:r)’

x ~

N~ is symmetric on L2(H, va). Moreover No~o¢du~ = - ~
PROOFWe first

recall

(2.9)

that 1

Nowif ~ E C~ (H) we have

= - ~ (D~, D log p~ +

=- 2

PROPOSITION 2.3 PROOF

D~, De)d,c.

No is symmetric on L2(H, v).

Let ~, ¢ e D(No). Since

we have lim am/HN~99¢e-2U°"~)#(dx) = a /H N°~P¢e-2u(’~’ #(dx)’

a.-+O

by the dominated convergence theorem. We have moreover lira

f (D~,D*)dv~ : lira

a~O J H

~0

a~ f (D~,D~)~(dx). J tl

Nowthe conclusion follows from (2.9). Weprove now the main result of the paper. THEOREM 2.4 Under assumptions 1.1,.1.2, on L~(H, ~). WeshM1set P~ = du, t ~ 0.

and 2.1, No is essentially

self-adjoint

488

Da Prato

PROOFSince No is dissipative, we have only to show that (A - No)(D(No)) is dense on L2(H, it) for A > 0. Let f E C~(H) and let ~o~ be the solution to (2.5). By Lemma2.1 we know that ~,~ e D(No) and Acp~ - No~ = f + (Eo(x) - E~(x),Dcp,~>. It follows

f

(2.10)

HI(Eo(x) Ea(x)

1

<- X allfll~

lEo(x) - Ea(x)l’2e-2uO’)dp

< X @fill lEo(x)

(2.11)

- E~(x)12dl~

The following result is useful to showthat the semigroupPt is strong Feller. LEMMA 2.5 Let f ~ Bb(H) and let

~ be the solution

to (2.5).

lira ¢4 = R(~, N)f.

o~---#0

PROOF we have

Let first f ~ C~ (H) and let ~oa be the solution to (2.5). Then by (2.10) ~ = R(A, N)f + R(A, N) [(Eo(X) E, ~(x), D~)].

Then the conclusion follows from (2.11). Let now f 6 B~(H), and let {fi,} be a sequence in C~(H) pointwise convergeut to f such that ][f,,[]0 ~ ~]f~]0. Set ~ = R(A,N)/, ~ R(A,N~)f, ~, = R(A,N)f,, ~.,~ = R(A, N~)fn. Weclaim that ~ ~ ~ in L~(H, v). Wehave in fact

But

afH

afH

Consequently, for all ~ > 0 there exists ’n~ such that

Dirichlet Operatorsfor Dissipative GradientSystems

489

for all n > n~. Nowby (2.12) we have

Nowthe claim is proved by recalling the first part of the proof. 1 Wecan prove now the strong Feller property. THEOREM 2.6 The semigroup Pt, t > 0 is strong ~ E Bb(H) we have

Moreover for any

Feller.

lPt~(x) Pt~o(Y)I _< t-1/21z- yll l~ll0. PROOFBy Lemma2.5 and the Trotter.-Kato

(2.13)

theorem we have

lim P~ = Pt~, V (p e L2(H, v). o~-.,, 0

Thus there exists a sequence aj -~ 0 such that the limit above hods ~-almost surely. Nowrecalling [10] we have

~ almost surely. |

3

CHARACTERIZATION

OF THE DOMAIN OF N

Westart with an integration by parts formula. Let {eta} be a complete orthonormal system on H and {Ak} a seqnence of positive numbers such that Qe~ = A~ek, k ~ N. Let xa = (x, e~), k 6 N, and let D~ be the derivative on the direction e~,. Let finally ~(H) be the linear span of all exponential functions x ~ i(h’’~), h~ H.Clearly $(H) C D(No). The following result is easy to check. LEMMA 3.1 For any ¢p,¢ ~ $(H) we have 1 fI~ Dh~¢dv = - fft

q~Dh,~dv + 2 /H (E0(x),

eh)q~¢dv + ~h /Hxh~¢dv’ (3.1)

Wedefine now Dh on ~(H) as the p~rtial derivative following result is a standard consequenceof (3.1). PROPOSITION3.2

Dh

on the direction

eh. The

is closable.

Wecan now define spaces W~"O(H,v), k = 1,2, in the usual way. Wealso set W~’2(H,p)= {qoE W~t’2(H, tt) Nowby [8] we obtain the result.

f. I(-

A)~/2Dqol’2dv< +c~.}

490

Da Prato For any +2,~b E L2(H,~+) we have

PROPOSITION3.3

N+2¢du = - -~ (D+2, D~p)du.

(3.2)

Finally, proceeding as in [8] the following result can be proved. THEOREM 3.4 Under assumptions 1.1, D(N) = {~ ~ W~’~(H,p)

4

ASYMPTOTIC

1.2,

~ W~’~(H,,)

BEHAVIOUR

and 2.1,

we have

: ~(Eo(x)D~,D~d~

(3 .3)

OF Pt

For any +2 E L2(H, u), we shall denote by ~ its mean:

Westart with an useful identity that follows easily from (3.2). PROPOSITION4.1 Let +2 ~ L2(H,u), +2 ~ L2(H, ~+) we have

and set u(t,x)

= Pt+2(x).

Then for

1

i.. ,.,,,x,,’.,’., +io S. ,’.,’,x,,’,.,’., :i.. ,.,..,,’.,’ Nowwe study strongly mixing of ~,. Werecall that ~, is said to be strongly mixing if lim (Pt+2,@) = ~, for all +2,~,b ~ L2(H,u).

t --~ c~

(.4.2)

Obviously strongly mixing implies ergodicity of ~,. The following result can be proved as in [11]. PROPOSITION4.2 ~ is strongly

mixing.

