Progress in Nonlinear Differential and Their :Applications
Nonlinear Analysis and its Applications
to Differential Equations M.R. Grossinho M. Ramos
C. Rehelo L. Sanchez Editors
Birkhauser
Progress in Nonlinear Differential Equations and Their Applications Volume 43
Editor Haim Brezis Universit6 Pierre et Marie Curie Paris
and Rutgers University New Brunswick, N.J.
Editorial Board Antonio Ambrosetti, Scuola Normale Superiore, Pisa A. Bahri, Rutgers University, New Brunswick Felix Browder, Rutgers University, New Brunswick Luis Cafarelli, Institute for Advanced Study, Princeton Lawrence C. Evans, University of California, Berkeley Mariano Giaquinta, University of Pisa David Kinderlehrer, Carnegie-Mellon University, Pittsburgh Sergiu Klainerman, Princeton University Robert Kohn, New York University P. L. Lions, University of Paris IX Jean Mawhin, Universitd Catholique de Louvain Louis Nirenberg, New York University Lambertus Peletier, University of Leiden Paul Rabinowitz, University of Wisconsin, Madison John Toland, University of Bath
Nonlinear Analysis and its Applications to Differential Equations
M.R. Grossinho M. Ramos C. Rebelo L. Sanchez Editors
Birkhauser Boston Basel Berlin
M.R. Grossinho CMAF University of Lisbon 1649-003 Lisbon Portugal
M. Ramos CMAF University of Lisbon 1649-003 Lisbon Portugal
C. Rebelo CMAF University of Lisbon 1649-003 Lisbon Portugal
L. Sanchez CMAF University of Lisbon 1649-003 Lisbon Portugal
Library of Congress Cataloging-in-Publication Data Nonlinear analysis and its applications to differential equations / M.R. Grossinho..[et
all, editors. p. cm. - (Progress in nonlinear differential equations and their applications ; 43) Includes bibliographical references. ISBN 0-8176-4188-2 (alk. paper) - ISBN 3-7643-4188-2 (alk. paper) 1. Differential equations. 2. Nonlinear functional analysis. 1. Grossinho, M. R. (Maria do Rosirio), 1956- II. Series. QA372.N65 2000 515'.35-dc21
00-048563
CIP AMS Subject Classifications: Primary: 34B15, 34C25, 35J25, 47H10 Secondary: 35B40, 35Q30, 49J40, 58E05, 70645 Printed on acid-free paper.
0 2001 Birkhauser Boston
Birkhduser
All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhduser Boston, c/o Springer-Verlag New York, Inc., 175 Fifth
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SPIN 10764729
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Reformatted from editors' files by TEXniques, Inc., Cambridge, MA. Printed and bound by Hamilton Printing Company, Rensselaer, NY. Printed in the United States of America.
987654321
Contents Preface Part 1. Short courses .
. ix
.1
An Overview of the Method of Lower and Upper Solutions for ODES
C. De Coster, P. Habets
......................................... 3
On the Long-time Behaviour of Solutions to the Navier-Stokes Equations of Compressible Flow
I. Ppirnisl
24
Periodic Solutions of Systems with p-Laplacian-like Operators J. Mawhin
37
Mechanics on Riemannian Manifolds 65
W. M. Olives
Twist Mappings, Invariant Curves and Periodic Differential Equations R. Ortega ...................................................... 85 Variational Inequalities, Bifurcation and Applications K..Srhmitt ............
113
Part 2. Seminar papers
145
Complex Dynamics in a Class of Reversible Equations F. Alessio, M. Calanchi, E. Serra
.
147
Symmetry and Monotonicity Results for Solutions of Certain Elliptic PDEs on Manifolds L. Almeida, Y. Ge ............................................
161
Nielsen Number and Multiplicity Results for Multivalued Boundary Value Problems J. Andres
.
............ 175
vi
Contents
Bifurcation Theory and Application to Semilinear Problems near the Resonance Parameter .189
D. Arcoya, J.L. Gdmez Orientation and Degree for Fredholm Maps of Index Zero Between Banach Spaces
P. Benevieri ...................................................201
On the Method of Upper and Lower Solutions for First Order BVPs A. Cabada, E. Liz, R.L. Pouso ................................ 215
Nonlinear Optimal Control Problems for Diffusive Elliptic Equations of Logistic Type
A. Canada, J.L. Gdmez, J.A. Montero
221
.
On The Use of Time-Maps in Nonlinear Boundary Value Problems
A. Capietto
.231
Some Aspects of Nonlinear Spectral Theory P. Drdbek .....................................................243
Asymmetric Nonlinear Oscillators
C. Fabry, A. Fonda
.253
Hopf Bifurcation for a Delayed Predator-Prey Model and the Effect of Diffusion T. Farin ...--..---.-
....
257
Galerkin-Averaging Method in Infinite-Dimensional Spaces for Weakly Nonlinear Problems M. Fe6kan
.269
PBVPs for Ordinary Impulsive Differential Equations D. F1anco, J.J. Nieto .......................................... 281
Homoclinic and Periodic Solutions for Some Classes of Second Order Differential Equations
M.R. Grossinho, F. Minh6s, S. Tersian
.289
Global Bifurcation for Monge-Ampere Operators J. Jacobsen ....................................................299
Contents
vii
Remarks on Boundedness of Semilinear Oscillators M. Kunze ..................................................... 311
The Dual Variational Method in Nonlocal Semilinear Tricomi Problems D. Lupo, K.R. Payne .......................................... 321
Symmetry Properties of Positive Solutions of Nonlinear Differential Equations Involving the p-Laplace Operator F. Pacelln
A Maximum Principle with Applications to the Forced Sine-Gordon Equation A.M. Robles-Phy z
.339
347
Lipschitzian Regularity Conditions for the Minimizing Trajectories of Optimal Control Problems A. V. Sarychev, D.F.M. Torres
.357
Abstract Concentration Compactness and Elliptic Equations on Unbounded Domains I. Schindler, K. Tintarev ...................................... 369
Preface In this book we present a significant part of the material given in an autumn
school on "Nonlinear Analysis and Differential Equations," held at the CMAF (Centro de Matematica e AplicaCOes Fundamentais), University of Lisbon, in September-October 1998. The school consisted of short courses
and a seminar, thus offering participants a systematic approach to some classes of problems and an opportunity to be in contact with the most recent developments in several areas. Two of the courses presented here deal with ordinary differential equations: one updating the technique of upper and lower solutions, the other dealing with continuation theorems applied to quasi-linear systems (involving operators of p-Laplacian type). Both ordinary and partial differential
equations appear in the scope of a course on bifurcation for variational inequalities. Another course deals with a geometric approach to dynamical systems in the plane via twist theorems. Two other important and useful topics (although not in the mainstream of the volume) have also been included: asymptotic behavior and periodic solutions for Navier-Stokes equations, and a mathematical formulation of mechanics in Riemannian manifolds. The papers originating from the seminar include a range of topics repre-
sentative of the recent research of participants. With respect to the area of ordinary differential equations, we include papers on spectral theory for the one-dimensional p-Laplacian, the use of the time-map in boundary value problems, and boundary value problems for impulsive and discontinuous equations.
Nonlinear oscillations occur in the study of asymmetric systems, for which a notion of nonlinear resonance is proposed, and in variational approaches to homoclinics and heteroclinics to periodic solutions. The important problem of boundedness of solutions deserves some attention too. Partial differential equations is a recurrent theme. Thus for equations involving the n-dimensional p-Laplacian operator, symmetries of positive solutions are studied. Symmetries are also considered for some PDEs on manifolds. Bifurcation theory is applied to nonlinear elliptic equations and
to the Monge-Ampere operator. The dual method is used in the study of Tricomi problems. A telegraph equation is analyzed by means of a maximum principle. An abstract concentration-compactness method is proposed in connection with elliptic equations in n-dimensional space. Degree
x
Preface
and index theories are represented by studies on the Nielsen number, and the degree on Banach manifolds. The calculus of variations and optimal control appear explicitly both in the context of ODEs (Lipschitz regularity of minimizers) and PDEs (optimal control for some elliptic equations of logistic type). The dynamics of delayed systems appears in connection with asymptotic expansions and Hopf bifurcation. All papers have been refereed. It is our pleasure to express our thanks to all the invited lecturers and participants of the autumn school; their presence and their work at the CMAF was the main contribution to the success of the meeting. We believe they had a pleasant and scientifically rewarding time at the University of Lisbon.
We warmly thank our colleagues Bento Louro and Luis Trabucho for their valuable help in the preparation of this volume. A special aknowledgement is due to Fundacao pare a Ciencia e a Tecnologia for its sponsorship (as part of a program funding afforded to CMAF). Our gratitude goes also to CIM (Centro Internacional de Matematica) for its support of the event and publication of this volume.
M.R. Grossinho M. Ramos C. Rebelo L. Sanchez Editors
Autumn School
Nonlinear Analysis and Differential Equations Lisbon, 14 September - 23 October 1998
Scientific Committee: - Djairo DE FIGUEIREDO - Jean MAWHIN
- Rafael ORTEGA - Luis SANCHEZ - Fabio ZANOLIN
Organizing Committee: - Maria do Rosario GROSSINHo - Miguel RAMOS
- Carlota REBELO - Lufs SANCHEZ
List of Participants: Luis ALMEIDA, CMLA-Cachan, France Doherty ANDRADE, Universidade Estadual de Maringy, Brazil Jan ANDRES, Palaky University, Czech Republic David ARCOYA, Universidad de Granada, Spain Pierluigi BENEVIERI, University di Firenze, Italy Bernard BROGLIATO, LAG, UMR CNR-INPG, France Antonio CANADA, Universidad de Granada, Spain Anna CAPIETTO, University di Torino, Italy Nicolai CHEMETOV, Universidade de Lisboa, Portugal Marielle CHERPION, University Catholique de Louvain, Belgium Monica CONTI, Istituto Politecnico di Milano, Italy Walter DAMBROSIO, University di Torino, Italy
xii
Autumn School Proceedings
Colette DE COSTER, Universite du Litoral, France Djairo DE FIGUEIREDO, Universidade de Campinas, Brazil Dan DOBROVOLSCHI, Institute of Meteorology and Hydrology, Romania Ana Rute DoMINGOS, Universidade de Lisboa, Portugal Pavel DRABEK, University of West Bohemia, Czech Republic Teresa FARIA, Universidade de Lisboa, Portugal Michal FECKAN, Comenius University, Slovakia Eduard FEIREISL, Institute of Mathematics, Czech Republic Alessandro FONDA, University di Trieste, Italy Daniel FRANCO, Universidad de Santiago de Compostela, Spain Pedro FREITAS, Universidade Tecnica de Lisboa, Portugal Fortios GIANNAKOPOULOS, University of Colonia, Germany
Pedro GIRAO, Universidade Tecnica de Lisboa, Portugal Jean-Pierre GOSSEZ, Universite Libre de Bruxelles, Belgium Maria do Roserio GROSSINHO, Universidade Tecnica de Lisboa, Portugal Patrick HABETS, Universite Catholique de Louvain, Belgium Jon JACOBSEN, University of Utah, USA Markus KUNZE, Universitat Koln, Germany Susana Lois, Universidad de Santiago - Lugo, Spain Rodrigo LoPEZ PoUSO, Universidad de Santiago de Compostela, Spain Daniela Lupo, Istituto Politecnico di Milano, Italy To Fu MA, Universidade Estadual de Maringa, Brazil Manuel Monteiro MARQUES, Universidade de Lisboa, Portugal Rogerio MARTINS, Universidade Nova de Lisboa, Portugal Julia MATOS, Universidade de Lisboa, Portugal Jean MAWHIN, Universite Catholique de Louvain, Belgium Feliz MINHds, Universidade de Evora, Portugal Daniel NUNEZ-LoPEZ, Universidad de Granada, Spain Walter OLIVA, Universidade Tecnica de Lisboa, Portugal Rafael ORTEGA, Universidad de Granada, Spain Filomena PACELLA, University degli Study di Roma La Sapienza, Italy Kevin PAYNE, University of Miami, USA Miguel RAMOS, Universidade de Lisboa, Portugal Carlota REBELO, Universidade de Lisboa, Portugal Aureliano ROBLES-PEREZ, Universidad de Granada, Spain Carlos ROCHA, Universidade Tecnica de Lisboa, Portugal Paula C. Patricio RODRIGUES, Universidade Nova de Lisboa, Portugal Luis SANCHEZ, Universidade de Lisboa, Portugal Andrei V. SARYCHEV, Universidade de Aveiro, Portugal Klauss SCHMITT, University of Utah, USA Enrico SERRA, Politecnico di Torino, Italy Boyan SIRAKOV, Universite de Paris 6, France Valentina TADDEI, Italy Massimo TARALLO, University degli Studi di Milano, Italy Susanna TERRACINI, Istituto Politecnico di Milano, Italy
Autumn School Proceedings
Stepan TERSIAN, University of Rousse, Bulgaria Kyril TINTAREV, Uppsala University, Sweden Laura TONEL, University di Torino, Italy Delfim TORRES, Universidade de Aveiro, Portugal Christophe TROESTLER, University de Mons-Hainaut, Belgium Salvador VILLEGAS, Universidad de Granada, Spain Michel WILLEM, University Catholique de Louvain, Belgium Fabio ZANOLIN, University di Udine, Italy
xiii
Part 1
SHORT COURSES
An Overview of the Method of Lower and Upper Solutions for ODEs C. De Coster P. Habets 1
Introduction
The method of lower and upper solutions deals mainly with existence results for boundary value problems. In this presentation, we will restrict attention to second order ODE problems with separated boundary conditions. Although some of the ideas can be traced back to E. Picard [16], the method of lower and upper solutions was firmly established by G. Scorza Dragoni [20]. This 1931 paper considered upper and lower solutions which are C2; in
1938, the same author extended his method to the L1-Caratheodory case [21]. Upper and lower solutions with corners were considered by M. Nagumo in 1954 [13]. Since then a multitude of variants have been introduced. The
Definitions 2.1 and 3.1 we present here tend to be general enough for applications and simple enough to model the geometric intuition built into the concept. Sections 2 and 3 present the basic existence results, respectively of C2 and W2,1-solutions, for the boundary value problem
u" = f(t,u,u'), aiu(a) - a2u'(a) = A, bju(b) + b2u'(b) = B. We also present some results concerning the structure of the set of solutions. Existence of extremal solutions was studied by G. Peano [14] and O. Perron [15] for first order ODE, by K. Ako [1] for elliptic PDE and W. Mlak [11] for parabolic problems. We refer to K. Ako [11 or K. Schmitt [19] for the proof of existence of C2-extremal solutions. Such a proof uses the maximum
of lower solutions and the minimum of upper ones. Extension to W2,1_ solutions follows using ideas of Troianiello [22]. Section 4 is concerned with a priori bounds on the derivative of solutions. Such results go back to S. Bernstein [4]. In 1937, M. Nagumo [12] generalized these ideas by introducing the so-called Nagumo condition which is
4
C. De Coster, P. Habets
both simple and very general. This is basically our Proposition 4.1. Later, I.T. Kiguradze [8] observed that, for some boundary value problems, the Nagumo condition can be restricted to be one-sided (see Proposition 4.2). In the same paper, this author extended the Nagumo condition so as to deal with W2"-solutions as we did in Theorem 4.7. The idea of the proof of this last theorem goes back to I.T. Kiguradze [9]. Dirichlet problems are studied in Section 5. As already noticed by A. Rosenblatt in 1933 [18], these problems can be studied for more singular nonlinearities than V'-Caratheodory functions. In 1953, G. Prodi [17] used lower and upper solutions for such singular problems. Here we follow P. Habets and F. Zanolin [7].
The idea to associate a degree with a pair of strict lower and upper solutions is developed in Section 6. It was already used in 1972 by H. Amann [2]. The three solutions theorem (Theorem 6.5) seems to be due to Y.S. Kolesov (10] but a proof using degree theory can be found in H. Amann [2]. The first result concerning non-well-ordered lower and upper solutions is due to H. Amann, A. Ambrosetti and G. Mancini [3] in 1978. Our presentation follows C. De Coster and M. Henrard [6].
Throughout the paper we have illustrated the theory with examples. Additional examples and more elaborate applications can be found in C. De Coster and P. Habets [5]. We also refer to this paper for additional references and corresponding results on other boundary value problems.
2
C2-solutions
Consider the problem
u" = f(t,u,u'), alu(a) - a2u'(a) = A,
(2.1)
blu(b) + b2u'(b) = B,
where A, B ER, a1, b ER,a2,b2ER+,ai+a2>0andbi+b2>0. This problem is called the separated boundary value problem and contains as special cases the Dirichlet problem
u" = f (t, u, u'),
u(a) = 0, u(b) = 0,
and the Neumann problem
u" = f (t, u, u'),
u'(a) = 0, u'(b) = 0.
(2.2)
If the function f is continuous, solutions of (2.1) are in C2 ([a, bl) and an appropriate notion of lower and upper solution is the following.
Definitions 2.1. A function a E C([a, b]) is a C2 -lower solution of (2.1) if
An Overview of the Method of Lower and Upper Solutions for ODES
5
(a) for any to E ]a, b[, either D_a(to) < D+a(to),
or there exist an open interval to C ]a, b[ with to E to and a function ao E C'(Io) such that : (i) a(to) = ao(to) and a(t) > ao(t) for all t E Io; (ii) a0 "(to) exists and a0 "(to)
f (to, ao(to), ao(to));
(b) ala(a) - a2D+a(a) < A, bla(b) + b2D_a(b) < B.
A function /3 E C([a, b]) is a C2-upper solution of (2.1) if (a) for any to E ]a, b[, either D-/3(to) > D+/3(to),
or there exist an open interval to C ]a, b[ with to E to and a function /3o E C1(Io) such that : (i) 3(to) =,6o(to) and 3(t) 5 Qo(t) for all t E Io; (ii) /30 (to) exists and /30 (to) S f (to, Qo(to), 00' (to)); (b) aif3(a) - a2D+A(a) ? A, b1Q(b) + b2D-Q(b) > B.
These definitions can be understood geometrically if we assume the inequalities (ii) to be strict. The graphs of a and /3 are then curves such that solutions cannot be tangent to a lower solution from above or tangent to an upper solution from below.
We can build lower and upper solutions from the maximum of lower solutions and the minimum of upper solutions.
Proposition 2.2. Let ai E C([a, bJ) (i = 1, ..., n) be C2 -lower solutions of (2.1). Then the function
a(t) = Max ai(t), 1
t E [a, b]
is a C2-lower solution of (2.1). Similarly, if /3j E C([a, b]) (j = 1, ..., m) are C2 -upper solutions of (2.1), then the function
/3(t) = min /3j(t), 1<j<-
t E [a, b]
is a C2 _upper solution of (2.1).
A first existence result for solutions of (2.1) is the following.