Moreover

lira Pt+2 = ~, in L2(H, u).

(.4.3)

PROOF It is enough to show (4.2) for +2 D(N~). Inthi s cas e let us s et u(t) = Pt+2, v(t) = Dtu(t) By (4.1) we have that

II~(t)ll~ + IlVu(s)ll2ds = 11+2112, t _>0,

(4.4)

and

2, t _>0, IIv(t)ll2 ÷ [{Dv(s)ll2ds IIN+2]I

(4.5)

Dirichlet Operatorsfor Dissipative GradientSystems

491

whereII"II denotes the normon L’2(H, v). ConsequentlyIlu(’)llis nonincreasing and

+~IIDu(s)ll2ds

IIDv(s)ll2ds < ÷o~.

(4.6)

Set ~(t) -- IIDu(t)ll ~, t _> o. Then a is differentiable

and we have

a’(t) = 2{Du(t), Dv(t)} <_IIDu(t)ll 2 ~. + IIDv(t)ll By (4.5) it follows that c~ ~ W1’1(0, +o~), so lira ~(t)= lim ]lDu(t)[[~=

(4.7)

Let now t. ~ +~ be such that u(t,) converges weakly to some function ¢ L~(H, v). Then by (4.7) it follows that Du(t~) ~ in L~(H, v). Since D is closable ¢ is constant and coincides with ~. This implies that u(t) converges weakly to ~ when t ~ +~ in Le(H, v). The last statement follows easily from the symmetricity of Pt. 1 Weprove finally spectral gap. For this it is enough to show a Poincar~ inequality that we prove as in [11]. PROPOSITION4.3 For ay ~ ~ L~(H,~) we have

/H ’~(x)

- ~’~v(dx)

(4.s)

_ ~ I~(x)l~(dx). 1 ~

PROOFLet p ~ L~(H, ~). Then taMng into account identity

(4.1) we find,

1 d

(4.9) = -~2 IDPt~(x)lUv(dx)

>- -~ IDPt~(x)12v(dx)"

Recalling (2.5) we have for any ~ >

~ : I~(DX~ ~ IDP~(~)I (t, ~)D~(X~(t, ~ [~(DX~(t, x)l 2 I~(D~(X~~t, x))[2 Letting a tend to 0, we have iDP~(z)i ~ ~ ¢-~Pt(ID~i:)(~). Integrating in H and taking into account the invariance of u we find

492

Da Prato

Nowby (4.9) it follows

l2 dtd fH IPt~(x)l~’(dx)

l

> --~

ID~(x)l’2~’(dx)"

Finally, recalling (4.5) and letting t --+ +o0, we conclude the proof.

REFERENCES

5. 6. 7.

11. 12. 13.

14. 15.

16.

V. Barbu and A. Rascanu, Parabolic variational inequalities with singular imputs, Differential Integral Equations, 10, 67-87, (1997) A. Bensoussan and A. Rascanu, Stochastic variational equations in infinite dimensional spaces, Numer. Funct. Anal. Optimiz., 18, 19-54, (1997) E. Cepa, Multivalued stochastic differential equations, C.R. Acad. Sci. Paris, Ser 1, Math. 319, 1075-1078, (1994). S. Cerrai, A Hille-Yosida theorem for weakly continuous semigroups, Semigroup Forum, 49, 349-367, (1994). S. Cerrai, Elliptic and parabolic equations in I~’~ with coefficients having polynomial growth, Comm.Partial Diff. Eq. 21 (1 & 2)~ 281-317, (1996). G. Da Prato, Applications croissantes et (!quations d’6volutions dans les espaces de Banach, Academic Press, (1976). G. Da Prato, Regularity results for Kolmogorovequations on L~(H, #) spaces and applications, Ukrainian Mathematical Journal, m.49, n. 3, 448-,157, (1997). G. Da Prato, Characterization of the domainof an elliptic operator of infinitely many variables in Lz(#) spaces Rend. Mat. Acc. Lincei, s.9, v. 8, 101-105, (1997). G. Da Prato, The Ornstein-Uhlenbeck generator perturbed by the gradient of a potential, Bollettino UMI,(8) I-B, 501-519, (1998). G. Da Prato,D. Elworthy and J. Zabczyk, Strong Feller property for stochastic semilinear equations, Stochastic Analysis and Applications, vol. 13, n.1, 35--46, (1995). G. Da Prato and B. Goldys, Invariant mesures of nonsymmetric dissipative stochastic systems, S.N.S. Preprint n.1, (1999). G. Da Prato, and L. Tubaro, A new method to prove self--adjoiutness of some infinite dimensional Dirichlet opera,or, submitted, (1998). G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, London Mathematical Society Lecture Notes, n.229, Cambridge University Press, (1996). U. G. Haussman and E. Pardoux, Stochastic variational inequalities of parabolic type, Appl. Math. Optimiz. 20, 163-192, (1989). V. Liskevich and M. RSckner, Strong uniqueness for a class of infinite dimensional Dirichlet operators and application to stochastic quantization, MSRI preprint No. 1998-005, (1998). Z. M. Ma and M. RSckner, Introduction to the Theory of (Non Symmetric) Dirichlet Forms, Springer-Verlag, 1992).