Theorem 2.3. Let A, B E R, a1, b1 E R, a2, b2 E R+ with al + a2 > 0, bi + b2 > 0. Assume a and /3 E C([a,b]) are C2-lower and upper solutions of the problem (2.1) such that a <,3. Let E := {(t, u, v) E [a, b] x R2 1 a(t) < u < /3(t)),
(2.3)
and assume the function f : E - R is continuous and bounded. Then the problem (2.1) has at least one solution u E C2([a, b]) such that for all t E [a, b], a(t) < u(t) 5 /3(t).
6
C. De Coster, P. Habets
Sketch of proof. Consider the modified problem u" - u = f (t, y(t, u), u') - -Y (t, u), u(a) - a2u'(a) = A + (1 - al)y(a,u(a)),
(2.4)
u(b) + b2u'(b) = B + (1 - b1)y(b, u(b)), where
y(t, u) = max{a(t), min{u, Q(t)}}.
(2.5)
Claim 1 - The problem (2.4) has at least one solution. To prove this claim,
we write (2.4) as Lu = Nu, where Lu = (u" - u, u(a) - a2u'(a), u(b) +
b2u'(b)) and Nu = (f(t,y(t,u),u')-y(t,u),A+(1-al)y(a,u(a)),B+(1b1)-y(b, u(b)). The operator L can be made invertible. Hence the problem (2.4) reads u = L-1 Nu and we can apply Schauder's Theorem.
Claim 2 - Solutions u of (2.4) are such that for all t E [a, b],
a(t) < u(t) :5,3(t)-
(2.6)
Let us assume on the contrary that u - a has a negative minimum at some point to. If to E ]a, b[, we have D-a(to) > D+a(to) and using the definition of a lower solution, we obtain the contradiction 0 < u"(to) - a0"(to) = f (to, ao(to), ao(to)) + u(to) - ao(to) - ao (to) < 0.
In case to = a (a similar argument holds for to = b), we have u(a) D+a(a) > 0 and obtain the contradiction
0 = A+(1-a1)y(a,u(a)) -u(a)+a2u'(a) > A-ala(a)+a2D+a(a) > 0. Conclusion - The equation (2.4) has a solution which satisfies (2.6) and therefore is a solution of (2.1). 0 Example 2.4. We can apply this result to singular perturbation problems. A model example is E2u" = u - ItI, u(-1) = 1, u(1) = 1. The function a E C([-1, 1]) defined by a(t) = Itl is a C2-lower solution and /1(t) = Itl + Eexp(-Its/E) a C2-upper solution. Hence, we deduce from Theorem 2.3 the existence of a solution u together with the asymptotic estimate u(t) = Itl + 0(E).
Remark. Theorem 2.3 can be interpreted as an intermediate value theore
for the operator Tu := L-1Nu. It is easy to see that Ta _> a and TO < p. Theorem 2.3 proves then the existence of an intermediate value u E [a, p] such that Tu = u. This intermediate value property depends strongly on the ordering a _< ,0. In case this ordering is not satisfied, the result does not hold as such. Consider for example the problem
An Overview of the Method of Lower and Upper Solutions for ODEs
7
u" + 4u = sin 2t, u(0) = 0, u(ir) = 0. It follows immediately from the Fredholm alternative that this problem has no solution, although a(t) = sin t and /3(t) = - sin t are lower and upper solutions. Example 2.5. Another example of application of Theorem 2.3 is the piecewise linear problem
u" + max{-u - 2, min{u, -u + 2}} = 0, u(O) = 0, u(27r) = 0.
It is easy to see that a(t) = -2 is a lower solution, 3(t) = 2 is an upper solution and that the functions u(t) = k sin t, with Ikj < 1, are solutions that lie between a and 0. Further, we deduce from Proposition 2.2 that J sin tj = max{sin t, -sin t} is a lower solution. Hence this problem has a positive solution u such that, for all t E [0, 27r], sin tl < u(t) < 2. This example shows that solutions between lower and upper solutions are not necessarily ordered. However they admit extremal solutions as follows from the next result.
Theorem 2.6. Under the assumptions of Theorem 2.3, the problem (2.1) has solutions Umin, Umax E C2([a, bJ) between a and /3,
a < umin 5 Umax "- Q,
which are such that any other solution u of (2.1) with a < u < 0 satisfies Umin <- u < umax.
The solutions Umin and Umax given in Theorem 2.6 are called minimal and maximal solutions. In Example 2.5, we can compute explicitly sin t
/
umax(t) = -umin(t) = sin to
= 2 - cosh(t - 7r)
_
cosh(to - ir)
sin t sin to
if t E 10, to],
if t E [to, 27r - to], if t E [21r - to, 27r],
where to is the first positive root of tan to tanh to + 1 = 0.
3
W2'1-solutions
In this section, we consider nonlinearities f which are LP-Caratheodory, i.e.,
8
C. De Coster, P. Habets
(i) for a.e. t E [a, b], the function f (t, , ) is continuous, (ii) for all (u, v) E 1R2, the function f u, v) is measurable and (iii) for all r > 0, there exists h E LP(a, b) such that for a.e. t E [a, b] and all (u, v) with I(u, v)I < r, we have If (t, u, v)I < h(t). With such an assumption, the solutions of the boundary value problem (2.1) are in W2,p(a, b) (and hence in W2,1 (a, b)) and we can adapt accordingly the definitions of lower and upper solutions.
Definitions 3.1. A function a E C([a, b]) is a W2,1 -lower solution of (2.1) if (a) for any to E ]a, b[, either D_a(to) < D+a(to),
or there exists an open interval Io C ]a, b[ such that to E Io, a E W2"1(10) and, for a.e. t E Io,
a"(t) > f (t, a(t), a'(t));
(b) ala(a) - a2D+a(a) < A, bla(b) + b2D_a(b) < B. A function 3 E C([a, b]) is a W2"1-upper solution of (2.1) if (a) for any to E ]a, b[, either D-,3(t0) > D+,Q(t0),
or there exists an open interval Io C ]a, b[ such that to E 10, p E W2,1 (10) and, for a.e. t E Io, A11 (t) < f (t, Q(t), #1 (0);
(b) a1Q(a) - a2D+,3(a) ? A, b1Q(b) + b2D-,3(b) > B.
In these definitions we did not use, as for C2-lower solutions, auxiliary functions ao. This is due to the fact that for W2.1-lower solutions we cannot work pointwise, and the natural modification to be used in the proof does not allow us to use information along the curve u = ao(t).
Notice also that, with such definitions, we cannot prove that the maximum of lower solutions is a lower solution. Nevertheless, we can prove existence of a solution between the maximum of lower solutions and the minimum of upper solutions. Here, to simplify, we present an extension of Theorem 2.6 which uses only one pair of lower and upper solutions.
Theorem 3.2. Let A, B E R, a1, b1 E R, a2, b2 E R+, ai + a2 > 0 and bi +b2 > 0. Assume a and /3 E C([a, b]) are W2"1-lower and upper solutions
of problem (2.1) such that a < 3. Let E be defined in (2.3) and f : E -> R be an L'-Caratheodory function such that for some function h E L1(a, b) and all (t, u, v) E E, If (t, u, v) I < h(t). Then, the problem (2.1) has solutions umin and Umax E W2,1 (a, b) which
are such that a < umin S Umax < Q, and any other solution u of (2.1) with a < u < ,6 satisfies umsn <_ U <_ Umax
An Overview of the Method of Lower and Upper Solutions for ODES
9
Example 3.3. Consider the problem u" + sin u = h + h(t), u'(0) = 0, u'(ir) = 0, where h E R and h E L'(0, 7r) has mean value zero. Let w be the solution with mean value zero of
w" = h(t),
W'(0) = 0, w'(7r) = 0.
(3.1)
Then, if for all t E [0, 7r], IhI < sin(2 + w(t)) (notice that this condition is satisfied for IhI < 1 and IIhlILl small enough), the functions a(t) = 2 +w(t) and Q(t) = s2 + w(t) are respectively lower and upper solutions. Hence, there is a solution u such that 3 2 + w(t) < u(t) < + w(t). As we will see later, the existence of a second solution v # u mod 27r can be obtained using further results on lower and upper solutions.
4 The Nagumo condition To prove existence results for (2.1) it is essential to work out a priori bounds on the derivative. These follow, in previous results, from the boundedness
of the nonlinearity f but can be generalized to nonlinearities which do not grow too quickly with respect to the derivative. A basic result in this direction is the following.
Proposition 4.1. Let a and /j E C([a, b]) be such that a < ,3. Define E C [a, b] x R2 from (2.3) and let cp : R+ -' IR be a positive continuous function satisfying 00
J
s) > max,0(t) - mina(t),
(4.1)
where r = max{ bb-'a a , A 6_n b } > 0. Then there exists R > 0 such that for every continuous function f : E --+ R which satisfies b(t, u, v) E E,
If (t, u, v) I <_ W(I VI)
and for every solution u of
u" = f(t,u,u') on [a, b] such that a < u < p, we have
IIu'II. < R.
(4.2)
C. De Coster, P. Habets
10
Sketch of proof. Define R to be such that
IR sds > maxf(t) - mina(t) Wks)
t
(4.4)
t
and consider a solution u of (4.3) such that a < u < ,Q. Observe first that there exists r E [a, b] with Iu'(r)I < r. Consider next an interval I = [to, t1] or [t1, to] such that u'(t) > r on I and u'(to) = r. The proof follows then from the computation ru'(tl) s ds / `' u'(t)u,.(t) f tl u'(t)f (t, u(t), u'(t)) fu-(to)
cs)
-
(t))
dt -
< lu(t1) - u(to)I
to
v(u'(t))
dt
0
max/3(t) - min a(t)
Notice that the growth assumption
If(t,u,v)I
plies to functions u such that some a priori bound on the derivative is known at the end points of the interval of definition. This is clearly the case for a Neumann problem.
Proposition 4.2. Let a and ,3 E C([a, b]) be such that a < /3. Define E from (2.3) and let r > 0 and V : R+ -+ R be a positive continuous function satisfying (4.1). Then there exists R > 0 such that for every continuous function f : E -R which satisfies V(t, u, v) E E,
f (t, u, v) < w(I vI)
(4.5)
and every solution u of (4.3) on [a, b] such that a < u < /3, u'(a) < r and u'(b) > -r, we have Ilu'll> < R. In the above result, the assumption (4.5) is called a one-sided Nagumo condition. Such a condition applies for a problem such as u" + uj32 + g(t, u) = 0,
u'(a) = 0, u(b) = 0.
Similar results hold if we modify the inequality as follows: (a) (b) (c)
V(t, u, v) E E, f (t, u, v) > -W(I vI ), u'(a) > -r and u(b) V(t, u, v) E E, sgn(v) f (t, u, v) < cp(Ivl) and lu'(a) 1 < r,
V(t, u, v) E E, sgn(v) f (t, u, v) > -V(Ivl) and lu'(b) I < r.
< r,
(4.6) (4.7) (4.8)
An Overview of the Method of Lower and Upper Solutions for ODES
11
Remark. Nagumo conditions, and more specifically one-sided Nagumo conditions, have a nice geometric interpretation. Assume for simplicity a and 3 are constant and f (t, a, 0) < 0 < f (t, /3, 0). Consider then the assumption f (t, u, v) < y>(v) for v > 0. In the phase space (u, u') we define the curve u' = v(u), where v is the solution of the problem
v'_W(V),
v(a)=vo.
If we assume the Nagumo condition (4.1) and choose vo E [0, r] it is easy the vector field corto see that v is defined on [a,,0]. Also, if u E responding to the differential equation (4.3) points downward along that curve. An assumption such as f (t, u, v) >
for v < 0
furnishes a curve u' = µ(u) < 0 so that the vector field points upward. More generally, the vector field cannot be tangent from the inside along the boundary of the set S := {(u,u') I a < u < /3, µ(u) < u' < v(u)}. This geometry is essential to obtain existence of solutions from a degree argument or a modification method as we did in this paper. Notice that it is not essential for the vector field to point upward along the curve u' = p(u) and downward along the curve u' = v(u). Other geometries are possible. These underlie the various one-sided Nagumo conditions we mentioned above. Using Nagumo conditions we can prove existence theorems.
Theorem 4.3. Let A, B E R, al, bl E R, a2, b2 E R+ with ai + a2 > 0, bi + b2 > 0. Assume a and /3 E C([a, b]) are C2-lower and upper solutions of the problem (2.1) such that a < /3. Let E be defined by (2.3), r > 0, W : R+ - R be a positive continuous function satisfying (4.1), and let f : E --4R be a continuous function which satisfies (4.2). Assume there exists N > 0 such that for all t E [a, b],
f (t, a(t), D+a(t)) > -N and f (t, /3(t), D+/3(t)) < N.
(4.9)
Then the problem (2.1) has at least one solution u E C2([a, b]) such that for all t E [a, b], a(t) < u(t) < /3(t).
Sketch of proof. Let R be large enough so that (4.4) is satisfied. Increasing the value of N if necessary, we can assume N > maxlo,Rl
then the modified problem
12
C. De Coster, P. Habets
u" = 1(t, u, u') + arctan(u - y(t, u)), u(a) - a2u'(a) = A - aly(a, u(a)) + y(a, u(a)), u(b) + b2u'(b) = B - biy(b, u(b)) + y(b, u(b)), where 1(t, u, v) := max{min{ f (t, -y(t, u), v), N}, -N} and y(t, u) is defined in (2.5). The proof follows now as for Theorem 2.3. Notice that (4.9) is satisfied if a, Q E W 1 ,°° (a, b) or if f does not depend on derivatives.
Example 4.4. Consider the Dirichlet boundary value problem u" = u'21n(u'2 + 1) - t, u(0) = 0, u(1) = 1. It is easy to see that a(t) = 0 is a lower solution, (3(t) = 2t is an upper solution and f (t, u, v) satisfies a Nagumo condition. From Theorem 4.3, this problem has a solution u such that, for all t E [0, 1], 0:5 u(t) < 2t. If the nonlinearity depends on u', the existence of a well-ordered pair of lower and upper solutions alone, i.e., without a Nagumo condition, does not guarantee the existence of a solution. Such a situation holds true in the following example.
Example 4.5. Consider the problem
u"=g(u')+u+1, u(0) =0, u(T) =0, where g is continuous, g(v) is positive if v # 0, g(0) = 0 and 1/g E L'(R).
Observe that -1 and 0 are lower and upper solutions of this problem. Further every solution is a convex function such that u E [-1, 0] and u" > g(u'). As 1/g E L1(R) we have a contradiction for T large enough. In case a2 or b2 is non-zero, we can use one-sided Nagumo conditions. For example for the Neumann problem (2.2), the following theorem holds.
Theorem 4.6. Let a and /3 E C([a, b]) be C2-lower and upper solutions of the problem (2.2) such that a <,3. Let E be defined by (2.3), r > 0, p : R+ -+ R be a positive continuous function satisfying (4.1), and let f : E -- R be a continuous function which satisfies one of the conditions (4.5), (4.6), (4.7) or (4.8). Assume there exists N > 0 such that for all t E [a, b], (4.9) holds. Then the problem (2.2) has at least one solution u E C2([a, b]) such that for all t E [a, b),
a(t) < u(t) < Q(t). Any LP-Caratheodory function f that satisfies a Nagumo condition is L°°-Caratheodory. In the next result, we extend the Nagumo condition so as to deal with LP-Caratheodory functions which are not LOO-Caratheodory and obtain solutions of (2.1) in W2,p(a, b).
Theorem 4.7. Let A, B E R, ai, bi E R, a2, b2 E R+, a2 + a2 > 0, b2 + b2 > 0 and p E [1, oo]. Assume a and 6 E C([a, b]) are W2'1-lower
An Overview of the Method of Lower and Upper Solutions for ODES
13
and upper solutions of problem (2.1) such that a < 6. Let E be defined in (2.3) and f : E - R be an LP-Carathdodory function. Suppose there exists N E L1(a, b), N > 0 such that, for a.e. t E [a, b], f (t, a(t), D+a(t)) > -N(t) and f (t, /3(t), D+/3(t)) < N(t). Assume moreover that there exist E C(R+, Ro) and t/J E LP(a, b) satisfying
(a)
f
00
1/q
g W()
where
ds > II1IILr(max/3(t) - mina(t))1/q, t t r=max{pn-ab (3b-Qa } -n ' b-a ' P-1'
(b) for a.e. t E [a, b] and all (u, v) such that (t, u, v) E E, If (t, u, v)I <'(t)co(IvI) Then the problem (2.1) has at least one solution u E for all t E [a, b], a(t) < u(t) < /3(t).
b) such that
Sketch of proof. Step 1 - Consider the modified problem nu = in (t, u, u'), u(a) - a2u'(a) = A - a1ry(a, u(a)) +7(a, u(a)), u(b) + b2u'(b) = B - b1-y(b, u(b)) +,y(b, u(b)) where (a) fn(t, u, v) is defined by
1(t, a(t), D+a(t)),
if u < an (t),
n(u - an (t)) f (t, a(t), v)
+n(a(t) - u) f (t, a(t), D+a(t)), _/ f (t) u, v), n(/3n(t) - u) f(t, qq/ N(t), v) R+n(u - 3(t))f (t, A(t), D+,3(t)),
f(t,,3(t), D+Q(t)),
if an (t) < u < a(t), if a(t) < u < /3(t), if /3(t) <- u < an(t), if On(t) < u,
(b) 1(t, u, v) := max{min{ f (t, u, v), N(t)}, -N(t)}, (c)
for all n E N\101, an(t) = a(t) -
n
,
3n(t) _ /(t) + n,
(d) R > 0 and N(t) are assumed large and ry(t, u) is defined by (2.5). Step 2 - Existence of a solution un of the modified problem follows from Schauder's Theorem. Next, we prove from an usual maximum argument that this solution un satisfies an < n < on. As f is bounded, we also have
IIunIk < R. Step 3 - From the Arzela-Ascoli Theorem, the sequence (un),, converges in C'([a, b]) to some function u. Next, we obtain from Lebesgue's dominated un, u'n) = j(-, u, u') in L1(a, b). At last, convergence theorem, limn -= using the closedness of the derivative we have that u is a solution of (2.1).
14
C. De Coster, P. Habets
Example 4.8. Consider the boundary value problem
u"_ - U'a+u+t, u'(0)=0, u'(1)=0, W2,2-`(0,1), fore > 0 small where 1 _< a < 3/2. Existence of a solution in enough, follows from Theorem 4.7 with a(t) = -1, 8(t) = 0, p = 2a < 2, tli(t) _ and cp(y) = ya + 1. Notice that
77
f 5
00 S11q
joo
ds=
Sa-1 Sa
V(S)
ds=oo. 1
Dirichlet problem
Dirichlet boundary value problems
u" = f(t,u),
u(a) = 0, u(b) = 0
(5.1)
can be studied for more general nonlinearities than LP-Caratheodory functions. This remark follows from the analysis of the linear problem
u" = h(t),
u(a) = 0, u(b) = 0.