Generators of Feller of LP-sub-Markovian

Semigroups as Generators Semigroups

NIELSJACOBUniversit~t der Bundeswehr - Miinchen, Fakult~t fiir hfformatik, Institut fiir Theoretische Informatik und Mathematik, Werner-Heisenberg-Weg39, 85577 Neubiberg, Germany

0

INTRODUCTION

Generators A(2) of L2 - sub - Markoviansemigroups are characterized to be Dirichlet operators, see N. Bouleau and F. Hirsch [1] and Z.-M. Ma and M. RSckner [12], i.e. they satisfy (A(’~)u)((u 1)+) dx<_ O, (0.1)

/

whereas generators A(~) of Feller semigroups have to satisfy the positive maximum principle: u(xo) = sup u(x) _> 0 implies A(~)u(xo) < O. (0.2) For operators satisfying the positive maximumprinciple we have well-known structure results by Ph. Courr~ge [2]. Unfortunately such results are unknown for Dirichlet operators. In various situations it was however possible to construct (non-symmetric) sub-Markovian se~nigroups by starting with Feller generators, see W. Hoh [4]-[6] and [7]-[9]. The purpose of this note is to extend the underlying abstract principle of these considerations to LP-spaces. In Section 1 we handle the LP-analogue of Dirichlet ope;ators, Section 2 gives the relation of Feller generators and LP-Dirichlet operators. The results in Section 2 are our main purpose. For this we introduce the notion of LP-Dirichlet operators in Section 1. A brief commenton the origin of these results seems to be in order. Westarted these considerations about two years ago when working on a book manuscript [10] and it seems to us that these are rather straightforward generalizations once knowing the L-%case and for example the monograph of N. Varopoulos, L. Saloff-Coste and T. Coulhon [16]. Later we learnt of rather close results due to 493

494

Jacob

E.-M. Ouhabaz [13]-[14] and in discussing these results with him in Oberwolfach in July ’98 it was clear that he was in principle also well aware of these generalizations. At the same time we learnt about a characterization of generaters of LP-sub-Markovian semigroups due to V. Liskevich and Yu.Semenov[11]. Further, in the middle of the year 1998 I received the thesis of A. Eberle [3]. In this thesis he also used the Lp -variant of Dirichlet operators for abstract diffusions admitting a carr~ -du-champ. This situation shows that there is ~nuch interest in an pL theory of Dirichlet operators and therefore we made up our mind to include these results. Thus we consider Section 1 as an exposition to results (partly) known and obtained independently by several colleagues in different situations.

1

SUB-MARKOVIAN

SEMIGROUPS

ON LP

In the following Lp stands for LP(R"; R), 1 < p Westart with DEFINITION1.1 A. A linear bounded operator S : Lp ~ L~ is called subMarkovian whenever 0 <_ u <_ 1 a.e. implies 0 <_ Sn <_ 1 a.e. B. We call S positivity preserving if 0 <_ u a,e. implies 0 <_ S u a.e. C. A strongly continuous contraction semigroup (Tt)t>_o on ~ i s c alled s ub-Markovian semigroup if each of the operators Tt is sub-Markovian. If each Tt is positivity preserving we call (Tt)t>_o positlvity pr eserving se migroup. The following lemmasare easily seen: LEMMA 1.2 A sub-Markovian operator is positivity preserving, Markoviansemigroup is also a positivity preserving semigroup.

hence a sub-

Let us denote by (R~)~>0the resolvent corresponding to (T~)t>0, i. R~u =

e-~Ttudt.

(1,1)

LEMMA 1.3 A strongly continuous contraction semigroup (Tt)t,>o on L~ is subMarkovian if and only if for any A > 0 the operator AR~is sub-Markovian. It is positivity preserving if and only if Rx is for all A > 0 positivity preserving. DEFINITION1.4 ~Ve call (R~)x>0 a sub-Markovian resolvent any A > 0 a sub-Markovian operator. PROPOSITION 1.5 vfor all u ~ L

if AR~ is for

Let S : L~ - L~ be a sub-Markovian operator. Then we have

/i~,(

Su - u)((u 1)+)~-~dx _<

PROOFSince S is positivity preserving it follows that u _< v a.e. Su <_ Sv a.e. For u ~ Lp we find ++uA1, u=(u-1)

(1.2) impliies (1.3)

495

Generators of Feller Semigroups

O<_MAI-uA1

(1.4)

and (1.5)

0 <_lul A1 <_1. Thus we get Now.,for u E Lv it follows that ((u - 1)+)v-1 E Lp’, p~~ +~p = 1, and ’ I1((u- 1)+)P-IIIL.’ = I1(u-I"~+IIP/P ~’ "L~"

0.6)

Using the sub-Markovianand the contraction property of S we find

/~o(s~)((~~)+)~-~dx = f~,(S(u 1)+)((u - 1)+)"-~dx + fa ~(S(uA 1))((u -

1)+)~-’dx

= /~.(u-1)+((u-1)+)P-’dz+ f~.((u-

whereweused in the last step that (u - 1)+((u- ’-~ = (u- 1)( (u- 1)+) ~-x Hence we have proved /p~,(Su)((u-1)+)p-ldx

< frt~u((u-1)+)P-’dx,

whichimplies (1.2). In the followingwedenote by (A, D(A)) the generator of the semigroup(Tt)t>_o. FromProposition 1.5 we derive immediately THEOREM 1.6 Let (Tt)t>_0 be a sub-Markovian semigroup on v with g enerator (A, D(A)). Then for all E D(A) wehave .(Au)((u-

1)+)P-~dx

(1.7)

496

Jacob

DEFINITION1.7 A closed, densely defined linear operator (A, D(A)) ~ i s called an LP-Dirichlet operator if for all u E D(A) relation (1.7) holds. Let (A, D (A)) be an LP-Dirichlet operator u ~ D(A)Since for an y k > 0 we have (ku - 1) + = k(u - ~.)+, it follows that (1.7) is equivalent