(5.2)
Its solution can be written b
u(t) = fa G(t, s)h(s) ds, provided h E A:= {h E LtoC(]a, b[) I (s - a)(b - s)h(s) E L1(a, b)}. In that case, solutions u are in C([a, b]) n WWl ()a, b[). For example, the problem
t(1 - t)u" = 1,
u(0) = 0, u(1) = 0,
has a solution u(t) = t log t+(1-t) log(1-t) which is in C([a, b])nW o' (]a, b[) but not in C'([a, b]). Notice that the condition b(t - a)(b - t)lh(t) I dt < +oo
Ja
is necessary if we want the solutions of (5.2) to be continuous on [a, b]. This suggests how to generalize the LP-Caratheodory conditions to deal with the Dirichlet problem. Let us first introduce the following definition.
A function f : E c [a, b] x R -+ R is said to be an A-Caratheodory function if : (i) for a.e. t E [a, b], the function f (t, ) is continuous, (ii) for all u E R, the function f (., u) is measurable and (iii) for any r > 0 there exists h E A such that for a.e. t E [a, b] and all u with (t, u) E E and Jul < r, we have If (t, u) I < h(t). Observe that L' (a, b) C A, so that this definition generalizes the classical L1-Caratheodory conditions on f.
An Overview of the Method of Lower and Upper Solutions for ODES
15
We can now state the main result for the Dirichlet problem (5.1).
Theorem 5.1. Assume that a and p are W2,1-lower and upper solutions of (5.1) such that a < 6. Let E _ {(t, u) E [a, b] x R I a(t) < u < Q(t)} and f : E -' R be an A-Caratheodory function. Then the problem (5.1) has at least one solution u E W2,A(a, b) such that for all t E [a, b], a(t) < u(t) :5,3(t).
The argument used to prove this theorem follows the line of the proof of Theorem 2.3.
Example 5.2. Consider the boundary value problem
u" + ui1/2 _Ie =- 0, u(O) = 0, u(7r) = 0.
It is easy to see that ,0(t) = 0 is an upper solution and a(t) = t In - t is a lower solution. Hence we have a solution u such that for all t E [0, ir],
tIn I -t
Observe that, in this example, the function f (t, u) is not L1-Caratheodory.
Remark. In the same way, we can consider the problem
u" = f(t,u), u(a) = 0, biu(b) + b2u'(b) = 0.
Here we look for solutions in C([a, b]) n C' (]a, b]) so that the boundary conditions make sense. A basic assumption on f : E --, R is that, for a.e. t E [a, b] and all u with (t, u) E E, If (t, u)I < h(t), where h : [a, b] -' R+ is some measurable function such that jb (s - a)h(s) ds < oo.
6
Relation with degree theory
In this section, for simplicity, we restrict attention to the Neumann problem
u" = f (t, u, u'),
u'(a) = 0, u'(b) = 0,
(6.1)
which can be written as a fixed point problem
u = Tu, where T : C '(1a, b]) - C1([a, b]) is defined from b
( Tu)(t) :=
f
G(t, s)[ f (s, u(s), u'(s)) - u(s)J ds
and G(t, s) is the Green function of the corresponding linear problem
6.2)
16
C. De Coster, P. Habets
u" - u = h(t),
u'(a) = 0, u'(b) = 0.
Let us associate to a pair of lower and upper solutions a and /3 the set 0 = {u E C' ([a, b]) I Vt E [a, b], a(t) < u(t) < Q(t), tu'(t) I < R} and consider the degree deg(I - T,11).
To this end we will have to reinforce the concepts of lower and upper solutions so that the boundary of S2 does not contain solutions of (6.1). This amounts to imposing that R be an a priori bound on the derivative of solutions and that these solutions cannot be tangent to the curves u = a(t) and u =,3(t) respectively from above or from below. This motivates the definitions we introduce in the following section.
6.1
The strict lower and upper solutions
Definitions 6.1. A lower solution a of (6.1) is said to be strict if every solution u of (6.1) with u > a is such that u(t) > a(t) on [a, b]. Similarly, an upper solution ,6 of (6.1) is said to be strict if every solution u of (6.1) with u < 0 is such that u(t) < 13(t) on [a, b].
The classical way to obtain such a notion in the case of a continuous f and a, 6 E C2 ([a, b]) is described in the next proposition.
Proposition 6.2. Let f be continuous and a E C2([a, b]) be such that (a) for all t E [a, b], a"(t) > f (t, a(t), a'(t)); (b) a'(a) > 0, a'(b) < 0.
Then a is a strict C2-lower solution of (6.1). If f is not continuous but LP-Caratheodory, this last result does not hold anymore. In fact, even the stronger condition for a.e. t E [a, b], a"(t) > f (t, a(t), a'(t)) + I
does not prevent solutions u of (6.1) to be tangent to the curve u = a(t) from above. This is, for example, the case for the bounded function
f(t,u,v)
-1 u2 + cost 1 + cost
if u < -1, if - 1 < u < cost, t # ir,
_ - cost if cost < u, if we consider a(t) -1, u(t) = cost, a = 0 and b = 21r. This remark motivates the following proposition.
Proposition 6.3. Let f : [a, b] x R2 - JR be an L' -Caratheodory function. Assume that a E C([a, b]) is not a solution of (6.1) and (a) for any to E [a, b], either D- a(to) < D+ a(to) or there exist an interval Io and co > 0 such that to E intlo (or to E
An Overview of the Method of Lower and Upper Solutions for ODES
17
{a, b} n 10), a E W2" (10) and for a.e. t E Io, for all u, v, with a(t) < u < a(t) + co, al(t) - co < v < a'(t) + Eo, we have a"(t) > f (t, u, v); (b) a(a) > 0, a'(b) < 0.
Then a is a strict W2"1-lower solution of (6.1).
Using the same ideas we can obtain corresponding results for upper solutions.
6.2
Existence and multiplicity results
Now we can state the key result of this section.
Theorem 6.4. Let a and 0 E W1"0°(a,b) be strict W2"1-lower and upper solutions of problem (6.1) such that on [a, b], a(t) < 0(t). Let E be defined by (2.3), p E [1, oo) and q E [1, oo] with 1 + p = 1. Assume f : E -+ R satisfies an LP-Caratheodory condition and there exist cp E C(R+, Ro ), 7i E LP (a, b) and R > 0 such that
f R -(3) ds > II0IILP(maX,6(t) t
mina(t))1/e.
Suppose moreover the function f satisfies one of the one-sided Nagumo conditions (a) V(t,a,v) E E, (b)
d(t,u,v) E E,
(c) d(t, u, v) E E, (d) V(t, u, v) E E, Then
f(t,u,v) 5 ip(t)sv(Iv1), f(t,u,v) > -V)(t)W(Ivl), sgn(v) f (t, u, v) < ,y(t)W(IvI ),
sgn(v) f (t, u, v) > -0(t)cp(IvI).
deg(I-T,cl) = 1, when T : C'([a, b]) -+ C1([a, bJ) is defined by (6.2) and S2 = {u E C1([a,b]) I Vt E [a,b], a(t) < u(t) < Q(t), Iu'(t)I < R}. In particular, the problem (6.1) has at least one solution u E W2'P(a, b) such that for all t E [a, b],
a(t) < u(t) < 0(t). Sketch of proof. Let R > R be such that R _> maxf1la'11., 110'11.1 and consider the boundary value problem u" - u = .f (t, 7(t, u), d y(t, u)) - 'y(t, u), u'(a) = 0, u'(b) = 0,
where I (t, u, v) = f (t, u, max{min{v, f?), -R}) and y(t, u) is defined in (2.5). This problem is equivalent to the fixed point problem
u=Tu, where T : C 1(f a, b]) -+ C 1([a, b]) is defined by
18
C. De Coster, P. Habets
f
(Tu)(t) =
b
G(t, s) [1(s, 7(s, u(s)), dt7(s, u(s))) - 7(s, u(s))]ds.
a
Observe that T is completely continuous and there exists R large enough so that 12 C B(0, R) and T(C1([a, b])) C B(0, R). Hence we have, by the properties of the degree, deg(I - T, B(0, R)) = 1. We know that every fixed point of T is a solution of (6.3). Arguing as above, we see that a _< u _< /3 and, as a and /3 are strict, a < u < /3. Next, arguing as in Proposition 4.2 we have IIu'II.,, < R. Hence, every fixed point of T is in Sl and by the excision property we obtain deg(I - T, 11) = deg(I - T, SZ) = deg(I - T, B(0, R)) = 1. 0 Remark. The same result holds true if we replace a by the maximum of a finite number of lower solutions and /3 by the minimum of a finite number of upper solutions. The simplest multiplicity result that we can deduce from Theorem 6.4 is obtained when we have two pairs of lower and upper solutions.
Theorem 6.5. Assume a 1, i3 and C(2, /32 E W 1 "O° (a, b) are two pairs of W2,1-lower and upper solutions of (6.1) such that a1 < /31 < /32i al _< a2 < /32i and a2 ¢ /31. Assume further 81 and a2 are strict upper and lower solutions.
Let E be defined by (2.3) (with a = a1 and /3 = 02), p and q E [1, oo] with p + .1 = 1. Suppose f : E -' R is an L"-Caratheodory function and there exist cp E C(R+, Ro ), 1/i E L'(a, b) and R > 0 such that R 31/4
f
ds > Ik&IILP(max/32(t) - mina,(t))1/q
W(s)
holds and for a. e. t E [a, b] and all (u, v) such that (t, u, v) E E,
If(t,u,v)I <- 0(t)W(IvI) Then, the problem (6.1) has at least three solutions u1i U2, u3 E W2,p(a, b) such that
a1
u3(tl) > ,31(t1), u3(t2) < a2(t2)-
Sketch of proof. Consider the modified problem u"(t) - u(t) = f (t, 712(t, u), dt712(t, u)) - 7'12(t, u),
u'(a) = 0, u'(b) = 0, where 712(t, u) = max{min{/32(t), u}, al (t)}.
Step 1 - Solutions u of the modified problem are such that al < u < 82.
An Overview of the Method of Lower and Upper Solutions for ODES
19
Step 2 - There exist three solutions u; (i = 1, 2,3) of the modified problem
such that
a, :5
l
and there exist t1, t2 E [a, b], with u3(ti) > 31(tl), i!3(t2) < a2(t2).
This follows from Theorem 6.4 and the subsequent remark applied successively with n1,1 = {u E Cl ([a, b]) I Vt E [a, b],
al(t) - E < u(t) < min{p1(t),
(t) + E}, IIu'II,,. < R},
p2,2 = {u E Cl ([a, b]) I Vt E [a, b],
max{al(t) - E, a2(t)} < u(t) < i32(t) + E, IIu'II0 < R} and
521,2 = {u E C1([a,b]) I dt E [a,b], ai(t)-E < u(t) <,62(t)+E, IIu IIoo < R}.
Step 3 - There exist solutions u= (i = 1, 2, 3) of (6.1) such that
al < u1 <31, a2 81(tl), u3(t2) < a2(t2)
Here we take ul and u2 to be respectively minimal and maximal solutions between al and ,02 and u3 = u3. l7
Remark. In this theorem, we can use one-sided Nagumo conditions as in Theorem 6.4.
Example 6.6. Consider again the Example 3.3, u" + sin u = h + h(t), u'(0) = 0, u'(7r) = 0, where h E R and h E L' (0, 7r) has mean value zero. Let w be defined from (3.1) and assume that for all t E [0, 7r], IhI < sin(2 + w(t)). In this case, the functions a = z + w and a + 27r are strict lower solutions, and the functions ,0 = 32 + w and 3 + 27r are strict upper solutions. Further, a < 3 < a + 27r < /3 + 27r. Hence, we can apply Theorem 6.5 and deduce the existence of a second solution v u mod 27r.
6.3
Non-well-ordered lower and upper solutions
The existence of lower and upper solutions such that a > 3 is not sufficient to guarantee the solvability of this problem. This is clear from the example u" + u = sin t, u'(0) = 0, u'(ir) = 0,
20
C. De Coster, P. Habets
which has no solution although a(t) = 1 and /3 = -1 are respectively lower and upper solutions. The difficulty comes here from the interference of the nonlinearity with the second eigenvalue of the problem \2 = 1. In the theorem we present in this section, a boundedness assumption will imply the nonlinearity remains "below this second eigenvalue".
Theorem 6.7. Let a and /3 E W1"OO(a,b) be W2,1-lower and upper solutions of (6.1) such that a ¢ 0. Assume f : [a, b] x R2 -+ R is an L1Carath6odory function such that for some h E L1(a, b), If (t, u, v) I < h(t) on [a, b] x R2. Then, there exists a solution u of (6.1) in S :_ {u E C1([a,b]) I 3t1 E [a, b], u(tl) > /3(tl), 3t2 E [a, b], u(t2) < a(t2)}.
Sketch of proof. Step 1 - Assume /3 is not a strict upper solution. There exists then a solution u of (6.1) such that u < 8 and for some tl E [a, b], u(t1) = 0(t1). As further a ¢ /3, there exists t2 E [a,bJ such that a(t2) > /3(t2). It follows that a(t2) > u(t2) and u is a solution of (6.1) in S. A similar argument holds if a is not strict. Step 2 - Suppose now that /31 = /3 and a2 = a are strict upper and lower solutions. For each r > 1, we define fr(t,u,V) = f(t,u,v), if Jul < r, = (1 + r - IuI) f (t, u, v) + (Jul - r) u, if r < Jul
u" = fr(t, u, u'), u'(a) = 0, u'(b) = 0.
(6 .4)
Claim - There exists k > 0 such that, for any r > k, solutions u of (6.4), which are in S, are such that IIuII,, < k. Conclusion - Consider the problem (6.4), with r > max{IIafl., II)3IIoO,k}.
It is easy to see that a1 = -r - 2 and /32 = r + 2 are lower and upper solutions. We deduce now from Theorem 6.5 the existence of three solutions of (6.4)
one of them, u, being such that for some t1i t2 E [a, b], u(t1) > /3(t1) and u(t2) < a(t2). Hence, u E S and from the claim, IIuII,,. < k. This implies that u solves (6.1) and proves the theorem. O
Example 6.8. Consider the case where a Landesman-Lazer condition is satisfied at the right of the first eigenvalue. Assume there exists a function k E L1(a, b) and R > 0 such that b
f (t, u, v)sign(u) < k(t)
and
k=
b
1
aj
k(t) dt < 0, a
An Overview of the Method of Lower and Upper Solutions for ODES
21
for a.e. t E [a, b], every u with Jul > R and every v E R. Define then w to be the solution with mean value zero of w" = k(t) - k, w'(a) = 0, w'(b) = 0.
It is easy to see then that for K > 0 large enough, a = K + w and p = -K - w E W1"°°(a,b) are respectively W2"1-lower and upper solutions of (6.1) such that 3 < a. If further, f : [a, b] x R2 --+ R is an L'-Caratheodory function such that for some h E L1(a, b),
If(t,u,v)I < h(t) on [a,b] x R2, then Theorem 6.7 applies and there exists a solution u of (6.1).
REFERENCES [1] K. Ako, On the Dirichlet problem for quasi-linear elliptic differential equations of the second order, J. Math. Soc. Japan 13 (1961), 45-62.
[2] H. Amann, On the number of solutions of nonlinear equations in ordered Banach spaces, J. Funct. Anal. 11 (1972), 346-384. [3]
H. Amann, A. Ambrosetti and G. Mancini, Elliptic equations with noninvertible Fredholm linear part and bounded nonlinearities, Math. Z. 158 (1978), 179-194.
[4] S. Bernstein, Sur certaines equations differentielles ordinaires du second ordre, C. R. A. S. Paris 138 (1904), 950-951. [5)
C. De Coster and P. Habets, Upper and lower solutions in the theory of ODE boundary value problems: classical and recent results. In Nonlinear Analysis and Boundary Value Problems for Ordinary Differential Equations (F. Zanolin, ed.), C.I.S.M. Courses and Lectures 371, Springer-Verlag, New York (1996), 1-79.
[6] C. De Coster and M. Henrard, Existence and localization of solution for elliptic problem in presence of lower and upper solutions without any order, J. Differential Equations 145 (1998), 420-452.
[7] P. Habets and F. Zanolin, Positive solutions for a class of singular boundary value problem, Boll. U.M.I. 9-A (1995), 273-286. [8] I.T. Kiguradze, A priori estimates for derivatives of bounded functions satisfying second-order differential inequalities, Diferentsial'nye Uravneniya 3 (1967), 1043-1052.
[9] I.T. Kiguradze, Some singular boundary value problems for ordinary nonlinear second order differential equations, Diferentsial'nye Uravneniya 4 (1968), 1753-1773.
22
C. De Coster, P. Habets
[10] Y.S. Kolesov, Periodic solutions of quasilinear parabolic equations of second order, Trans. Moscow Math. Soc. 21 (1970), 114-146. [11] W. Mlak, Parabolic differential inequalities and Chaplighin's method, Ann. Polon. Math. 8 (1960), 139-152. [12] M. Nagumo, Uber die differentialgleichung y" = f (t, y, y'), Proc. PhysMath. Soc. Japan 19 (1937), 861-866.
[13] M. Nagumo, On principally linear elliptic differential equations of the second order, Osaka Math. J. 6 (1954), 207-229.
[14] G. Peano, Sull'integrabiliti delle equazioni differenziali di primo ordine, Atti Acad. Torino 21 (1885), 677-685. [15] O. Perron, Ein neuer existenzbeweis fur die integrale der differentialgleichung y' = f (x, y), Math. Ann. 76 (1915), 471-484.
[16] E. Picard, Sur l'application des methodes d'approximations successives h 1'etude de certaines equations differentielles ordinaires, J. de Math. 9 (1893), 217-271. (17] G. Prodi, Teoremi di esistenza per equazioni alle derivate parziali non lineari di tipo parabolico, Nota I e II, Rend. Ist. Lombardo 86 (1953), 1-47.
[18] A. Rosenblatt, Sur les theoremes de M. Picard dans la theorie des problemes aux limites des equations differentielles ordinaires non lineaires, Bull. Sc. Math. 57 (1933), 100-106. [19] K. Schmitt, Boundary value problems for quasilinear second order elliptic equations, Nonlinear Anal. T.M.A. 2 (1978), 263-309. [20] G. Scorza Dragoni, II problema dei valori ai limiti studiato in grande per gli integrals di una equazione differenziale del secondo ordine, Gior-
nale di Mat. (Battaglini) 69 (1931), 77-112. [21] G. Scorza Dragoni, Intorno a un criterio di esistenza per un problema
di valori ai limiti, Rend. Semin. R. Accad. Naz. Lincei 28 (1938), 317-325.