/R,

(AI~)((t~ ~)+)p-ldx <_

for all k > 0. Now,if k tends to infinity we find frt, function -u we get

(Au)(u+)p-ldx < O. Taking instead of u the

/rt

(Au)(u+)P-~dx <_ and /r t (Au)(u-)~’-~dx >-0

(1.8)

for any Dirichlet operator. PROPOSITION 1.8 Suppose that an L~ - Dirichlet operator (A, D (A)) generates a strongly continuous contraction semigroup (Tt)t>_o on Lp. Then (Tt)t>_o is a sub-Markovian semigroup. PROOF In virtue of Lemma1.3 it is sufficient to prove that the resolvent (R~)~>0 is sub-Markovian. To see this let u ~. L~ and set v :=/~R~u ~ D(A). Suppose that u
(~’~v)((~,-

or

g)((v - 1)+)Pdz <_ /~. ¢. (Av)((v 1)+)~-~dx.

Taking for ~) a sequence ,~)~ ~ C~C(Rn),o_< ~. _< 1, which tends pointwise to 1, finally arrive at

fR"

((v -- 1)+)Pdx<_

Generatorsof Feller Semigroups

497

since A is an LP-Dirichlet operator. But now we may deduce that v _< 1 a. e. For u >_ 0 a.e. it follows that- ku_< 1 a.e. for all k ~ N, which gives v >_ -~1 a. e. and in the limit we have v >_ 0 a.e. and the proposition is proved. DEFINITION 1.9 Let (A,D(A)) be an operator definite in Lp if for all u ~ D(A)

Lp. We call A neg at ive

/~tn(

Au)( sign u)lulP-ldx <_

(1.9)

PROPOSITION 1.10 A. Let (,4, D(A)) be an Lp - Dirichlet operator. Then A is negative definite. B. Anynegative definite operator in Lp is dissipative. +-u-,w PROOF efi A. nd ForueD(A),u=u ]~. (Au)(sign u)lulP-ldz =/ft. (Au)(sign u)(lu+lP-’

where we used (1.8) B. For u e D(A) and A ~ 0 it follows

II(A- A)ullL,~L,P/P~: I]( A -

A)ullL’ll(sig n ’u)lulP-illL,

E fa~ ((A - A)u)(sign

or

(1.1o) REMARK 1.11 A The proof of Proposition 1.10 does not require that A is closed. B. Note that for u ~ D(A) we

/

p~

(sign u)lule-~.dx =

I1~11~.,

and we may deduce by the general theory, see A. Pazy [15] Section 1.4, that our definition of negative definiteness is already equivalent to the dissipativity of A. Combiningour results with the Hille-Yosida and Lumer-Phillips theory we find THEOREM 1.12 Let (A, D(A)) be ~ - Dir ichlet oper ator with the p rope rty that R(,~ - A) = p for s ome ~> 0. Then A generates a s ub-Markovian semigroup p. in L

498

Jacob

THEOREM 1.13 Let (A,D(A)) be a densely defined operator on Lp such that (1.7) holds for all u E D(A) and assume that for some A > 0 we have R(A - A) p. ~’. Theu .4 is closable and its closure generates a sub-Markoviansemigroup on L The following result is proved iu [10] THEOREM 1.14 Let (Tt)t>_o be a strongly continuous contraction semigroup on L~ with generator (A, D(A)). The semigroup is positivity preserving if and only if

/~t n

(1,11)

(Au)(u+)~-~dx

holds for all u e D(A).

2

SUB-MARKOVIAN

SEMIGROUPS

AND FELLER

SEMIGROUPS

Wewant to study the relation between generators of Feller semigroups and generators of sub-Markovian semigroups. Recall that a Feller semigroup is a strongly continuous contraction semigroup on Coo(Rn) which is positivity preserving. Note that unlike in the case of LP-Dirichlet operators we have a nice structure theorem for generators (A(~}, D(A(~))) of Feller semigroups. Since they have to satisfy the positive maximum principle, a result due to Ph. Courrhge [2] says that on ’~) C~(R (if contained in D(A(~)))we A(~)u(x) = -q(x,D)u(x)

= -(2~)-’/~fa"

eiX(q(x,~)~(~)d~

(2.1)

where q : Rn x Rn -~ C is a function such that q(., ~) is measurable and locally bounded and q(x, .) is continuous and negative definite. Recall that ¢ : Rn -+ C is a continuous negative definite function if and only if we have the L~vy-Khinc]hin representation / -i~’~ (1-e a-\{o}

¢(()=c+id.(+Q(()+

(2.2)

with c _> 0, d ~ R", Q is a symmetric non-negative definite quadratic form and v ~s a bounded Borel measure on Rn \ {0}. THEOREM 2.1 Let (A(~’I,D(A(~))) be the generator of a Feller semigroup (Tt(~))t>_0. Moreover suppose that D((° °)) is a dense subspace of Lp. If A(~)lu extends to a generator A(p) of a strongly continuous contraction semigroup (Tt!Pl)t>_o on Lp such that V := (A- A(P))-~U is an operator core A(p) then (A(p), D(A(p)) is an Lp -Dirichlet operator and hence (T~P))t_>o is sub-Markovian. PROOF For (A-A(~))-~f=

anyf~VandA>0wehave e -~

T(~)fdx

and

()~ - A(~))-~]