[22] G.M. Troianiello, On solutions to quasilinear parabolic unilateral problems, Boll. U.M.I. 1-B (1982), 535-552. Colette De Coster
Universite du Littoral Cote d'Opale LMPA J. Liouville 50 rue F. Buisson, BP 699 62228 Calais Cedex, France
decosteralmpa.univ-littoral.fr
Patrick Habets Universite Catholique de Louvain Inst. de Math. Pure et Appliquee Chemin du Cyclotron 2 1348 Louvain-la-Neuve Belgique Habetstanma.ucl.ac.be
On the Long-time Behaviour of Solutions to the Navier-Stokes Equations of Compressible Flow Eduard Feireisl 1
Problem formulation
Let SZ C R3 be a spatial domain filled with a fluid. We shall assume that the motion of the fluid is characterized by the velocity u = u(t, x) of the particle moving through x E St at time t E I C R1. Moreover, for each time t, we shall suppose the fluid has a well-defined mass density p = p(t, x). The behaviour of p, u is determined by the Navier-Stokes system of equations:
Conservation of mass: Otp + div(p u) = 0,
(1.1)
8t(pu) + div (pu ® u) + Vp = div T(u) + of
(1.2)
Balance of momentum:
for all tEI,xEQ. We assume the fluid is Newtonian, i.e., the stress tensor T is given by the formula T(u) = µ(Vu + (Vu)t) + A div u Id (1.3) where A, µ are the viscosity coefficients, assumed constant and satisfying
µ>0, A+p>0.
(1.4)
Moreover, we suppose the fluid is isentropic, i.e., the pressure p and the density p are functionally dependent and the relation between them is given by the equation of state
p = p(p), p E C[0, oo) fl Cl (0, oo), p'(p) > 0 for all p > 0.
(1.5)
Work supported by Grant 201/98/1450 of GA OR and A1019703 of GA AV OR
24
E. Feireisl
The boundary 812 will be smooth and solid in the sense that the fluid cannot cross it, i.e., u(t,x).n(x) = 0 for all t E I, X E 812
(1.6)
where n denotes the outer normal vector. In addition, we shall consider either the no-slip
[u(t,x)],=0 foralltEI, xE812
(1.7)
[T(u(t, x)).n(x)]r = 0 for all t E I, X E 812
(1.8)
or the no-stick
boundary conditions where [v], denotes the tangential component of a vector v. The fluid is driven by a given external force f = f(t, x). The main questions we want to address here read as follows:
Assume f is a gradient of a time independent potential F, i.e.,
f(t,x) = VF(x) for all t, x.
(1.9)
The vorticity component of f being zero, one can anticipate the solutions of the problem will behave like those of gradient-like systems, in particular, one should have
o(t) --+ o Lou - 0 in an appropriate topology as t -+ 00 where o, is a solution of the stationary problem Vp(e) = oVF(x) in 12.
(1.10)
As we will see such a conclusion holds even for weak solutions of the problem, provided certain additional hypotheses are imposed on F and the form of the state equation (1.5). The situation becomes particularly interesting if 12 is unbounded due to the possible "loss of mass" at infinity (see Section 4). Next, we examine the case when f depends on t in a periodic way, i.e.,
f(t + w, x) = f(t, x) with a certain period w > 0.
(1.11)
We shall show that the problem (1.1), (1.2) complemented by the no-stick boundary conditions possesses at least one globally defined (weak) solution with the same property, i.e., periodic in t with the period w. It should be pointed out that we look for solutions with a given positive mass 0
<M=fo(t) dx. in
Navier-Stokes Equations of Compressible Flow
25
To obtain such a result, some restrictions are needed concerning the state equation (1.5), specifically, the growth of p for large values of
e should be fast enough - a condition which also appears in the hypotheses of the known existence theorems.
2
The variational formulation
In accordance with the boundary conditions (1.6), we introduce a class of functions E [C-(f2) n C2(12)]3 I ¢ satisfies (1.6)}
D. (0) and
E D,, 10 satisfies (1.7)}.
DntW)
Multiplying, formally, the continuity equation (1.1) by the expression Y(p) we get 8bQO)
+ div(b(e)u) + W(e)e - b(o)]div u = 0 .
(2 . 1)
This motivates the following definition. We shall say that p, u is a renormalized solution of the equation (1.1) if the identity
/,I
[P
-
u'8x; cp) - k(1Q,k] o 9.,,,u' (P dx dt = 0
(2.2)
holds for any k > 0 and any test function cp E D(I x R3). Here (and in what follows), the summation convention is used. The fact that we consider
cp with compact support in R3 rather than in fI reflects the boundary conditions (1.6) imposed on u. Multiplying, formally again, the equation (1.2) by u:, integrating by parts and making use of the boundary conditions (1.6), (1.7) or (1.6), (1.8) respectively, we obtain the energy inequality
dEddtt)
+Jn
µlVu(t)I2+(a+µ)Idiv u(t)I2 dx < I a f(t).u(t) dx for all t E I n
where the energy E is defined as
E(t)
= Ekin(t) + Eint(t),
Ekin(t)
=
21
E=nt
=
L1Qtdx
n
e(t)lu(t)12 dx,
(2.3)
E. Feireisl
26
with P (0)
We shall say that Q, u is a finite energy solution of the equation (1.2) if (2.3) holds in V(I) and the identity
f f gu'(At + eu'u'a=jco) - jiaz,u'a,,co
( 2.5)
+[p(P) - (A + p) aZj uj)ax; cp' + Af' p' dx dt = 0 is satisfied for any cp E D(I; Dnt(SZ)) or any cp E D(I; Dn(S2)) in accordance
with the boundary conditions (1.6), (1.7) or (1.6), (1.8) respectively. Finally, we shall define the Sobolev spaces W01'2(11)
- a completion of Dnt(Sl) in the norm of the space [W12(n)]3
and
a completion of Dn in the norm of the space
[W1'2(SZ)]3
Definition 2.1. Let I C R' be an open interval. We shall say that a pair of functions p, u is a weak solution of the problem (1.1), (1.2), (1.5) complemented by the boundary conditions (1.6), (1.7) or (1.6), (1.8) respectively if . The density 2 > 0 belongs to the class
,o E L' (I; L' (0)), P(e) E L' (I; L'(11)); The velocity u satisfies u E L10j, ; Wo'2(12)) or u E L C(I; W,i,'2(S2)) respectively, L01 u12
EL
(I; L1(S2));
The couple Q, u is a renormalized solution of the equation (1.1) and a finite energy solution of (1.2).
3 On the zero velocity stationary solutions Consider the case when f = VF(x). Obviously, the energy inequality (2.3) now reads d
r
p(t)F dx] < - jn pIu(t)12 + (A + p) Idiv u(t)12 dx dt [E(t) - J to be understood in D'(I).
(3.1)
27
Navier-Stokes Equations of Compressible Flow
Consequently, the quantity on the left-hand side of (3.1) represents a sort of Lyapunov function for the problem (1.1), (1.2) and one may anticipate that any global solution, provided it exists, will converge to a stationary state of zero velocity, i.e.,
p(t) - o, in L' (Q), olul -+ 0 in L'(Sl) as t - 00 where o, solves the equation C7x, p(o(x)) = o(x) 8x, F(x), i = 1, 2,3 for all x E Q.
(3.2)
Thus it is of interest to examine nonnegative solutions of the problem (3.2) "normalized" by the condition
.(x) dx = M > 0.
(3.3)
,o
We report the following result (see [8, Theorem 1.1, Theorem 1.2]):
Theorem 3.1. Let Il C R3 be an arbitrary domain. Assume p(o) = ap with a > 0, y > 1.
Let F be a locally Lipschitz function on n. If y > 1 assume, in addition, that the level sets [F(x) > k] = {x E Q I F(x) > k} are connected in n
for any k < SUP-En F(x). (1) Then given M > 0, there exists at most one function o > 0 such that
Bx,P(o)=oB3,F, i = 1,2,3 in V'(0),
J in
odx=M.
Vii) Moreover, if such a function exists, it is given by the formula
o(x) = exp(F(x
K
a-
if 'y = 1
or
LO (x) _
(yay l)
([F(x) - K)+)
for y > 1
where K is a certain constant depending on M. Corollary 3.2. Let y = 1, F locally Lipschitz and bounded from below on St and on meas(1t) = oo. Then there is no solution of the problem (3.2) with a finite and strictly positive mass M.
28
4
E. Feireisl
Convergence of global trajectories
In this section, we state a result on convergence of global solutions, provided they exist, of course. Unfortunately, we are able to treat only the case when
the equation of state takes the form
p(o)=ap
,
a>0, 7>
3
2
We remark, however, that the case -y E [1, 2] could be handled in exactly the same way, provided suitable a priori estimates of the density o were available.
Theorem 4.1. Let Il C R3 be a domain with compact Lipschitz boundary. Let f = VF where F is bounded and Lipschitz continuous on U. Moreover, if Il is unbounded, assume
lim ess sup (IF(x)I + IVF(x)I) = 0. r-oo I xI>r Let the level sets [F > k] be connected for any k < supxEn F(x). Suppose p(o) = aory with a > 0, ry > 2
Finally, let o, u be a weak solution of the problem (1.1), (1.2) and (1.5) complemented by the no-slip boundary conditions (1.6), (1.7) defined on an interval I = (to, oo). Then there exists a unique function oq E Lry n L'(0), a solution of the stationary problem (3.2), such that
o(t) -' o, strongly in L"(11), o(t)u(t) -' 0 strongly in [L'(11)]3 as t - oo. (4.1)
The proof is given in [6, Theorem 1.1].
Remark. Note that existence of the weak solutions for p as in Theorem 4.1 has been proved only recently in [6].
5 On the existence of time-periodic solutions To conclude, we shall address the question of the existence of time-periodic solutions. To this end, assume that f E [L°O(R' x St)]' and
f (t + w, x) = f (t, x) for a.e. t, x
(5.1)
for a fixed positive period w. Mainly for technical reasons, we shall restrict ourselves to the case when
Il = (0, a) x (0, b) x (0, c) C R3
and we consider the no-stick boundary conditions (1.6), (1.8).
(5.2)
Navier-Stokes Equations of Compressible Flow
29
Theorem 5.1. Let f C R3 be of the form (5.2) and assume
p(p)=ap'' where a>0, .y> 5
(5.3)
Let f E L°O(R1 x St) be a time-periodic function, i.e., f satisfies (5.1) for a certain w > 0. Then given M > 0 there exists a time-periodic (with the same period w > 0) weak solution p, u of the problem (1.1), (1.2), (1.5) satisfying the no-stick boundary conditions (1.6), (1.8). For the proof see [5, Theorem 1.1].
Dependence on the data
6
In this section, we consider a general barotropic case, i.e., the pressure p and the density p are functionally dependent and the relation between them is given by the equation of state p = p(p), p non-decreasing and locally Lipschitz continuous on [0, oo), p(0) = 0. To simplify the presentation, we consider only solutions p, u periodic in x = (Xi, x2, x3) with a period, say, w = 27r. Accordingly, a suitable function space framework is provided by functions defined on the set
(t,x)EQ=IxT where T = ([0, 21r]I{o,2,,))3 is topologically equivalent to a torus and I = (0, T) is a bounded time interval. We concentrate on the following problem: consider a family Lo", u" of weak solutions of (1.1), (1.2) with p = p. (Lo.), f = fa such that
Pn(pn) is bounded in L°°(I;L'(T)),
(6.2)
pnI unI is bounded in LO°(I;L2(T)),
(6.3)
u;, are bounded in L2(I; W1,2 (T)) for i = 1, 2, 3
(6.4)
pn pn(pn) is bounded in L1(Q) for a certain 0 > 1
(6.5)
and 5
where the functions Pn are determined by (2.4).
Now, since the quantities pn, f approach some limit values p, f, the main issue we intend to discuss here is to find sufficient conditions so that p,, u, may converge to a solution p, u of the limit problem.
E. Feireisl
30
To formulate our main result, some hypotheses concerning the structural
properties of the functions pn are needed. In addition to (6.1), we shall assume there exist positive constants c1, i = 1, 2, 3 such that Clzry - C2 .5 pn(z) < pn(2z) < c3(pn(z) + 1) for all z > 0
(6.6)
holds with the exponent ry,
Moreover, we suppose there exists a function k : [0, oo) ,-+ [0, oo) such
that 0 < p,, (z) < k(Y) for a.e. z E (0, Y).
(6.8)
As for the sequence fn, we require
IfnI bounded in L' (Q) and f,', -- f` strongly in L'(Q), i = 1,2,3. (6.9) Our main result reads as follows:
Theorem 6.1. Let pn be a sequence of functions satisfying the hypotheses (6.1), (6.6)-(6.8) with the quantities ci, i = 1, 2, 3 and k independent of
n. Let on > 0, un solve the equations (1.1), (1.2) in D'(Q) with P = pn(pn) and f = fn satisfying (6.9). Moreover, let the estimates (6.2)-(6.5) be satisfied. Finally, let at least one of the following conditions hold: Either on(0, .) go strongly in L1(T), (6.10)
or 60n0, .) = on(T,.) for all n,
(6.11)
div un -+ 0 strongly in L2(Q).
(6.12)
or Then, passing to subsequences if necessary,
pn -+ p uniformly on compact sets of (0, oo),
(6.13)
and on
o weakly in L2(Q),
(6.14)
p n (pn ) -' p(o) weakly in L'(Q) ,
(6 . 15)
u;, -+ u' weakly in L2(I; W1.2(T)), i = 1, 2, 3
(6.16)
where o, u = [u1, u2, u3], P = p(p) and f satisfy (1.1), (1.2) in 1Y(Q).
Navier-Stokes Equations of Compressible Flow
31
The proof of Theorem 6.1 is given in [8, Theorem 1.1).
Remark 6.1. Following the ideas of DiPerna and Lions [4) one can show that on E C(I; L1(T)) and, consequently, (6.10), (6.11) make sense. Remark 6.2. The last inequality in (6.6) is nothing else but the tion used in the theory of Orlicz spaces. It makes it possible to estimate pn(Pn) in terms of the "potential energy" Pn(on)
i2-condi-
Remark 6.3. The validity of (6.2)-(6.4) along with the hypothesis (6.10) may be easily justified for the solutions of the Cauchy problem with fixed (or compactly varying) initial density and bounded initial energy. The estimates (6.2)-(6.4) also hold for any finite mass time-periodic solution satisfying (obviously) the hypothesis (6.11) (see (5, Lemma 4.2]). The estimate (6.5) may be formally deduced applying a Bogovskii type multiplier A` i%, [go], i = 1, 2, 3 to the equation (1.2). Such a procedure may be rigorously justified both for the Cauchy problem (cf. Lions [15])
and the time periodic case (see [5, Lemma 4.2]). In fact, the only reason to assume the lower bound (6.7) is that it yields, along with (6.5), (6.6), boundedness of on in the space L2(Q). This in turn implies that on is a renormalized solution of the equation (1.1) in the sense of DiPerna and Lions [4], i.e., on E C(I;L1(7)) and b(en)t + div(b(on)un) + (b'(en)en - b(on))div un = 0 in 7Y(Q)
(6.17)
for any b E C' [0, oo) globally Lipschitz on [0, oo).
Remark 6.4. The hypothesis (6.12), no matter how strong it seems, is satisfied in a number of important cases. Suppose, e.g., we have a global weak solution of (1.1), (1.2) with
f (t, x) = g(t, x) + VF(x) where sup max Ige(t, x) I E L1(0, oo). xET
i
Now, the energy E may be modified to contain an additional term fT Fo dx and it is easy to deduce from the energy inequality (2.3) that IVuI2 dx dt is finite.
Consequently, the sequence of the time-shifts on (t, x) = o(t + n, x), un(t, x) = u(t + n, x)
32
E. Feireisl
will satisfy the hypotheses of Theorem 6.1 for any finite interval I, say, I = (0, 1). Accordingly, we obtain
Pn - A, p(P,) - p(p) weakly in L1(Q) where a is a solution of the stationary problem
Op(A)=QVFonT. More precise results as well as a thorough discussion of this problem including the question of uniqueness of the stationary states and development of vacua for large times can be found in Stra§kraba [18] (N=1) and [7] (general case).
Remark 6.5. The hypothesis (6.9) of the strong convergence of f may be omitted (i.e., replaced by the weak convergence of a subsequence) if, for
instance, p = p and p is strictly increasing.
Theorem 6.1 under the hypothesis (6.10) and for p = p, f = f was proved by Lions [15, Chapter 1]. His proof is based on regularity of the commutator u'R,,i[guu] - Ri,i[Lou'u-'] where Re,, = 8x,08=,.
More specifically, by virtue of the results of Coifman and Meyer [3], the above quantity belongs to the Sobolev space W ',q provided u` E W1,2,
pu3ELp,p>2and 1
1
q
p+2
1
(cf. Lions [15, Chapter 1, Step 3]). This argument is correct provided P is bounded in L' with ry > 3, since uu E W1,2 C L6. No indication is given how to carry over this step for general ry > s . As shown in [8], a very elementary proof of Theorem 6.1 may be given based solely on compensated compactness arguments, namely, on the weak continuity of the bilinear operator Qi,j [v, w] = v(8x, A` a-!) [w] - w(8-; A -
) [v]
(6.18)
on the product L1(T) x LQ(T) with
-+-<1. p q 1
1
As a matter of fact, the time-periodic case covered by the hypothesis (6.11) seems more delicate than the Cauchy problem. Consider, for example, a very naive "phase transition" model with
fn=f, fIB=0 for some BC
Navier-Stokes Equations of Compressible Flow
33
where
p'(z) = 0 on some interval z E [rl, r2] independent of t, supported in B, and rangObviously, any function ing in the interval [rl, r2] satisfies the hypotheses of Theorem 6.1 (specifi-
cally (6.11)) with u = 0. Clearly, and in contrast to the Cauchy problem, a family of oscillating solutions may be constructed which is not (strongly) compact in L1(Q).
7
Comments, remarks, literature
The mathematical theory of viscous fluids is far from being complete. Even if the flow is incompressible and, consequently, supposed to be governed by the classical Navier-Stokes system of equations, it is still a major open problem whether or not global classical solutions exist in three space dimensions for general (not necessarily small) initial data. We refer the reader to the monographs of Ladyzhenskaya [131 or Temam [19] for more information on this subject. A more recent theory of incompressible flows may be found in the books of Lions [141 or Galdi [10]. In spite of the enormous amount of the existing literature, the question of uniqueness of solutions (in the same class where existence is known) for given initial data in three space dimensions remains completely open. Clearly, the problem becomes even more involved when general compressible fluids are considered. The question of existence of (weak) solutions for the initial value problem corresponding to (1.1), (1.2), (1.5) has been largely settled in the case of space dimension one (see e.g., Antontsev et al. [11, Serre [17]). The spherically symmetric flows even with large data
are treated by Hoff [11]. The truly multidimensional case is much more involved. There is a vast amount of literature originated by the paper of Matsumura and Nishida [16] dealing with small and smooth initial data. The requirement of smoothness (but not of smallness) was subsequently relaxed by Hoff [12]. Global existence of weak solutions for large data in two space dimensions was proved by Vaigant and Kazhikhov [20] on condition
that the viscosity coefficient A depends on the density in a very specific way.
The obvious mathematical difficulties of the problem led to the development of new concepts such as that of renormalized solutions introduced by DiPerna and Lions [4] and subsequently adapted by many authors in rather different contexts. The crucial question of compactness of the set of bounded solutions has been positively resolved by means of recent results in the compensated compactness theory, namely, new weakly convergent quantities called paracommutators have been identified by Coifman et al. [21. Finally, Lions ([15]) presented a theory giving positive existence results
34
E. Feireisl
in any space dimension and for general initial data under certain restrictions on the state equation similar to (5.3).