499

Generatorsof Feller Semigroups which implies for A > 13 that

f

oo~ e-~t(T~) f - T(tV) f)dt

since for f ~ V we have (X - A(~))-lf = (X - A(P))-~f .Thus we have T~)f T/V)f a. e. by the uniqueness of the Laplace transform. For f ~ V we find further usin~ the calculation of the proof of Proposition 1.5 tha~ fR =(~v,’)((’-’)+)~-td=

fRn(~,’)((,-1)+)~-1~= ~

]; n

By our assumption we have A(P)= li ra Tt(p) f - f t-~o ~ and for f 6 I/it follows that

f~tn( A(")f)((f-

l)÷)"-Id~: [ (lira JR> t~o

T~P)f - f )((f i) +)P-’dz t

= lira [ ( Tt(p)-f - f t )((f 1)+)"-Xdz ~ O. t-~OJR, Since V is an operator core of (A(~), D(A(")) it follows that

f

R.(A(P)u)(( u - 1)+)P-~dx 5

holds for all u 6 D(A(~)). For p = 2 Theorem2.1 was proved in [9] . A closer look at the proof of Theore~n 2.1 yields COROLLARY 2.2 Let A(v) be the generator of a sub-Markovian semigroup (T(tV))t>o on Lp. Moreover suppose that U C D(A(p)) is a dense subspace of Lq, 1 < q < c¢. If A(V)lv extends to a generator A(q) of a strongly continuous contraction semigroup (T~q))t>o on Lp and if V := (,~ - A(q))-IU is an operator core for A(~), then A(q) is a Dirichlet operator and (Tt(q))t>o is a sub-Markovian semigroup. n) REMARK 2.3 Theorem 2.1 says in particular that op&,ators defined on C~°(R and satisfying the positive maximum principle are also candidates for pre-generators of sub-Markovian semigroups. For p = 2 the results of Section 2 are applied in several situations by W. Hoh [4] - [6] and the author [7] -[9] to concrete pseudo differential operators. Corollary 2.2 and some interpolation theory yields that these pseudo differential operators generate also L~-sub-Markovian semigroups.

500

Jacob

REFERENCES 1. N. Bouleau and F. Hirsch. Formesde Dirichlet g6n6rales et densit6 des variables al~atoires r~elles sur l’espace de Wiener. J. Funct. Anal. 69 (1986) 229-25’9. 2. Ph. Courr~ge. Sur la forme int6gro-diff~rentielle des op6rateurs de Cff dans C satisfaisant au principe du maximum.S6m. Th6orie du Potentiel 1965/66, Expos6 2, 38 pp. 3. A. Eberle. Uniqueness and non-uniqueness of singular diffusion operators. Dissertation. Universit~t Bielefeld, 1998. 4. W. Hoh. Feller semigroups generated by pseudo differential operators. In: Intern. Conf. Dirichlet Forms and Stoch. Processes, Walter de Gruyter Verlag, Berlin 1995, 199-206. 5. W. Hoh. Pseudo differential operators with negative definite symbols and the martingale problem. Stoch. and Stoch. Rep. 55 (1995) 225-252. 6. W. Hoh. A symbolic Calculus for pseudo differential operators generating Feller semigroups. Osaka J. Math. (in press) 7. N. Jacob. Feller semigroups, Dirichlet forms and pseudo-differential operators. Forum Math. 4 (1992) 433-446. 8. N. Jacob. A class of Feller semigroups generated by pseudo-differential operators. Math. Z. 215 (1994) 151-166. 9. N. Jacob. Non-local (semi-)Dirichlet forms generated by pseudo-differential operators. In: Intern. Conf. Dirichlet Forms and Stoch. Processes, Walter de Gruyter Verlag, Berlin 1995, 223-233. 10. N. Jacob. Pseudo differential operators and Markovprocesses. Vol. 1: Fourier analysis and semigroups. (Monographin preparation) 11. V.A. Liskevich and Yu. A. Semenov. Some problems on Markov semigroups. In: SchrSdinger Operators, Markov Semigroups, Wavelet Analysis, Operators Algebras. Mathematical Topics Vol. 11, AkademieVerlag, Berlin 1996, I63217. 12. Zh.-M. Maand M. R6ckner. An int{’oduction to the theory of (non-symmetric) Dirichlet forms. Universitext-Springer Verlag, Berlin 1992. 13. E.-M. Ouhabaz. L°° -contractivity of semigroups generated by sectorial forms. J. London Math. Soc. (2) 46 (1992) 529-542. 14. E.-M. Ouhabaz. L~ contraction semigroups for vector valued functions. Pr~publications de l’Equipe d’Analyse et de Math~matiques Appliqu~es 10/98, Universit6 de Marne-la-Vall~e. 15. A. Pazy. Semigroupsof linear operators and applicatious to partial differential equations. Applied Mathematical Science Vol. 44. Springer Verlag, NewYork 1983. 16. N. Varopoulos, L. Saloff-Coste and ~r. Coulhon. Analysis and geometry on groups. Cambridge Tracts in Mathematics Vol. 1(10, Cambridge University Press, Cambridge 1992

A Note on Stochastic

Wave Equations

ANNAKARCZEWSKA Institute of Mathematics, Maria Curie-Sktodowska versity, Pl. M. Curie-Sktodowskiej 1, 20-031 Lublin, Poland, email: [email protected]

Uni-

JERZYZABCZYK Institute of Mathematics, Polish Academy of Sciences, deckich 8, 00-950 Warszawa, Poland, email: [email protected]

~nia-

1

INTRODUCTION

It is well-known,see e.g. [1], that the stochastic wave equation 02u.

OW(t,O),

-g-~(t,o) = zx~(~,o)

t > O, O ~

u(O,O) = O, OeD, where D = ~d and DY owis a space-time white noise, has a function-valued solution if and only if the space dimension d = 1. In recent papers [2] and [3] the spacetime white noise has been replaced by the space-correlated noise with the spatially homogeneouscovariance function..More precisely, let Wr(t,0), t >_ 0, 0 E D, a (distribution-valued) Wiener process with the covariance function F of the form F(0) = f(10[), 0 E EWr(t,O)Wr(s,~) = t A sF(O-~/), 0,r/ ~ D. It has b een shown in [2] that if the space dimension d = 2 and f is a non-negative function, continuous outside 0, then the stochastic wave equation

0%(t, o)A~ (t, O)+ O W(t, r ~ at

--g-i-" ’

u(O,O) = O, O e Research supported by KBNGrant No. 2PO3A082 08 501

t>O,

O6D

(1)

502

Karczewskaand Zabczyk

has a function-valued, locally square integrable solution if and only if 1 I
(2)

The proof in [2] was based on ingenious but long calculations and used explicit representation of the fundamental solution of deterministic waveequation in dimension d = 2. In the present paper we treat the case of general dimension and the correlated, spatially homogeneousnoise of the more general form. Because we are interested in the local properties of the solutions we restrict our attention to the stochastic wave equation on the d-dimensional torus, D = T and consider :D’(T)-valued Wiener process with the general covariance kernel r. Our starting point is an explicit expression for the semigroup solving an equation of the second order in time, in terms of a negative operator determining the drift. Weuse systematic approach to stochastic evolution equations which goes back to [4] and was applied recently in [5] and [6] to stochastic heat equations. This approach reduces the considered problem to elementary questions in harmonic analysis and leads to complete answers. Wefind necessary and sufficient conditions on the kernel P such that the equation (1) has an L~(T)-valued solution. In the case when d = 2 and the positive definite distribution F has a non-negative density f, our condition reduces to (2). Wealso describe the spaces in which the equation (1), driven by the space-time white noise: F = a{0}, has a solution. The paper is a rewritten version of the report by Karczewskaand Zabczyk [711.

2

FORMULATION

OF THE RESULT

The d-dimensional torus T will be identified with the product [-~r, ~)d and regarded as a group with the addition modulo 2rr (coordinate-wise). Weassume that W a 7)~(T)-valued process spatially homogeneous,that is, for each fixed t >_ 0 the law of W(t) is invariant with respect to all space T-translations in the space :D~(T) distributions on T. The value of a distribution ( ~ 79’(T) on ~o ~ :D(T) will denoted by ({, ~p). An arbitrary 7P~(T)-valued spatially homogeneousWiener process Wis uniquelly determined by a positive definite distribution F ~ "D’(T) according to the formula:

~(w(t),~) (W(s),~;) = ~/~s (r, ~o¯ ~s),

(3)

In (3), ~o and ¢ are arbitrary functions in :D(T) and ~he function ¢~ denotes symmetrization of ¢: ¢~(0) = ¢(-0) for 0 It is well-known,see e.g. [8], that an arbitrary real-valued distribution ( e :D’(T) can be uniquelly expanded into its Fourier series:

((0) = ~(’~’°)’,,,~, 0 eT, dn~Z

Stochastic WaveEquations

503

convergent in 79’(T), with the coefficients such that ~.~ = (_,~, n d, and <+°%

Z ~

n~.~+ I~lr

for

(4)

somer>0.

Let us notice that. the Fourier coefficients of the positive definite kernel F E 79~ (T) satisfy evidently condition (4) and are non-negative. Denote Z~s= Nand,byinduction, --szd+~ = Zsx~ZaU{(0,n); n ~ Z~}.If~n = ~n, then ~(0) = ~ ’(n’°)

= ~o+ 2~ (~cos(n ,0) - ¢~

si n(n ,0)),

where ¢0 = ¢~, ~ = 0 m~d ~.n = ¢~+i~, n ~ Z~. Moreover,

for ~ ~ ~’(T),

~ e ~(T) (¢,~)=~o~o+2~

~ ’ ~ ~

n~Z~

Let A be a non-negative operator on a separable Hilbert space H. Denote by ~ the Cartesian product ~(I ~ A)~ ~ H. The following result is well-known, see

PROPOSTION 1

The operator .4: A=

(o

defined on the Hilbert space 7/= 79(1 .+ A)½x H, with the domain D(A) := 79(1 + A) x 79(1 generates a strongly continuous group of transformations S(t), t >_ Moreover S(t)

cosA~t, = -A ½ " smart,

A-~ s~n A~t’~ " cosA~t ’ ) ’

t~l~.

The group S(t), t E ll~, defines a mild solution of the following system: du dv d-~ = v, dt

with u(0) = Uo, v(0)

by the formula:

v(t)

vo ’

tEIIL

504

Karczewskaand Zabczyk

Denote by Hc’ = Ha(T) and H-~ = H-a(T), ct ¯ N+, the real Sobolev spaces of order a and -a, respectively. The norms are expressed in terms of the Fourier coefficients, see [10]

and

I111.- =

(

(1

= I,,i

an.~Z

: I~ol 2 -a +2 E (1+ln[2)

where (n = (1~ + i~n2, ~n = ~-n, n ¯ d. The Laplace operator A reduced to the space H-~(T) will be denoted by A~. It is non-negative with the domain "D(Aa) = H-~+~(T). Let A~ = -A~. Z)(I + A~) -~ +2 and~D(I+ A~)½ = H-~+~. The semigroup generated by the deterministic wave equation on H~ := H-a+~ x -~ H is S~. In the present paper we answer the following question. QUESTION equation "

Under what conditions

dZ(t) = A~Z(t)dt

on the covariance kernel F the stochastic

0 ) z(0)=0

dWr(t) ’

has a weak solution taking values in the space "Hi := L=(T) x H-I(T). Equation (5) is a special case of the following one:

dZ(t)

= A~Z(t)dt

+ dWr(t

on the space ‘H~ := H-~+1 -~. x H Our first result can be formulated as follows.