Acknowledgement. This paper represents a series of lectures of the author at the "Nonlinear Analysis and Differential Equations" autumn school organized in Lisbon and sponsored by Fundacao para a Ciencia e a Tecnologia and Programa PRAXIS XXI. He is grateful to the organizers for their hospitality as well as generous financial support.
REFERENCES [1] S.N. Antontsev, A.V. Kazhikhov and V.N. Monachov, Krajevyje zadaci mechaniki neodnorodnych zidkostej, Novosibirsk, 1983.
[2] R. Coifman, P.L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. 72 (1993), 247-286. [3]
R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc. 212 (1975), 315331.
[4] R.J. DiPerna and P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), 511-547. [5] E. Feireisl, 9. H. Petzeltova and I. Straakraba, On the motion of a viscous compressible flow driven by a time-periodic external flow, Arch. Rational Mech. Anal. 149 (1999), 69-96. [6] E. Feireisl, A. Novotny and H. Petzeltova, On the existence of globally
defined weak solutions to the Navier-Stokes equations of isentropic compressible fluids, J. Math. Fluid Mech. 2000, submitted. [7]
E. Feireisl and H. Petzeltova, On the long time behaviour of solutions to the Navier-Stokes equations of compressible flow, Arch. Rational Mech. Anal. 150 (1999), 77-96.
[8] E. Feireisl and H. Petzeltova, On compactness of solutions to the Navier-Stokes equations of compressible flow, J. Differential Equations 163 (2000), 57-75. [9]
E. Feireisl and H. Petzeltova, On the steady state solutions to the Navier-Stokes equations of compressible flow, Manuscripta Math. 97 (1998), 109-116.
[10] G.P. Galdi, An Introduction to the Mathematical Theory of the NavierStokes Equations, I, Springer-Verlag, New York, 1994.
Navier-Stokes Equations of Compressible Flow
35
[11] D. Hoff, Spherically symmetric solutions of the Navier-Stokes equations for compressible, isothermal flow with large, discontinuous initial data, Indiana Univ. Math. J. 41 (1992), 1225-1302. [12] D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data, Arch. Rational Mech. Anal. 132 (1995), 1-14.
[131 O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969. [14] P.L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 1, Incompressible Models, Oxford Science Publication, Oxford, 1996. [15] P.L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 2, Compressible Models, Oxford Science Publication, Oxford, 1998.
[16] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible and heat conductive fluids, Comm. Math. Phys. 89 (1983), 445-464. [17] D. Serre, Solutions faibles globales des equations de Navier-Stokes pour un fluide compressible, C. R. Acad. Sci. Paris 303 (1986), 639-642.
[18] I. StraAkraba, Asymptotic development of vacuum of 1-dimensional Navier-Stokes equations of compressible flow, Nonlinear World 3 (1996), 519-533.
[19] R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam, 1977.
[20) V.A. Vaigant and A.V. Kazhikhov, On the existence of global solutions to two-dimensional Navier-Stokes equations of a compressible viscous fluid, Sibirskij Math. Z. 36 (1995), 1283-1316. (In Russian).
Eduard Feireisl Institute of Mathematics AV CR, 2itng 25, 11567 Praha I
Czech Republic feireislCmath.cas.cz
Periodic Solutions of Systems with p-Laplacian-like Operators Jean Mawhin 1
Introduction
A great deal of attention has recently been given to extending spectral, bifurcation or existence results for semilinear equations of the second order,
in both ordinary and partial differential cases, to the case of nonlinear perturbations of the so-called p-Laplacian operator u --* Apu, defined by Opu = (lu'IP-2u')' if N = 1, and Opu = div (Igrad ulp-2grad u) if N _> 2, or of some suitable generalization. The aim of this presentation is to concentrate on problems of the form
(O(u'))' = f (t, it, u'),
u(0) = u(T), u'(0) = u'(T),
(1.1)
where f : [0, T] x RN x RN -, RN is continuous (or satisfies Caratheodory conditions), and 0 : RN -+ RN belongs to a suitable class of homeomorphisms containing the mapping t/ip defined by 1/Tp(u) 'V l 1 =
Iulp-2u,
if u # 0,
(1.2)
4/lp(0) = 0, 1/2
where u = (u1i u2, ... , UN) E R', Jul = (Eiu) , and p > 1. Of course, for p = 2, (1.1) with 0 = 02 reduces to the classical problem
u' = f (t, u, u'),
u(0) = u(T), u'(0) = u'(T).
The choice of the periodic boundary conditions is motivated by the supplementary difficulty they introduce with respect to the Dirichlet conditions
u(0) = u(T) = 0. Indeed, it can be shown that the Dirichlet problem (V,,,(u ))' = h(t),
u(0) = u(T) = 0
(1.3)
is uniquely solvable for each continuous (or even L') h : [0, T] - RN and the solution defines a nice nonlinear operator Tp : C([0, T), RN) C0 'Q0, T], RN). Consequently the corresponding nonlinear problem
(Op(u'))' = f (t, u, u'),
x(0) = x(T) = 0
is equivalent to the fixed point problem in C' ([0, TI, RN)
u=TpoNf(u),
(1.4)
J. Mawhin
38
where Nf is the Nemytski operator associated to f defined by Nf(u)(t) = f (t, u(t), u'(t)). Furthermore, one can show that Tp o Nf is completely continuous, which allows the use of Leray-Schauder theory [5]. The reader can consult [13] for references to this Dirichlet problem. As in the classical case p = 2, the periodic problem
(ip(u'))' = h(t),
u(0) = u(T), u'(0) = u'(T)
(1.5)
is not solvable for each h E C([0, T], RN), and, when solvable, has no unique
solution: for any c E RN, u(t) + c is a solution together with u(t), and a trivial necessary condition for the solvability of (1.5) is that h :=
TJT h(t) dt = 0.
(1.6)
J0
Of course, as in the classical case, one could replace (1.1) by the equivalent problem (cb(u'))' + g(u) = f (t, u, u') + g(u),
u(0) = u(T), u'(0) = u'(T), (1.7)
where g : RN -+ RN has been chosen in such a way that the problem
(c(u'))' + g(u) = h(t),
u(0) = u(T), u'(0) = u'(T)
(1.8)
has a unique solution for each h E C((O,TJ,RN), and proceed as in the Dirichlet case. By analogy with the classical case p = 2, one can try for example, when ¢ = Ypp, g(u) = -au or g(u) = -aipp(u) for any a > 0. In this case, (1.8) is easily seen to be the Euler-Lagrange equation associated to the respective action integrals
IT
[IU,(t)lp +a'u(2)"- h(t)u(t)J A
(1.9)
J(u) = Jjp 1 u't)'p + a 1u(p)1p - h(t)u(t)J dt.
(1.10)
J(u) = and
It is a classical problem of calculus of variations (see e.g. [19]) to prove that each of those action integrals J is a strictly convex weakly lower semicontinuous coercive function over the space WT''([0,T],RN) of functions
u in Wl,p([O,T],RN) such that u(0) = u(T), and that the corresponding unique minimum is a (classical) solution of (1.5) with respectively g(u) = -au and g(u) = -aiip(u). However, in the more general case of a homeomorphism 0 having no gradient structure, the study of the unique solvability of a problem like (O(u'))' - au = h(t),
u(0) = u(T), u'(0) = u'(T)
39
Periodic Solutions of Systems with p-Laplacian-like Operators
is more complicated. This is the reason why we shall instead follow [13] and reduce (1.1) to a fixed point problem in a way which avoids the introduction
of some shift g in the equation, but is based upon the direct study of problem (1.8) with g = 0. Throughout the paper I I will denote absolute value, and the Euclidean Also norm on RN, while the inner product in RN will be denoted by
for N > 1 we will set I = [0, T], C = C(I, RN), C1 = C' (I, RN), CT = {u E C I u(0) = u(T)}, C. = {u E C' I u(0) = u(T),u'(0) = u'(T)}, LP = LP(I, RN), W 1,p = W 1,P(I, RN), and W1T = WT,P(I, RN), p > 1. The norm in C and CT will be denoted by IIuIIo = maxtE[o,T) Iu(t)I, the norm in C1 and CT by (lull, = IIuIIo + Ilu'llo, the norm in LP by IIuIILP =
2/P 1/2
LEN
1
(fo Iuj(t)IPdt)
,
and the norm in W1,P and W1'P
J
by IIuIIWI.P = IIUIILP + IIU'IILP
Each u E L' can be written u(t) = u + ii(t), with
U:= T
J0
T u(t) dt,
J 0
u(t) dt = 0.
We will use the following Sobolev inequality: for each absolutely continuous
u e CT, one has (1.11)
1P110 :5 Ilu'IILI
To prove (1.11), we notice that, as uj = 0 for each 1 _< j < N, there exists tj E [0, T] such that iij (tj) = 0. Now, if r E [0, T[ is such that 1u(r) I = I Iui (o,
then 1/2
N IIuIIo =
j u,' (s) ds
1 uj(T)I2 j=1
[N(JT)2] <
1/2
= Ilu'IILI
The sections are organized as follows. In Section 2, we begin by establish-
ing the monotone type conditions on the function 0 that we will consider and show some important examples of functions ¢ which verify those conditions. In Section 3, we then develop the machinery which allows us to solve
the forced Laplacian with periodic boundary conditions (1.5). In Section 4, we use the previous results to reduce the periodic boundary value for nonlinear perturbations of the O-Laplacian to a fixed point problem in CT. In Section 5, combining Leray-Schauder degree theory with the results of Section 4, we state and prove a first general existence theorem for problem (1.1). This result generalizes to our situation some well-known continuation theorems [16, 17, 221, obtained in the framework of coincidence degree
40
J. Mawhin
for nonlinear perturbations of linear differential operators with periodic boundary conditions. Indeed our approach can be viewed as an extension of coincidence degree to some quasilinear problems. The existence theorem also generalizes a result proved in [9] for nonlinear perturbations of the one-dimensional p-Laplacian and p > 2, to a much wider class, which includes arbitrary homeomorphisms in the scalar case and the operator (Op(u'))', for any p > 1, in the vector case. Moreover, our generalization is obtained by using classical Leray-Schauder degree theory, instead of the more sophisticated degree theory for mappings of type (S)+ used in [9]. In Section 6, we apply this first continuation theorem to the study of the periodic boundary value problem for some O-Laplacian generalizations of Lienard systems. In Section 7, we apply those results together with the modification method to generalize to the vector p-Laplacian case some existence conditions of Hartman-Knobloch type. In Section 8, using degree theory for compact vector fields which are invariant under the action of S', as developed in [1], we extend to our quasilinear situation the continuation theorem of [3] and [1], in which a homotopy is made to an arbitrary autonomous system. An application is given to a perturbation of (op(u'))' by an asymptotically autonomous and (p - 1)-positive homogeneous system. A special case of this result provides results for the spectral theory of the vector p-Laplacian with T-periodic boundary conditions, which is discussed in Section 9. The reader can also consider, in the light of those sections, the T-periodic problems for scalar equations considered in [2, 7, 6, 8, 9, 11].
2
Some monotone mappings
Let 0 : RN
RN be a continuous function which satisfies the following
two conditions: (H1) For any x1, x2 E RN, xl 0 x2,
(41x1) - 0(x2), x1 - X2) > 0,
(H2) there exists a function a : [0, +oo[-+ [0, +oo[, a(s) -+ +oo as s +oo, such that (O(x),x) > a(IxD)IxI,
for all x E RN.
It is well known that under these two conditions 0 is a homeomorphism from RN onto RN, 0-1 satisfies (H1) and that 10-1(y)I -' +oo as jyj -+ +oo (see [5], ch. 3). Example 2.1. Let 0 be a homeomorphism from R onto R. Then 0 is either increasing or decreasing. Clearly in the first case 0 satisfies (H1) and (H2) while in the second case -0 does.
Periodic Solutions of Systems with p-Laplacian-like Operators
41
Example 2.2. For p > 1, let Op : RN - RN be defined in (1.2). Then t1', is a homeomorphism from RN onto RN with inverse t/ip. (x) = I xI ' -2x, where p" = pp 1. Let now x, y E RN; from the inequality (V)P(x) - 1/ip(y),x - y) > (IxIp-1 -
IyIp-1)(IxI
- IiI)
>- 0,
it follows immediately that (Op(x) - t',,(y), x - y) = 0 implies x = y, and thus (H1) holds. Also (H2) follows from (t[ip(x),x) = Ixlp = Ixlp-IIxl. Example 2.3. More generally, we can consider any q5 = V , with E C1(RN, R) strictly convex, such that ¢ satisfies (H2). An interesting example is given by fi(x) = el,12-Ix12-1, for which (04(x), x) = 2(e1"1 -1)1x12. Example 2.4. Further examples can be obtained from the following proposition. proved in (13J.
Proposition 2.1. For i = 1,
k, let Ni E N and Wi : RN, - RNc satisfy
the following conditions.
denoting the inner product (i) (t/)i(z) - 0j (w), z - y) i > 0, (with in RN.) for any z, y E RNI, with equality holding true if and only if
z=y;
(ii) there exists a function , : (0,+oo) -' [0, +oo), ai(s) - +oo as s - +oo, such that (z), z)i > ai(lzl)lzl, k
k
i=1
i=1
for all z E RN'.
Then the function W : 11 RN. --+ fi RN', x = (x',
,xk)
W(x) _ k
(1/)1(x1),
3
,tpk(xk)), satisfies conditions (H1) and (H2) with N = E Ni. i=1
Forced q5-Laplacians with periodic boundary conditions
Let us now consider the simple periodic boundary value problem
(O(u'))' = h(t),
u(0) = u(T), u'(0) = u'(T),
(3.1)
where h E L' is such that h = 0, and let u be a solution to (3.1). By integrating from 0 to t E I, we find that 0(u'(t)) = a + H(h)(t), where H(h)(t) =
r
h(s)ds,
(3.2)
42
J. Mawhin
and a E RN is arbitrary. The boundary conditions on u' imply that 1
J
T
T 0-1(a
+ H(h)(t)) dt = 0.
For fixed l E C, let us define G1 (a)
jT
1
¢-1(a + l(t)) dt.
(3.3)
We have
Proposition 3.1. If 0 satisfies conditions (H1) and (H2), then the function G1 has the following properties :
(i) For any fixed l E C, the equation Gi(a) = 0
(3.4)
has a unique solution a(l).
(ii) The function a : C -' RN is continuous and sends bounded sets into bounded sets.
Proof. (i) By (H1), it is immediate that (GI(a,) - G1(a2),al - a2) > 0,
for
al 54 a2,
and hence if (3.4) has a solution, then it is unique. To prove existence we show that (GI(a),a) > 0 for IaI sufficiently large. Indeed we have jT
(C1(a),a) =
(0-1(a+l(t)),a+l(t))dtfT(O_
1(a+1(t)),1(t))dt,
T
(GI(a),a)
>T
jT(O_j
( a+l(t)),a+l(t))dt
- -- fo T I¢-1(a + l(t))I dt. 111110
(3.5)
Now from (H2), for any y E RN, we have that
(&-1(y),y) >
a(Io-1(y)I)Io-1(y)I.
(3.6)
Thus from (3.5) and (3.6), fT
(GI (a), a) >
T
(-' (a + l(t))I) - IIIIIo) I o-1(a + l(t))I dt.
(3.7)
Periodic Solutions of Systems with p-Laplacian-like Operators
43
Since Ial - oo implies that I0'1(a + 1(t))I -a oo, uniformly for t E I, we find from (3.7) that there exists an r > 0 such that (Gi (a), a) >0 for all a E 1RN with j al =rIt follows by an elementary topological degree argument that the equation GI(a) = 0 has a solution for each l E C, which defines a function a : C --+ RN which satisfies
r
T
-'(a(1) + 1(t)) dt = 0, for any l E C.
(3.8)
0
To prove (ii) let A be a bounded subset of C and let l E A. Then, from (3.8)
rT
+I(t)),a(l)) dt = 0,
J and hence
+ 1(t)), a(1) + 1(t)) dt =
J0
J
1(t)),1(t)) dt. (3.9)
T
Assume next that {a(1), 1 E A} is not bounded. Then for an arbitrary A > 0 there is 1 E A with IIIIIo sufficiently large so that
A < a( I.-'(a(1) + l(t))I), uniformly in t E I. Hence by using (3.6) and (3.9), we find that
Af I0-1(a(1)
+ 1(t))I dt <
0
IT T a((a(I)
0
5 IIIIIo
f
+ 1(t))
I(t))I dt
T
+1(t))I dt.
Thus A < IIIIIo, a contradiction. Therefore a sends bounded sets in C into bounded sets in RN. Finally to show the continuity of a, let {1,J be a convergent sequence
in C, say I,, -+ 1, as n -+ co. Since {a(l,a)} is a bounded sequence, any subsequence contains a convergent one Let a(l,a,) - a, as j -, oo. By letting j - oo in
f
T
-'(a(1
)
0,
0
we find that
T
j-1(&+l(t))dt=0, 0
and hence a(l) = a, which shows the continuity of a.
O
J. Mawhin
44
Let now a : L1 -+ RN be defined by
a(h) = a(H(h))-
(3.10)
Then, it is clear that a is a continuous function which sends bounded sets of L' into bounded sets of RN, and hence it is a completely continuous mapping. We continue now with our argument previous to Proposition 3.1. By solving for u' in (3.2) and integrating we find
u(t) = u(0) + H {¢-1 (a(h) + H(h)]} (t).
(3.11)
Here 0-1 is understood as the the operator 0-1 : C -+ C defined by 0-1(v)(t) = 0-1(v(t)). 0-1 is continuous and sends bounded sets into bounded sets. Let us define the projectors P and Q respectively by
P:CT -+CT, us-+u(0), Q:L1-+L1, Then it is clear that if u E CT solves (3.1), then ti satisfies the abstract equation
u = Pu + Qh + )C(h),
(3.12)
where the (in general nonlinear) operator 1C : L' --+ C. is given by
1C(h)(t) = H {q-1 [a((I - Q)h) + H((I - Q)h)]} (t), for all t E I. (3.13) Conversely, since, by definition of the mapping a,
H {,0-1 [a((I - Q)h) + H((I - Q)h)]} (T) = 0,
it is a simple matter to see that if u satisfies (3.12), then u is a solution to (3.1).
Note that since a(0) = a(0), we have, by (3.13) and (3.8), that 1C(0) = 0.
Lemma 3.2. The operator 1C is continuous and sends equi-integrable sets in L' into relatively compact sets in CT I.
Proof. The continuity of 7C in C follows immediately by observing that this operator is a composition of continuous operators. Also, we have that 1C(h)'(t) = 0-1 [a((I - Q)h) + H((I - Q)h)] (t), which is also a composition of continuous operators and hence continuous.
Let now E be an equi-integrable set in L'. Then there is 77 E L' such that, if h E E, Ih(t)I < r7(t) a.e. in I.
Periodic Solutions of Systems with p-Laplacian-like Operators
45
We want to show that IC(C) C CT is a compact set. For this it suffices to is a subsequence in !C(E), then it contains a convergent prove that if be a sequence in L' such that v = subsequence in CT. Let
For t, t' E I, we have that
H(I -
H(I -
-
I
t
Je'
5 if` h(s)dsl + IQ(h)II t - t'I
ii(s)ds + It - t'I I
Jo
T ij(s)ds.
Hence the sequence {H(I-Q)(h,,)} is uniformly bounded and equicontinuous. By the Ascoli-Arzela theorem there is a subsequence of {H (I -Q) (hn) }, which we rename the same, which is convergent in C. Then, passing to a subsequence if necessary, we obtain that the sequence {a((I - Q)(hn)) +
H(I -
is convergent in C. Using that 0' : C -- C is continuous
it follows from
(K(hn))'(t) = 0-' [a((I - Q)(hn)) + H(I - Q)(hn)] that the sequence {(/C(hn))'} is convergent in C and hence the sequence {(1C(h,,))} is convergent as well.
0
4 An equivalent fixed point problem Let us consider the abstract functional differential periodic problem
(c(u'))' = N(u, A),
u(O) = u(T), u'(0) = u'(T),
(4.1)
where A E [0, 1], and N : CT x 10, 11 -' L' is continuous and sends bounded sets into equi-integrable sets. Thus defining q : CT. x [0, 1] --i CT by 9(U' A) := Pu + QN(u, A) + (IC o N)(u, A),
(4.2)
we obtain that C is a completely continuous operator. Furthermore, problem (4.1) is equivalent to the problem u = G(u, A).
(4.3)
In particular, if g : I x RN x RN x [0, 1] is Caratheodory, and if we denote by Ng : CT x [0, 1] -+ L' the Nemytski operator associated to g defined by N9(u, A)(t) = g(t, u(t), u'(t), A), a.e. on I,
then it is known that Ng is continuous and sends bounded sets into equiintegrable sets.
46
J. Mawhin
We will apply Leray-Schauder's degree theorem to (4.3) by choosing N 0) with respect to in such a way that the Leray-Schauder degree of I a suitable open bounded set of CT exists and is easier to compute. Recall that to any bounded open set Q in a normed vector space X, and to any completely continuous mapping .F : Sl -+ X such that u 4 .F(u) for each u E M, one can associate an integer dLs [I - .F, S , 0], the Leray-Schauder degree of I -Y with respect to 11 and 0, which is a type of algebraic count of the number of zeros of I - .F in Sl [5].
5
Homotopy to the averaged nonlinearity
We suppose in this section that 0 : RN . RN is continuous and satisfies the conditions (Hl)-(H2) of Section 2. Our aim is to extend a continuation theorem proved in [16] for semilinear equations (see also [17, 20, 21]) to the quasilinear problem (1.1), which we repeat here for convenience of the reader,
(q5(u'))' = f (t, u, u'),
where f : I x RN x RN
u(0) = u(T), u'(0) = u'(T),
(5.1)
RN is Caratheodory.
Theorem 5.1. Assume that Sl is an open bounded set in CT. such that the following conditions hold.
(1) For each A E ]0, 1[ the problem (O(u'))' = Af (t, u, u'),
u(0) = u(T), u'(0) = u'(T),
(5.2)
has no solution on all.
(2) The equation fT
F(a) := T
f (t, a, 0) = 0,
(5.3)
has no solution on Of) n RN.
(3) The Brouwer degree
dB[F,)nRN,0] 54 0.
(5.4)
Then problem (5.1) has a solution in N.
Proof. Let us embed problem (5.1) into the one parameter family of problems
(4(u ))' = AN1(u) + (1 - \)QNf(u),
u(0) = u(T), u'(0) = u'(T), (5.5)
Periodic Solutions of Systems with p-Laplacian-like Operators
47
where N f : CT. - IR" is the Nemytski operator associated to f. Explicitly,
Mu'))' = a f (t, u, u') + (1 - A) T
r
T
f (s, u(s), u'(s)) ds
,
u(0) = u(T), u'(0) = u'(T). For A E ]0, 1], observe that in both cases, u is a solution to problem (5.2) or u is a solution to problem (5.5), and we have necessarily j0T
f (s, u(s), u(s))ds = 0. It follows that, for A E 10, 1], problems (5.2) and (5.5) have the same so-
lutions. Furthermore it is easy to see that f Caratheodory implies that N : C7. x [0,1] -, L' defined by N(u, A) = ANf(u) + (1 - A)QNf(u) is continuous and takes bounded sets into equi-integrable sets. Also problem (5.5) can be written in the equivalent form
u = 9f (U,,\),
(5.6)
with Gf(u, A) = Pu + QNf(u) + (IC o [,\N f + (1 - A)QNf])(u) = Pu + QNf(u) + (IC o [,\(I - Q)Nf])(u).
We assume that for A = 1, (5.6) does not have a solution on 811 since otherwise we are done with the proof. Now by hypothesis (1) it follows that (5.6) has no solutions for (u, A) E 811x]0,1]. For A = 0, (5.5) is equivalent
to the problem
fT 0 ( cb(u'))'
=T
f (s, u(s), u'(s))ds,
u(0) = u(T), u'(0) = u'(T), (5.7)
and thus if u is a solution to this problem, we must have T
J0
f (s, u(s), u'(s))ds = 0.
(5.8)
Hence
u'(t) = 0-1(c), where c E RN is a constant. Integrating this last equation on I we obtain that 1-1(c) = 0, and thus u(t) = d, a constant. Thus, by (5.8) rT
J 0
f(s,d,0)ds = 0,
48
J. Mawhin
which together with hypothesis (2) imply that u = d ¢ 812. Thus we have proved that (5.6) has no solution (u, A) E 8f1 x [0, 1]. Then we have that for each A E [0, 1], the Leray-Schauder degree dLS[I - 9f (-'A), 12, 0] is well
defined and, by the properties of that degree, that dLS[I
-
0, 0] = dLS[I - 9f(-, 0), 12, 0].
(5.9)
Now it is clear that problem u = Qf(u,1) is equivalent to problem (5.1),
and (5.9) tells us it will have a solution if we can show that dLS[I 9f 0), 12, 0] -A 0. This we do next. We have that
ggf(u,0) = Pu+QNf(u)+1C(0) = Pu+QNf(u). Thus we obtain fT
u - 9f(u, 0) = u - Pu - T
f (s, u(s), u(s))ds.
Hence by the properties of the Leray-Schauder degree we have that dLS[I - 9f (', 0), St, 0] = (-1)NdB[F, c n RN, 0], where the function F is defined in (5.3) and dB denotes the Brouwer degree. Since by hypothesis (3) this last degree is different from zero, the theorem is proved.
6
Periodic solutions of some q5-enard systems
Let 0 satisfy the conditions (Hl) and (H2) of Section 2, F E C2(RN,R), A be a (N x N)-matrix and e E L1. We consider the T-periodic solutions of the following Lienard system with -Laplacian (0-6nard system)
(O(u'))' + (VF(u))' + Au = e(t),
u(0) = u(T), u'(0) = u'(T).
(6.1)
The corresponding classical case where 4' = I has been considered in [15].
Theorem 6.1. If A is negative definite, then, for each e E L1, problem (6.1) has at least one solution u such that u = A-1 e. Proof. To apply Theorem 5.1, we consider the family of problems (O(u'))' + A (VF(u))' + AAu = Ae(t),
u(0) = u(T), u'(0) = u'(T),
A E ]0, 1].
(6.2)
49
Periodic Solutions of Systems with p-Laplacian-like Operators
Let u(t) = u + u(t) be a solution of (6.2) for some A E 10, 1]. Then, integrating both members of (6.2) over [0, T], we get Au = e, and hence I1I =
IA-"91
(6.3)
< IA-IIIeI.
Taking now the inner product of (6.2) by u and integrating over [0, T], we get, after integration by parts and use of the boundary conditions, T
T
J 0 0
(¢(u'(t)), u (t)) dt-A f (Au(t), u(t)) dt = AT(e, V) -A 0
Jin0
(e(t), ii(t)) dt,
and hence, using (6.3), T
1
Mu'(t)), u'(t)) dt
<TIeI2IA-'I
+NII7ILIIP110.
) (6.4)
As (q5(s), s) > a(s)I sI with a(s) - +oo as IsI - oo, there exists R > 0 such that (6.5)
(O(s), s) ? (NIIe11LI + 1) ISI
whenever IsI > R. Separating the sets where Iu'(t)I < R and where Iu'(t)I > R in the integral of (6.4), and using (6.5), we obtain
rT (NIIeI ILI + 1)
Iu'(t)I dt < NIIe1ILI III(t)1 Io + TI-12IA-' I + Cl(-),
J and hence, using Sobolev inequality,
1101L1 < TIeI2IA-1 + CI (e) := R1.
(6.6)
Using (6.3), (6.6) and Sobolev inequality, we obtain 1)IFIIA--'I
IIuIIo 5 (TIeI +
+ CI(e) := R2.
(6.7)
Now equation (6.2) also implies
I(0(u'(t)))'I < IF"(u(t))u'(t)I + IAIiu(t)I +
N
Ie;(t)I,
for a.e. t E [0, T], and hence, using (6.7), N
I(-O(u'(t)))'I 5 C3(R2)
N
Iu,(t)I + IAIR2 + E Ie;(t)I .i='
;=1
Consequently, integrating and using (6.6),
I
0
T
I (.(u'(t)))' I dt 5 C3(R2)NIIu'IILI + IAIR2T + NIIeI ILI < C3(R2)NRI + IAIR2T + NIIeHILI := R3.
(6.8)
50
J. Mawhin
If we now decompose 0(u'(t)) = b + b(t), so that
u'(t) = and hence
' (b + b(t))
,
T
0-' (b + b(t)) dt = 0, 10
we deduce, from (6.8) and Sobolev inequality, that IIbIIo 5 R3, and from Proposition 3.1 that IbI = Ib(b)I < R. Consequently, II0(u )IIo < R3 + R4 = R5, and hence
IIu'Ilo <- R6,
for some R6 > 0 which is independent of u and ). Hence, there exists Ro > 0 independent of u and A such that
Ilulli
F(a):=Tf[e(t)_Aa]dt=_Aa, so that F(a) = 0 has the unique solution a = A-'-e, which trivially implies that dB [F, B(R), 0) is defined and equal to ±1 for all sufficiently large 0 R > 0, so that conditions (2) and (3) of Theorem 5.1 are satisfied. An immediate but useful consequence of Theorem 6.1 is the following corollary.
Corollary 6.2. If A is negative definite, then, for each e E L', the problem (cb(u'))' + Au = e(t),
u(0) = u(T), u'(0) = u'(T),
(6.9)
has a unique solution and, for each sufficiently large R > 0, the LeraySchauder degree over B(R) of the associated fixed point operator in CT. is equal to ±1.
Proof. Only the uniqueness has to be proved. Let u and v be solutions of (6.9). Then we have
(#(u'))' - (O(v'))' + A(u - v) = 0, u(0) = u(T), u'(0) = u'(T), v(0) = v(T), v'(0) = v'(T),
Periodic Solutions of Systems with p-Laplacian-like Operators
51
and hence, after scalar multiplication by u - v, and integration by parts over [0, T], we get
T(q(u'(t))-¢(v'(t)), I 0/
u'(t)-v'(t)) dt- f
T(A(u(t)-v(t)),
u(t)-v(t)) dt = 0.
o
Hence, both integrals, and therefore both integrands, have to be equal to zero, which easily implies that u = v. 0 One can deduce from Theorem 6.1 some existence results which cover the case where A is negative semi-definite.
Corollary 6.3. If A is negative semi-definite, then, for each e E L' such that e = 0, problem (6.1) has at least one solution u such that u = 0. Proof. We consider the family of problems
(,O(u'))' + (VF(u))' + Au
- 1u = e(t),
u(0) = u(T), u'(0) = u'(T), n E N.
(6.10)
Notice that A - II is negative definite for each n and that, by integrating the equation over [0, T], each solution u of (6.10) is such that Ii = 0. It follows therefore from Theorem 6.1 and its proof that, for each n E N`, equation (6.10) has at least one solution u and that I unI I 1 < R for some R > 0 and all n E N*. Those un are fixed points of the equivalent completely continuous fixed point operator, and hence the existence of a subsequence converging to a solution u of (6.1) with mean value zero is easily deduced.
0 This corollary in turn implies a necessary and sufficient solvability condition for (6.1) when A is negative semi-definite, together with a multiplicity result.
Theorem 6.4. If A is semi-definite, then problem (6.1) is solvable if and only if i3 E R(A). If it is the case, then, for each c E N(A), problem (6.1) has at least one solution u such that u = c. Proof. The necessary condition immediately follows by integrating both members of (6.1) over [0, T[. For the sufficiency, assume that e E R(A), let
d be the unique solution in N(A)1 of Av = e, and let c E N(A). Letting u(t) = c + d + v(t), with v = 0, our problem is equivalent to finding a solution v such that v = 0 of the problem (O(v'))' + (VF(c + d + v))' + Av = e(t), v(0) = v(T), v'(0) = v'(T). (6.11) As there are no assumptions upon F in Corollary 6.3, we can apply it to problem (6.11) to obtain the existence of the required solution v. 0
52
J. Mawhin
Other applications of Theorem 5.1 are given in [13], which extend results
of Ward, Canada, Martinez-Amores and Ortega for 0 = Id. For example they imply the existence of T-periodic solutions for the following equations or systems.
1. The problem
(Ix'IP-2xi)' + x2(1 + x2) + sinxl = el(t),
x1(0) = xl(T), x'(0) = xi(T), (I
x'IP-2x'2)'
- x1(1 + xi) + coSx2 = e2(t),
(6.12)
x2(0) = x2(T), x2(0) = x2' (T)
has a solution for each e = (e1, e2) E L1(I, R2).
2. The problem
J f(Iu'Ip-2ui)' _ - expu2 - ei(t),
f(Iu'Ir-2u2)'
= expul - e2(t),
ul(0) = ui(T), u2(0) = u2(T), ui(0) = ui(T), u'(0) = u2(T) has a solution if and only if ei < 0 and e2 > 0. 3. The problem
± ((exp Iu'I - 1) sgn u')'+expu = e(t),
u(0) = u(T), u'(0) = u'(T)
has a solution if and only if e > 0.
7
Systems of p-Laplacian type with a Hartman condition
Let f : R x RN -i RN, (t, u) '-4 f (t, u) be T-periodic with respect to t and continuous, and let us consider the existence of a T-periodic solution u (i.e., a solution such that u(t + T) = u(t) for all t E R) of the system
(iv(u'))' = f (t, u),
(7.1)
where p > 1 and ,p is defined in (1.2). Of course, those solutions are continuations over R, by T-periodicity, of the solutions over 10, T] such that
u(0) = u(T), u'(0) = u'(T), so that the previous theory can be applied. The following result extends to the p-Laplacian case some results of Hartman [10] for Dirichlet conditions and of Knobloch [12] for periodic solutions. The proof is inspired by that given in [18] for the method of upper and lower solutions associated to a second order equation.
Periodic Solutions of Systems with p-Laplacian-like Operators
53
Theorem 7.1. If there exists R > 0 such that
(f(t,u),u) > 0
(7.2)
for all t E R and all u E RN satisfying Jul = R, then system (7.1) has at least one T-periodic solution u such that lu(t)l < R for all t E R. Proof. Define PR : RN - RN by
PR(u)=uif lulR. Notice that PR is continuous and bounded (by R) over RN. Consider the modified problem (Op(u ))' - u = f (t, PR (U)) - PR (U)
(7.3)
fR(t, u).
Notice that this system is equivalent to (7.1) when Jul < R. By Corollary 6.2, the system (iip(u'))' - u = h(t) has, for each h E CT a unique T-periodic solution Sh, and hence finding the T-periodic solutions of (7.3) is equivalent to solving the fixed point problem in CT (whose elements are supposed to be extended to R by T-periodicity) u = S o N!R (u),
(7.4)
where NfR is the Nemytski operator associated to fR. Now, it is not difficult to show that the operator S o NfR is completely continuous and bounded
in CT, so that Schauder's fixed point theorem implies the existence of a solution u to (7.4), and hence of a T-periodic solution of (7.3). We show now that I u(t) l < R for all t E R, so that u is indeed a T-periodic solution of (7.1). Notice that PR(U(t)),
((lu'(t)Ip-2u'(t),u(t)))' = Iu'(t)Ip + (u(t) + (f(t,PR(u(t))),u(t)),
u(t)) (7.5)
and hence, for each t E R such that lu(t)l > R, we have lpR(u(t))l = u(t)' = R, and it follows from (7.2) and (7.5) that uRt ((Iu'(t)lp-2u (t),u(t)))'
> lu(t)I(lu(t)I - R) > 0.
(7.6)
If Iu(t)I > R for all t E R, then (7.6) holds for all t E R, which is not possible
for a T-periodic function. If lu(to)l < R for some to and lu(tl)l > R for some tl, then we can find r and v > r such that Iu(r)I = R,
lu(a)I = m ax
> R,
Iu(t)I > R, t E]r,a].
J. Mawhin
54
Thus, lu(a)I2
(
2
J
= (u(a),u'(a)) = 0,
(7.7)
and, from (7.6) we see that (lu' Ip-2u', u) is strictly increasing in ]r, a]. Therefore, using (7.7), we have, for all t E IT, a], (Iu'(t)Ip-2u'(t), u(t)) < (Iu'(a)I p-2u'(u), u(a)) = 0,
and hence (U' (t), u(t)) < 0.
But, then, R2 < Iu(a)I2 < Iu(r)I2 = R2,
0
a contradiction. The special case of a scalar equation is of interest.
Corollary 7.2. Assume that f : R x R - R, (t, u) ,-a f (t, u) is T-periodic with respect to t and continuous. If there exist R > 0 such that f (t, -R) < 0 < f (t, R) for all t E R, then equation (OP (u'))' = f (t, u)
has at least one T-periodic solution u such that -R < u(t) < R for all
tER. 8
Homotopy to an autonomous system
Let us consider now the problem (4.(u'))' = g(t, u, u', A),
u(0) = u(T), u'(0) = u'(T),
(8.1)
where A E [0, 1], and g : I x RN x RN x [0, 1] - RN is Caratheodory. The following result extends to our quasilinear situation a continuation theorem first proved in [3] for periodic solutions of semilinear systems. We follow here the simpler approach of [1] and [20]. In this continuation theorem, the homotopy is made to an autonomous system and one takes advantage of the Sl-invariance of the corresponding periodic problem to compute the associated Leray-Schauder degree through the following result.
Periodic Solutions of Systems with p-Laplacian-like Operators
55
Lemma 8.1. Let X be a normed linear space, 12 C X open and bounded and g : 12 -+ X be a compact perturbation of identity such that g-1 (0) n 812 = 0. Assume S' acts on X through linear isometries, {l C X is invariant under this action and g is equivariant. Let gs' : Sls' -+ Xs' denote 11s = St n Xs', Xs' = {x E X the restriction of g to the fixed point set
r*x=x forall7- ES'}. Then dLS[g,11, 01 = dLS[gS', SIS', 0].