(5)

Stochastic WaveEquations

505

THEOREM 1 The stochastic equation (1) has an -a+l-valued only if the Fourier coefficients (7,~) of the kernel F satisfy:

s olution i

f a nd

E (1 + Inl:) ~ < +co.

an~Z

Equivalently, equation (1) has an H-~+l-valued solution if and only if there exists a positive definite, continuous function ~o(0), 0 E T, and a constant 70 _> 0 such that

r(o)

+ o e T.

The proof of the theorem is postponed to the next section. REMARK 1 Assume that owr is the space-time white noise. This means that %~= 1 for all n E Zd. Thus the stochastic wave equation with F = (~{0} has on H-a+l solution if and only if d < 2c~. In particular, if c~ = 1 then H-z+~ = L2(T) and then d has to be 1. Recently, the existence of solutions to linear stochastic wave equation has been discussed in [11] and [12] with the explicit use of the fundamental solutions. To formulate our second result, which is a rather direct consequence of Theorem1, we denote by Gd the Green kernel of the operator (I - A~)-~. Thus

(I - ~)-~(~)

= £ Gd(~ -~)~(~)d~

for ~ e r.

If B is the standard Brownian motion on T and (Pt) is the transition of ~B, then e-t Pt~(()dt

semigroup

= Gd(~ - ~)~(~)dy for ~ e r.

It is well-knownthat Gd(~)

~e ~Z ~

dO

e-~

dt,

~

~

T.

~

If d = 1, then G~ is a positive continuous function and if d = 2, G~is continuous outside 0 with singularity at 0 of the form cln ~. If d > 2, then

ae(()

1

as

0.

Karczewskaand Zabczyk

506 In fact, +~

~

1

1

-t _~1~ .

where K~’) is the modified Bessel function of the third order, see [13, p. 371]. THEOREM 2 1) The stochastic wave equation (1) has an L~(T)-valued solution if and only % E 1 ÷ In[ ~ < 2) If the kernel F is a non-negative measure then the stochastic wave equation (1) has an L~(T)-valued solution if and only if (P, Gd) < +~.

(7)

REMARK If the kernel F is a non-negative measure and d = 1, the condition (7) is always satisfied. If d = 2 then (7) is equivalent

I~ll /lln

F(d{)<

÷oo

T

and for d >_ 3 the condition (7) is equivalent /

~F(d~)

< +~.

T

COROLLARY 1 Condition of (2).

(7) from Theorem 2 is the required

generalization

If { e D’(T), we denote by ~ = ({,~) the sequence of Fourier coefficients distribution {, that is, ~ = ~nEz~ei(n’O){n, 0 ~ COROLLARY 2 Assume that P ~ L2iT) and d = 1,2,3. wave equation (1) has a solution with values in L~(T).

of

Then the stochastic

Proof: Let us note that if F ~ L’~(T) then ~ = (%) ~/’~(Za). Consequently 1

~ 1 +

an~Z

Inl ~ -

(1 + 1.1=)

507

Stochastic WaveEquations But ~.eza’~ < +oe and , if d = 1,2,3, 1

(i

<+oo.

So, the result follows. The following result is a generalization of the previous one. COROLLARY 3 Assumethat for some 1 _< p _< 2, f" E /P(Zd). If d < ~2_--~-7, then the stochastic wave equation (1) has a solution in La(T). Proof:

Note that % /-~1 + Inl 2 <-

1 q (I+ ]nl~)

and if and only if2q-d+l>l.

But q = ~ and the result

follows.

REMARK 2 If in Corollary 3, p = 2, then by Plancherel’s theorem, [" E d) 12(~, if and only if F E L2(T). Consequently, Corollary 3 implies Corollary 2. If p = 1 or, equivalently, if F is a bounded and continuous, positive definite function then thc condition of Corollary 3 is satisfied for any d = 1, 2,... and the stochastic wave equatiou has a unique solution in L~(T) for any dimension d = 1,2,....

3 3.1

PROOF Proof

OF THE

THEOREMS

of Theorem 1

Let W(t), t >_ 0, be a cylindrical Wiener process on a Hilbert space U. Assume that S(t), t >_ 0, is a strongly continuous group of operators on a Hilbert space 7-/ generated by the operator .4 and 3" is a linear operator acting fl’om U into 7/. The following elementary result holds, see [14], Theorem9.2.1. PROPOSITION2 The stochastic

equation

dZ(t) = AZ(t)dt + ~TdW(t), Z(0) = 0, has a weak solution with values in 7/if and only if J is a Hilbert-Schmidt operator from U into 7/.