The corresponding continuation theorem for periodic solutions goes as follows.
Theorem 8.2. Assume that g(t, u, v, 0) = go (u, v)
(8.2)
is independent of t, and that 12 is an open bounded set in CT such that the following conditions hold. (1) For each A E [0, 1[ the problem (8.1) has no solution on 812.
(2) The Brouwer degree dB [go
0),12 n RN, 0] 0 0.
(8.3)
Then problem (8.1) with A = 1 has at least one solution in 12.
Proof. Problem (8.1) can be written in the equivalent form (4.3) i.e., u = 99(u, A),
(8.4)
where
gg(u,.1) = Pu + QN9(u,.1) + (IC o N9)(u, A),
and Ng : CT x [0, 1] -> L' is the Nemytski operator associated to g. We assume that for \ = 1, (8.4) does not have a solution on 812 since otherwise we are done with the proof. Now by hypothesis (1) it follows that (8.4) has no solutions for (u, A) E 812 x [0, 1]. Then, for each \ E [0, 1], the LeraySchauder degree dLS[I A), 12, 01 is well defined and
dLS[I -
0] = dLS[I -
(8.5)
0), 11, 0].
Now it is clear that problem
u=Qg(u,1)
(8.6)
is equivalent to problem (8.1) with A = 1, and (8.5) tells us that this problem will have a solution if we can show that dLS[I This we do next. We have that
9.(u,0) = Pu+QN9(u,0) + (1C o N.)(u,0),
0), 12, 0]
0.
56
J. Mawhin
where N9(u,O)(t) = go(u(t),u'(t)). Hence, because go is independent of t, Q9(., 0) is invariant under the action of the group S' acting on C7. through the linear isometry Tu = We can then use Lemma 8.1 to compute dLS [I - 9.9 0),Q, 0] when fl is invariant under the action of S1. If fl is not invariant we replace it by fl = {u E fl : dist(u, 0 n Fix Q9(., 0)) < E} with 0 < e < dist(fl n Fix 0), ofl). Here Fix Q9(., 0) denotes the set of fixed points of S1 is invariant since S' is path-connected, hence 8fl n Fix Q9 0) = 0 implies that S2 n 0) is invariant. We also used here that S' acts through isometries. Since Fix 0) n S C fl c fl, the excision property of the Leray-Schauder degree yields
dLS[I -
0), 5, 0].
0), cl, O] = dLS[I -
Now the fixed point set (CT)S1 = {u E CT : for all r E I} is the set of constant u in CT and, for such a constant c, Pc = c, QN9 (c, 0) go (c, 0), and
(K o Ng)(c,O)(t) = H{¢-1[a(O)]}(t) = tO-'[a(0)] But, by definition of a, we have 0 = Go(a(0)) = T
f
T
-1
[a(O)] dt = 0-1 [a(0)],
so that (1C o N9)(c, 0) = 0. Thus
c-Q9(c,0) = -9o(c,0) Consequently, as the Leray-Schauder degree in a finite dimensional space reduces to the Brouwer degree, we get, using Lemma 8.1 and excision,
dcs[I - 99(.,0),f1,01 = dLS[(I - Q9(',O))I
I
S1,fl n (CT)SI,O]
(-1)NdB[9o(',0)IRN,clnRN,O]
=
= (-1)NdB[90IRN,fl n RN,0],
as fl n RN contains all constant T-periodic solutions of (8.8) with A = 0 contained in fl, i.e., all zeros of By assumption (2), this last degree is different from zero, and the proof is complete. 0 As an application of Theorem 8.2, let us consider the problem
(O (u'))' = h(u, u') + e(t, u, u'),
u(0) = u(T), u'(0) = u'(T),
(8.7)
RN is continuous and e : I x RN x RN - RN is where h : RN x RN Caratheodory. The following result extends to the p-Laplacian case, Corollary 9 of [3].
Periodic Solutions of Systems with p-Laplacian-like Operators
57
Theorem 8.3. Assume that the following conditions hold.
(1) h(ku, kv) = kp-1h(u, v) for all k > 0 and all (u, v) E RN x RN. u'*`v) + v p-
(2) liml
-
_ 0 , uniformly a. e. in t E I .
(V,,(y'))' = h(y, y'),
y(O) = y(T), y'(0) = y'(T'),
has only the trivial solution y = 0.
(4) dB
0), b(Ro), 0] # 0 for some Ro > 0.
Then problem (8.7) has at least one solution.
Proof. We apply Theorem 8.2 to the homotopy (V)p(u'))' = h(u, u') + Ae(t, u, u'),
u(0) = u(T), u'(0) = u'(T),
A E [0, 1],
8.8)
and show that there exists some R > 0 such that, for each A E [0, 1] and each possible solution u of (8.8), one has Ilylll < R, with Ilylll = IIyIIo + Ily'IIo The result will then follow from Theorem 8.2 by taking 11 = B(R). If it is not the case, one can find a sequence {fin} in [0, 1] and a sequence {un} of
solutions of (8.8) with A = an such that Ilun II1 -' oo when n -+ oo. If we set
yn=
IIunII1'
n=1,2,...,
it follows from assumption (1) that, for each positive integer n, (+Gp(yn))= h(yn,yn)
+\ne(t, llu
lYn,
Ilunlllyn),
(8.9)
yn(0) = yn(T), yn(0) = yn(T). As IlynIli = 1 for all n, we can assume, going if necessary to a subsequence,
that yn -+ y uniformly in I, for some y E CT. Letting zn = l/ip(yn), it is clear that {zn} is bounded in CT and it follows from equation (8.9) and from assumption (2) that {z,,} is bounded in CT as well. Thus, up to a further subsequence, we can assume that {zn} converges uniformly on I to some z E CT. Notice that, then, {yn} converges uniformly on I to 1.Ip. (z),
and that Ilyllo +
(8.10)
1,
where p` is conjugate to p. Now, problem (8.9) is equivalent to yn
=h(yn,'ip-(zn))+Ane(t,Ilunlllyn,llunll1Op-(zn)),
zn
llunlli-1
(8.11)
58
J. Mawhin
y. (0) = y,,(T), zn(0) = z, (T), n = 1, 2,... .
Using the above convergence results and an integrated form of (8.11), it i easy to see that (y, z) will be a solution of the problem
z' =
y' =
y(O)
= y(T), z(0) = z(T),
and hence y will be a solution of the problem
(,(y'))'
= h(y,y'), y(0) = y(T), y'(0) = y'(T) But Assumption (3) implies that y = 0, and hence ?Pp- (z) = 0, contradict-
0
ing (8.10).
9
Spectral theory for the vector p-Laplacian with periodic boundary conditions
We summarize in this section some preliminary results obtained in [14]. Recall that p E R is said to be an eigenvalue of minus the vector p-Laplacian
u - - (1lip(u'))' with T-periodic boundary conditions if the problem
u(0) = u(T), u'(0) = u'(T),
( 4' (U'))' + AiPp(u) = 0,
(9.1)
has a non-trivial solution u, which is called a corresponding eigenfunction. Let E(p, N) denote the set of those eigenvalues. It is not empty, as 0 E E(p, N). In a way similar to linear theory, we can define the resolvant set p(p, N) of minus the vector p-Laplacian with T-periodic boundary conditions as the set of p E R such that the problem
(')p(u'))' + pp(u) = e(t),
u(0) = u(T), u'(0) = u'(T),
(9.2)
has at least one solution for each e E L'. We can also define the spectrum v(p, N) of minus the vector p-Laplacian with T-periodic boundary conditions as a(p, N) = R \ p(p, N). The following direct consequence of Theorem 8.3 shows that a(p, N) C E(p, N).
Corollary 9.1. If p is not an eigenvalue of minus the vector p-Laplacian with T-periodic boundary conditions, then, for each e E L1, problem (9.2) has at least one solution. It is therefore of interest to study e(p, N). We have seen that 0 E E(p, N). Moreover, if cp is an eigenvalue, taking the inner product of the equation in (9.1) by cp and integrating over [0, T] easily implies that
A=
fT o
I W'(t) I p dt
fo I v(t) I p dt
>
0,
(9.3)
Periodic Solutions of Systems with p-Laplacian-like Operators
59
i.e., that each eigenvalue is nonnegative, so that 0 is the smallest element of e(p, N). Furthermore, it is clear from the form of (9.1) that each eigenvalue of the scalar case N = 1 is also an eigenvalue of the vector case. In the scalar case, the direct integration of the differential equation in (9.1) allows the determination of all the eigenvalues, and they are given [6J by
C27,p) P
1)(TpP
= (P-
An(A1) =
1P
(ir/P)
=
(P(psin p)
P,
with, as usual, w = T and 7rp defined by 1rp = 2(p - 1)11P
I
L(1
(r"P) - tP)-1/Pdt = 2(p - 1)1/P sin(rr/p)
But, as observed by Del Pino (private communication), e(p, N) contains more elements when N > 1, for example the infinite sequence ((n.)'), n = 1, 2, ... , as one can immediately show by taking the corresponding eigenfunctions
V = (0,... ,0,cosnwt,0,... ,0,sinnwt,0,... ,0). Indeed, one can show more generally that, if k is a positive integer, each non-trivial solution u of u" + k2w2u = 0
such that (u(t), u'(t)) = 0 for all t E R is an eigenfunction associated to the eigenvalue (kw)P of minus the vector p-Laplacian with T-periodic boundary conditions. One can also show that if A E C7.(R, £(RN, RN)) is such that A"(t) + k2w2A(t) = 0, A(t)'A(t.) = I and A'(t)'A'(t) = k2w2 for some positive integer k, then, for each c E RN, A(t)c is an ezgenfunction associated to the eigenvalue (kw)P. Here M* denotes the transpose of any matrix M. The complete structure of e(p, N) for N > 1 is far from being understood. Now, if A E E(p, N) \ {0} and cp is an associated eigenfunction, then. integrating the differential equation in (9.1) over [0, TJ gives
f T l '(t) I p-2W(t) dt = 0.
(9.4)
To study the first positive element Al (p, N) of e(p, N), it is natural to consider the set
P; f C(P, N) = y E W1 T
0
T
l y(t)I P dt = 1,
f
0
T
(y(t)I
P-2y(t) dt = 0
J. Mawhin
60
and the functional T
Jp,N : WTp --+ R, y'-i fo Iy'(t)Ipdt.
If we define the set S(p, N) by S(p, N) = {Jp,N(y) : y E C(N, p)} , it follows from (9.3), (9.4) and the positive homogeneity of (9.1) that e(p, N) \ {0} C
S(p, N). In analogy with the linear theory, one considers the associated minimization problem min {Jp,N(y) : y E C(p, N)}
(9.5)
and proves the following result.
Proposition 9.2. Jp,N has a positive minimum A, (p, N) on C(p, N). An important consequence of Proposition 9.2 is the extension of Wirtinger's inequality to WT''.
Corollary 9.3. For each u E WT p, such that
LT
iu(t)Ip-2u(t) dt = 0, one
has
,\1(p,N)JT Iu(t)Ipdt <
fT
Iu'(t)Ipdt.
(9.6)
Another consequence of Proposition 9.2 is that, for p > 2, Al (p, N) E e(p, N).
Corollary 9.4. If p > 2, al (p, N) is the smallest positive element of e(p, N).
Proof. From the Lagrange-Caratheodory rule of the calculus of variations, it follows from Proposition 9.2 that if Jp,N reaches its minimum on C(p, N) a t y, and p > 2, then there exists (a, /j, 7i, ... , 7N) # 0 such that
aJrT(pI y'(t)Ip-2y(t)', z(t)') dt + O f T (pl y(t)I p-2y(t), z(t)) dt 0
o
N
+>
1k
k=1
[(p-2)Iy(t)Ip-4(y(t),z(t))yk(t)+Iy(t)Ip-2zk(t)] dt=0
JT (9.7)
for all z E WT '([0,T), RN). Taking z = ry = (ii, ... , yN), one gets
J[(p - 2)Iy(t)Ip-4((y(t),'r))2 + Iy(t)Ip-2b'I2] dt = 0, hence Iryl2 fo I y(t) l p-2 dt = 0, and -y = 0. Consequently, (9.7) becomes
a fT(I
J0
y (t)I
p-2y(t)', z(t)') + 0
f 0
T (I
y(t)IP-2y(t), z(t)) dt
= 0.
Periodic Solutions of Systems with p-Laplacian-like Operators
61
If a = 0, then 3 j4 0 and fT (Iy(t)IP-2y(t), z(t)) dt = 0, for all z E W "([0, T], RN). Taking z = y, we obtain f o f y(t)J dt = 0, a contradiction to f o I y(t) Ip dt = 1. Thus a 0 0, and we get T o
(I y'(t )jp-2 (t), z(t)') dt +
jT(ly(t) lp-2y(t),
z(t)) dt = 0,
(9.8)
for all z E W"([0, T], RN ), which shows that y is a weak, and hence classical, solution to the equation (Iy'Ip-2y')'
-
lylp-2y = 0,
i.e., an eigenfunction of (9.1). But, taking z = y in (9.8), we obtain JoT
l y'(t)I p dt +
10
= 0,
a i.e., - = al (p, N). Thus A(p, N) E e(p, N) \ {0} and is clearly the smallest one by (9.2).
We do not know a proof of the result for p E ]1, 2[ when N > 1. The regularization argument used in [23] for the scalar p-Laplacian on a compact manifold and by [4] for the scalar case the periodic case does not seem easy to be generalized to the vector case.
REFERENCES [1]
T. Bartsch and J. Mawhin, The Leray-Schauder degree of S'equivariant operators associated to autonomous neutral equations in spaces of periodic functions, J. Differential Equations 92 (1991), 9099.
[2] A. Cabada and R.L. Pouso, Existence result for the problem (4(u'))' = f (t, u, u') with periodic and Neumann boundary conditions, Nonlinear Anal. T.M.A. 30 (1997), 1733-1742.
[3] A. Capietto, J. Mawhin and F. Zanolin, Continuation theorems for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc. 329 (1992), 41-72.
[4] M. Cuesta, Etude de la resonance et du spectre de Fucik des operateurs laplacien et p-laplacien, Ph.D. Thesis, Universite de Bruxelles, 1993.
[5] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.
J. Mawhin
62
[6] M. Del Pino, R. Manasevich and A. Murua, Existence and multiplicity of solutions with prescribed period for a second order quasilinear o.d.e., Nonlinear Analysis T.M.A. 18 (1992), 79-92. [7]
H. Dang and S.F. Oppenheimer, Existence and uniqueness results for some nonlinear boundary value problems, J. Math. Anal. Appl. 198 (1996), 35-48.
[8] C. Fabry and D. Fayyad, Periodic solutions of second order differential equations with a p-Laplacian and asymmetric nonlinearities, Rend. Ist. Mat. Univ. Trieste 24 (1992), 207-227. [9]
Z. Guo, Boundary value problems of a class of quasilinear ordinary differential equations, Differential and Integral Equations 6 (1993), 705-719.
[10] Ph. Hartman, On boundary value problems for systems of ordinary nonlinear second order differential equations, Th"ans. Amer. Math. Soc. 96 (1960), 493-509.
[11] Y.X. Huang and G. Metzen, The existence of solutions to a class of semilinear equations, Differential and Integral Equations 8 (1995), 429-452.
[12] H.W. Knobloch, On the existence of periodic solutions for second order vector differential equations, J. Differential Equations 9 (1971), 67-85.
[13] R. Manasevich and J. Mawhin, Periodic solutions for nonlinear systems with p-Laplacian-like operators, J. Differential Equations 145 (1998), 367-393.
[14] R. Manasevich and J. Mawhin, The spectrum of p-Laplacian systems with various boundary conditions and applications, Advances in Differential Equations 5 (2000), 1289-1318. [15] J. Mawhin, An extension of a theorem of A.C. Lazer on forced nonlinear oscillations, J. Math. Anal. Appl. 40 (1972), 20-29.
[16] J. Mawhin, Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces, J. Differential Equations 12 (1972), 610636.
[17] J. Mawhin, Topological Methods in Nonlinear Boundary Value Problems, CBMS Regional Conf. Ser in Math., vol. 40, AMS, Providence, 1979.
Periodic Solutions of Systems with p-Laplacian-like Operators
63
[18] J. Mawhin, Points fixes, points critiques et problemes aux limites, Semin. Math. Sup., vol. 92, Universite de Montreal, 1985. [19] J. Mawhin, Problemes de Dirichlet variationnels non lineaires, Semin. Math. Sup., vol. 104, Universite de Montreal, 1987. [20] J. Mawhin, Topological degree and boundary value problems for non-
linear differential equations. In: Topological Methods for Ordinary Differential Equations (M. Furi, P. Zecca eds.), Lecture Notes in Math. 1537, Springer-Verlag, Berlin, 1993, 74-142.
[21] J. Mawhin, Continuation theorems and periodic solutions of ordinary differential equations. In: Topological Methods in Differential Equations and Inclusions (A. Granas, M. Fnigon eds.), NATO ASI Series C 472, Kluwer, Dordrecht, 1995, 291-375. [22] N. Rouche and J. Mawhin, Ordinary Differential Equations. Stability and Periodic Solutions, Pitman, Boston, 1980.
[23] L. Veron, Premiere valeur propre non nulle du p-Laplacien et equations quasi lineaires elliptiques sur une variete riemannienne compacte, C.R. Acad. Sci. Paris 314 (1992), 271-276.
Jean Mawhin Universite Catholique de Louvain Institut de Mathematique Pure et Appliquee Chemin du Cyclotron 2, 1348 Louvain-la-Neuve Belgique MawhinQamn.ucl.ac.be
Mechanics on Riemannian Manifolds W.M. Oliva
1
Introduction
The main point of this short course is to present a unified formalism for a series of classical mechanical problems. We will see a way that enables us to describe the dynamical systems generated by mechanical systems. Most of the proofs have been omitted here. For a more complete presentation of the subject we refer the reader to the notes on Geometric Mechanics by W.M. Oliva (preprint IST, 2000, see [0]) and to the references therein. A mechanical system on a CO° Riemannian manifold (Q, (,)) depends
on the knowledge of an external data, the field F of external forces. The manifold Q is the configuration space. The Riemannian metric is related with the given distribution of masses and corresponds to the kinetic def (vp, vp), where energy K : TQ --+ R, a smooth function given by K(vp) a is the tangent vp belongs to TQ, the phase space of velocities. (TQ, T, Q)
bundle, so r : TQ - Q, T(vp) = p E Q, relates the phase space with the configuration space. In the cotangent bundle (T*Q, T`, Q), r*: T'Q -+ Q T' (ap) = p E Q, relates the space of momenta T*Q with the configuration space; a momentum ap E T'Q is a linear form ap : TTQ - R.
A field of external forces F is a fiber preserving C1 map, that is, J r: TQ -+ T'Q is C1 and satisfies r` o .F = T. A conservative field of external forces is defined by a C2 function U: Q -- IR in the following way: FU(vp) 40f -dU(p)
for all vp E TQ
.