508

Karczewskaand Z~tbczyk

Wealso have the following description of the reproducing kernel Ur = S{~r of the Wiener process Wrwith respect to which the process Wris cylindrical. Let us recall that the stochastic integral fo t ¢(s)dW(s), t >_ 0, is well defined for exactly those operator-valued processes ¢(s) which satisfy the condition f~ I]go(.s)ll)tsds < +oo, t > 0, where II~b]]lts denotes the Hilbert-Schmidt norm from Ur into ~, see [4] and [15]. PROPOSITION 3 A real distribution ~ ~ D’(T) belongs to the space Ur it’ only if its Fourier coefficients (~n) satisfy the following conditions: d’s 1) ,~n=~n=0if%=0,

and

n~Z

Proof: By the very definition, see [4] and [16], the distribution ~ ~ 7)’(t) belongs to Ur if and only if, there exists a constant C >_ 0 such that I(~, ¢p)l < C(F, ~o * ~os) ½ for ~o ~/)(T). Thus, in terms of the Fourier coefficients we have

[{0~00 +

~

~

1 1

2 2

C

(8)

7o~2o

n~X~

This easily implies that if %= 0, then {~ = {~ = 0, n ~ Z~ U {0}. But the estimate (8) is equivalent to: I

) <_ C Iv,

ol 2 + 2 an~Z,

and the Riesz representation

+

theorem implies condition 2) of the proposition.

REMARK 3 We would arrive at the same conclusion noticing that Wv can be expanded into a series with respect to harmonics ei(n’e) or treating directly the equation (7) in the frequently domain. This would sTmphfythe considerations only slightly and could not be used in different situations. For the wave equation (1) the transformation y is of the form (~) where 27 is inclusion operator of Ur into H-~’(T). Consequently, the equivalent condition for the solution of the wave equation to be in H-a+I(T) is that Ur H-c’(T) with the Hilbert-Schmidt imbedding.

Stochastic WaveEquations

509

Consider the following orthonormal and complete basis { (cn, sin), n E Zsa U {0}, m e Zsd} in the space Ur:

co(0)=v’-~,

= ~/~ cos(n, 0),

sin(O) The Hilbert-Schmidt to

Z~.

norm of the imbedding of Ur into H-~(T) is therefore equal 7m dmEZ

and consequently the first part of the theorem follows. To show the second part it is enough to define ei(m,O) m~0 Then ~ is the characteristic functional of a finite non-negative me~ure and fi’om (6) we have, in the sense of distributions

(-~)~ ~(o) - ~ ~’<~’%m r(0) - ~o m~0 On the other hand, if ~ is a positive definite continnous function with Fourier coefficients ~. ~ 0 then, in the sense of distributions,

n¢o

Thus,

(-~x)~~ + 7o : ~ e~"’°)lnl2~. he0

and it is enough to define for n # 0. 3.2

Proof

¯

of Theorem 2

The first part is contained in Theorem 1. To prove the second one denote by pt, t > 0, the kernels of the transition operators Pt, t > 0. Then Ptq~(O) = fTq~(O- rl)pt(rl)do,

510

Karczewskaand Zabczyk

and the Fourier coefficients of the measures p~ are of the form:

fTei(°’°)pt(O)dO e -tlnl~,

t > O, n ~ zd.

Hence ~+ Inl ~ ~ 1 ~e-tl’~l~

= Z 7n \1 + Inl

= (r, Ga*Pt).

But Ga is a 1-excessive function for v~B(t), t > 0, and therefore e-tGa*pt <_ Ga and

e-tGa * pt ~ Ga

as t J¢ 0.

If F is a non-negative distribution on T, then it is a finite measure and

= (r,G~). ~ 1 + "r,~ In[ 2= l~!~0(r,G~,pt) . an~:Z

This finishes the proof,

REFERENCES 1.

Walsh, J., An introduction to stochastic partial differential equations, l~cole d’l~t~ de Probabilit~s de Saint-Flour XIV-1984, Lecture Notes in Math., Springer-Verlag, NewYork-Berlin 1180 (1986), 265-439. 2. Dalang, R. and Frangos, N., The stochastic wave equation in two spatial dimensions, The Annals of Probability No. 1 26 (1998), 187-212. 3. Mueller, C., Long time existence for the wave equation with a noise term, The Annals of Probability No. 1 25 (1997), 133-151. 4. It6, K., Foundations of Stochastic .Differential Equations in Infinite Dimensional Spaces, SIAM,Philadelphia, 1984. 5. Tessitore, G. and Zabczyk, J., Invariant measuresfor stochastic heat equations, Preprint 561, Institute of Math., Polish Acad. Sc., Warsaw(1996). 6. Tessitore, G. and Zabczyk, J., Strong positivity for stochastic heat equations, Preprint 561, Institute of Math., Polish Acad. Sc., Warsaw(1996). 7. Karczewska, A. and Zabczyk, J., A note on stochastic wave equations, Preprint 574, Institute of Math., Polish Acad. Sc., Warsaw(1997). 8. Kufner, A. and Kadlec, J., Fourier Series, Academia, Prague, 1971. 9. Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, NewYork - Berlin, 1983. 10. Adams, R., Sobolev Spaces, Academic Press, NewYork, 1975.

Stochastic WaveEquations 11. 12. 13. 14. 15. 16.

511

Gaveau, B., The Cauchy problem for the stochastic wave equation, Bull. Sci. Math. 119 (1995), 381-407. Martin, A., Waveequations driven by space-time white noise, Ma~ster Thesis, University of Warwick, 1995. Landkof, N.S., Foundations of modern potential theory, Springer-Verlag, Berlin, 1972. DaPrato, G. and Zabczyk, J., Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996. DaPrato, G. and Zabczyk, J., Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. Peszat, S. and Zabczyk, J., Stochastic evolution equations with a spatially homogeneousWiener process, Stochastic Processes and Applications 72 (1997), 187-204.