U is called the potential energy.
The Riemannian metric (,) establishes a smooth diffeomorphism µ TQ -+ T*Q which is a fiber preserving map: p(vp) def (vp, ). That map µ is called the Legendre transformation or mass operator. As is well known, the metric (,) defines, on the manifold Q, an affine connection V, the (Levi-Civita) Riemannian connection. An affine connection is a law that maps a pair of smooth vector fields X, Y into a smooth
66
W.M.Oliva
vector field VV Y such that
Vjx+gyZ = f VxZ+gVyZ
Vx(Y+Z) = V Y+VVZ,
Vx(fZ) = fVxZ+X(f)Z, for all X, Y, Z E X(Q), the set of all smooth vector fields on Q, and all f,g E V(Q), the set of all smooth real valued functions on Q. The Riemannian connection V is the unique affine connection which is symmetric (VxY -VyY = (X, Y), for all X, Y E X(Q)) and compatible with the metric (X (Y, Z) = (VxY, Z) + (Y, VxZ), for all X, Y, Z E X(Q)). The Levi-Civita connection is defined by the Koszul formula
2 (VyX, Z) = X(Y, Z) + Y(Z, X) - Z(Y, Z)
- ([X,Z],Y) - ([Y,Z],X) - ((X,Y],Z)
Covariant derivative, parallel transport and geodesics of an affine connection
2
Proposition 2.1. If V is an affine connection on Q, then i) If X or Y is zero on an open set S2 of Q, then VV Y = 0 on SZ.
ii) (VVY)(p) depends on the value X(p) and on the values of Y along a curve tangent to X at p, only. iii)
If X(p) = 0 then (VxY)(p) = 0.
Proposition 2.2. If V is an affine connection on Q, then there exists a unique law that to each differentiable vector field V along a differentiable curve c: I - Q (I C IR is an open interval) associates another vector field DV along c, called the covariant derivative of V along c such that: i)
ii)
iii)
d(V+W)= DV + Dt(fV) _ (g')V+f DV If V is induced by a vector field Y E X(Q), (that is V(t) = Y(c(t))) then de = VEY, where c is the velocity field of c.
Given an affine connection V on Q and a differentiable vector field V = V (t) along a differentiable curve c : t E I --' c(t) E Q, one says that V is
parallel along c if di = 0.
Mechanics on Riemannian Manifolds
67
Proposition 2.3. (Parallel transport) If V is an affine connection on Q, c = c(t) is a differentiable curve on Q, and Vo E T,(to)Q a tangent vector to Q at c(to), then there exists a unique parallel vector field V along c such that V(to) = Vo. A geodesic of an affine connection V on Q is a differentiable curve c = c(t)
on Q such that the corresponding velocity field V = t(t) is parallel along c, that is, De dt_0'
for all tEI.
Generalized Newton law
3
Given a mechanical system (Q, (,),F) on a configuration space Q, the acceleration of a C2 curve q = q(t) with values on Q is the covariant of the velocity vector field 4 of the curve. Here, the covariant derivative derivative is relative to the Levi-Civita connection defined by (, ). The generalized Newton law is the relation
d
u(dq) = F(q) and a motion of the mechanical system is a C2 curve q(t) E Q that satisfies that Newton law. In local coordinates (ft; ql,..., qn) of Q, the local smooth are well functions gij = (a4- , q-) and r given by VA-q- = >k= I' defined on the coordinate neighborhood ft. If q(t) has its velocity written, 1.7
in ft, as 4 = Dq -d t-
1
4i(t) '(q(t)), then
n
E [9k+ ri'gi qjJ k=1
a
n and
aqk
(d aK
µ ( dt)
841,
f7Q ) dqk
where K = K(qi, qi) is the local expression of the kinetic energy. The
-
generalized Newton law implies, locally, d OK aq = Fk, k = 1, ..., n, (Lagrange equations) where Fk = .Fk(q, 4) are the local components of F: F(q) = E+k=1 Xk (q, q) dqk
A conservative mechanical system is defined by the data (Q, (, ), 1= -dU).
The mechanical energy E,,,: TQ -+ R is the C2 map E,,, = K + U o r and it is constant along a given motion q = q(t): dt
Em(4) = dt [ 2
( 4,
4 ) + U(q)]
=
(dU(q))
dt ' q) + = (,.r'[_dU(q)],4) + (dU(q)) q = 0
(that is we have conservation of energy).
W.M. Oliva
68
4 The Jacobi metric Let (Q, (,), -dU) be a conservative mechanical system and h E R a regular value of E,,, such that E;' (h) 00. On the manifold Qh = {p E Q I U(p) < h} one can define the Jacobi metric gh associated to (,) and h: gh (p) (Up, Vp) = 2 (h - U(p)) (up, vp)
Proposition 4.1. (Jacobi) The motions of (Q, (,), -dU) with mechanical energy Em = h are, up to a reparametrization, geodesics of Qh with respect to the Jacobi metric.
5
Horizontal and vertical vectors. TQ as Riemannian manifold
Fix vp E TQ and define two lifting operators:
H,,: wp E TPQ -
(horizontal lifting) where
(TQ)
(wp) def (c(t), V
(t))'.0, c(t) being the geodesic characterized by c(O) = p, 6(0) = wp and V (t) is the parallel transport of vp along c, V(O) = vp.
One can see that dr(vp)(H ,,wp) = wp and that is linear and injective. The elements of HP (TpQ) are the horizontal vectors at vp. (Vertical lifting) where
C,,,,: Wp E TpQ -,
(TQ)
,
a=°f
(vp + t wp)i_o. The map C,,, is linear and injective and is the set of vertical vectors, since We (wp)
have then Tvv (TQ) = C., (TpQ) ® H., (TpQ)
The metric of TpQ induces a metric on
(TpQ) and a metric on
(TPQ). If we define (TpQ) orthogonal to (TpQ), we have an inner product on (TQ) so a Riemannian metric on TQ.
6
Mechanical systems as (2nd order) vector-fields
From the local expressions of , the equations for the geodesics are xk + Ei j r k xi xj = 0, k = 1, ..., n, or, as a first order system, we have:
=vk
r vi vj
'Uk = 1
13
i, j
.
Mechanics on Riemannian Manifolds
69
in these local coordi-
So, the geodesic vector-field vp i-+ S(vp) E
nates, is written as
S(v,) = ((xkvk),
vp = (xk,vk)
(vk,-Er7vivj))
Now, a mechanical system (Q, (, ), )c), with the Newton law: p( 2nd order vector field: or
) = F(q),
= µ'1defines the
vp = (xk, vk) ' --+ E(vp) = ((xkvk)) (vk,
-
vi vj + fk) )
i, j
where fk is characterized by A-1(F(vp)) = Ek fk Then E(vp)
(p).
S(vp) + CvP (A-1 (F(vP)))
since C",(µ-1(F(Vp))) = ((xk,vk),(0,fk))
7
The geodesic flow of an affine connection
The horizontal vector field of TQ that to vp E TQ associates S(vp)
'ef
is the geodesic flow S: vp i- S(vp) of the affine connection V.
8
Mechanical systems with holonomic constraints
Let (Q, (,),F) be a mechanical system. A holonomic constraint is a submanifold N C Q such that dim N < dim Q. A C2-curve q : I C R -+ Q is compatible with N if q(t) E N for all t E I, so q(t) E Tg(t)N for all t E I. In order to obtain motions compatible with N we need to introduce a field of reactive external forces R,
R: TN -+ T*Q depending on Q, (,), N and F, only, and write a generalized Newton law
µ(dt)
_
for motions compatible with N (q(t) E Tq(t)N, `d t E I). The constraint N is perfect or satisfies d'Alembert's principle if the field R is "orthogonal to N"
that is "µ-1 R(vq) is orthogonal to TqN for all vq E TN" (orthogonality
W.M.Oliva
70
with respect to the given metric (,)). Since TqN ® (TgN)1 = TqQ for all q E N we project the generalized Newton law and assume 4(t) j4 0: 1(V44)T = µ-1
[EL-' %7(4)] T
(*)
+
[A-1 F(4)]1
R(4) = (V 4)1 -
(**)
The affine connection V induces on the submanifold N C Q another condef [(VYX)(p)]T for all nection D : X(N) x X(N) -' X(N); (DyX)(p) X, Y E X(N), P E N and X, Y local vector fields on Q extending X and Y. The above definition does not depend on the extensions, so D is an affine connection on N. If V is the Levi-Civita connection of (Q, (,) ), D is the Levi-Civita connection of (N, ((,))) where ((,)) is the Riemannian metric on N induced by (,). Consider on N the mechanical system (N, ((, )), N ) where .FN(V4)
def AV
[(IL-1 f (V4))T ] +
Vg E TqN ,
AN being the mass operator of (N, ((,))). The equation (*) corresponds,
precisely, to the generalized Newton law for the mechanical system (N, ((,)),.FN). On the other hand, (**) gives a formula to compute the field of reactive external forces, also called the reaction of the constraint µ-' 1Z(4)
= V44 - (o44)T - [lt-
'f(4)] 1
= (044 - D44) - [µ-1g'(9)] 1 = B(4, 4) - [µ-' T(4)] where B is the second fundamental form of the imbedding i : N - Q; so A-1 R(Vq) = B(vq, vq) - [EL-' T(vg)]
1
for all v.ETQN.
9 The double pendulum A good example of a mechanical system with holonomic constraints is the mathematical double pendulum.
Mechanics on R.iemannian Manifolds
71
We have two mass points gl(m1) and q2 (M2), q; E R2, i = 1,2. So Q = R2 x R2 = R4 and N C Q is the subset defined by Iql I = el and
Iq2-q1I=e2
The metric (u, v) = m1 ui v1 +m2 u2 v2 on Q defined with u = (ui, u2) E R2 x R2 and v = (v1, v2) E R2 x R2 corresponds to the kinetic energy
K(4) =
2
(m141 - 41 + m2 42 42),
where 4 = (41, 42) E R4
.
The field of external forces F is defined through the classical forces F1 = (0, m1 g) and F2 = (0, m2 g) in the following way:
F: T(R2xR2)
-' T*(R2xR2),
.F(4)(ul,u2)4efF,(4)u,+F2(4)u2
where
.F(q)(ui, u2) represents the total work of the pair of forces F1 and F2 along the directions ui, and u2. The potential energy U : R2 x R2 -,, R is defined by def
U(gi,g2) _ -m1 9y1 - m29y2 where q1 = (xi, yl) and q2 = (X2, Y2)- It is clear that .F(vp) = -dU(p), V vp E TpR4.
The manifold N is a torus, parametrized by the local coordinates (gyp, 9); so we have to consider the restrictions of the potential energy U and of the kinetic energy K to the torus (still denoted by U and K):
U = -m1 gelcos9-m2g(elcos9+t2cos40)
K
2
,
(mi 41 41 + m2 42 42) = 2 (mi (x1 + i2) + m2 (t + 1!2))
where qi = (i1i y1) and 42 = (x2, y2) for
x1=e1sin0,
yi=21cos9,
xi = £j sing+£2sincp,
Y2 = 6 cos9+£2COSV .
W.M. Oliva
72
Then i = ei
8 Cos 0,
yl = -e19 sin 6 ,
22 = 0 COS 0+60 COS W,
02 = - i 6 sin 6 - e2 cp sin cp .
Now the Lagrange equations can be written
daK_OK_ au dt a9
To
W6
-(ml + m2) g li sin 8
,
daK_OK4U dt ao
19W
-
acp = m2 g 6 sin
and give us a system of two second order differential equations. The mechanical energy Em = K + U o r is conserved; then Em = 2 (MI + m2) P162 +
2
m2 e2
02 + m2 t i e2 6 0 COW
(p)
is a first integral of the flow defined on tangent space of the torus that has local coordinates (6, 9, cp, cp).
The critical points are the zero vectors Op E TpN such that dU(p) = 0, that is, p = (6, gyp) such that sin 6 = sin W = 0; so, one has four critical configurations on N: pi = (0, 0), p2 = (0, ir), p3 = (--7r, 0) and p4 = (ir, 7r). We saw, by the theorem of Jacobi, that for the mechanical energy E,,, = h > max{U(p) I p E torus} the motions are geodesics, up to reparametrization, and for any Riemannian metric on the torus, the Morse theory states that between all closed curves giving m tours along 6 and n tours along cp, there exists one curve with the smaller length and that one is a geodesic; so the double pendulum has at least one periodic motion (geodesic of the Jacobi metric) that gives m tours along 6 and n tours along gyp.
10
The kinematics of a rigid motion
Let K and k be two oriented Euclidian 3-dimensional vector spaces. An isometry M: K -p k is a distance preserving map, that is IMX - MYI _
IX - YIforallX,YEK The induced map M*: K - k defined by M'(X) = M(X) - M(O), X E K and 0 the zero vector (origin) of K, satisfies: 1. M` is modulus preserving;
2. M` preserves inner product and is linear; 3. M' is a bijection, so M is a bijective affine transformation;
Mechanics on Riemannian Manifolds
73
4. The inverse of M is an isometry;
5. If M' is orientation preserving, M' preserves the vector product. A rigid motion of the moving system K relative to a stationary system
k is a map t - Mt where Mt is an isometry (Vt) with MM : K -+ k an orientation preserving linear map. In this case there exists a unique w(t) E k such that
Mi (Mt)-I=w(t)x
.
This follows because k,* (Mt)-' : k -+ k is skew-symmetric. In fact, we have
(Mi .(Mt)-Ix,y)+(x,Mt wt)-Iy)
=0
for all x, y E k, because one can compute the derivative with respect to
tof (MMX,M=Y)=(X,Y)for all X,YEKand make X=(Mt)-lx, Y = (M')-Iy. 11
Rigid bodies
A rigid body S C K is a bounded connected borelian in K such that under
the action of any rigid motion t '-4 Mt : K - k we have, for any C E S, that Q(t, l;) _ C, and then q(t, l;) = Mt(Q(t, C)) = Mt 1; = Mt C + Mt (O) where 0 E K is the origin of K. We say that the motion of S has a fixed point 0 if Mt(O) = 0 for all t. In this last case, 9(t, ) = M i % = M i (Mt*)-' g(t, ) = w(t) xq(t, (w(t), if nonzero, is the instantaneous axis of rotation).
Distribution of mass on a rigid body S is a positive scalar measure m on K such that m(U) > 0 for all nonempty open sets U of S. The center of mass of S is the point G E K defined by
G=
1
M(S) s
m(s) = total mass, m(s) > 0
l;
.
The kinetic energy of the motion of a rigid body S is given by
Kc(t) =
2
fI(t,e)I2dm()
The angular momentum of the motion of S is the vector p(t) _ fs [q(t, ) x4(t, )] dm(C) or p(t) = f(M) x (w(t) x q(t, )) dm()
74
W.M. Oliva
The general equations of the motion of S are
f
dfrt() def P,
EG1) fJ(t7)dm(e) = s EG2)
f
s
- c) x 4(t, )] dm(l;)
[(q(t, )
= f (q(t, ) - c) x
def pect
for any x E k (see [0] the "pre-print" notes on Geometric Mechanics by W.M. Oliva, for the definition of the measure dit Xt(c)) A rigid body is said to be isolated if there are no external forces acting
on it.
Proposition 11.1. If S is an isolated rigid body, the kinetic energy and the angular momentum are constants of motion (four scalar quantities).
Proof.
Ki(t)
2 dt
f((t), Q(t, )> dm(C) _
(4(t, e), 4(t, C))
f (w(t) xq(t, C), 4t,E) dm(l;) = f (4(t, ) xq(t, C), w(t)) (w(t), f 4(t, £) xq(t, l;) dm(l;)) = 0
(by EG2 with c = 0)
.
The configuration space of a rigid body with a fixed point is SO(k; 3), the group of all proper rotations of k. Assume the rigid body has at least
three points not in a straight line and fix a proper isometry B : K -+ k. Then the set SO(k; 3) is diffeomorphic to the set of all proper isometries M: K --p k satisfying M(O) = 0. In fact the map (DB: M F-+ M*B-1 E SO(k;3)
(where M*(X) = M(X) - M(O), has an inverse WB:
Thus, one can say that a rigid motion M corresponds to h(t) E SO(k, 3) and so
Jq(t, l;) = Mtt = h(t) B 4(t, ') = h(t) B t;
.
Then Kc(t) and p(t) suggest the definition
K`: TSO(k,3)
R
p: T SO (k, 3) - k ,
,
Mechanics on Riemannian Manifolds
75
by setting
(h, s) E T SO (k, 3) -- K°(h, s) = 2 jIsBI2dm() (h, s) E T SO(k, 3) m--- p(h, s) =
Jis
(h B l; x s B e) dm(le)
.
Then, there are four scalar first integrals defined on the 6-dimensional manifold T SO(k, 3). In general, they are independent, and the (nonempty) inverse image of a regular value (Ko, po) is an invariant 2-dimensional sub-
manifold. If we take Ko > 0, the vector-field induced on that invariant and orientable 2-dimensional manifold does not have zeros, so the manifold is necessarily a torus or a finite number of tori (the torus is the only orientable compact surface that admits a nonvanishing vector-field). By an argument similar to the one used for the double pendulum, there exists, always, a periodic motion in the dynamics of an isolated rigid body with a fixed point.
12
Mechanical systems with non-holonomic constraints
Let (Q, (, )) be a C°O Riemannian manifold and E : q E Q -+ Eq be a distribution on Q with dimension m less than n = dim Q. E being CO° means that each point q E Q admits m C°° local vector fields Y1, ..., Y'", defined in an open neighborhood St of q, that generate E. in all the points x E St. The distribution E is called the constraint and non-holonomic means non-involutive, that is, non-integrable. One can talk about the orthogonal distribution, so, we have two complementary vector sub-bundles EQ and E--Q of the bundle (TQ, T, Q). The two
corresponding orthogonal projections P: TQ - EQ and Pl : TQ -+ E1Q are well defined. A mechanical system with constraints is defined by the data (Q, (, ), E,F)
where F E Fk, that is F: TQ -' T*Q is a Ck (k > 1) fiber preserving map. Assume G E Fk and C(v) = C(Pv) for all v E TQ. The set of those Ck (k > 1) maps C is denoted by F. A C2 curve t -+ q= on Q is compatible with E if 4(t) E Eq(t) for all t. In order to obtain motions compatible with E we write a generalized Newton law
JJ(dt
= (.F+R)4
where R E F, (k > 1), is a suitable field of reactive forces. The constraint E is perfect or satisfies the D'Alembert principle if for any .F E Fk the field R E FF satisfies µ-l1 (vq) E EQ for all vq E EQ.
W.M.Oliva
76
12.1
Examples of distributions as constraints
1. A planar disc of radius r rolls without slipping along another disc of radius R. Q = S1 xS' and E is given locally by the zeros of the 1-form w = rd81 - Rd82. E is involutive (integrable).
2. Motions of a vertical knife free to slip along itself and also free to make pivotations cp around the vertical line passing through a point P of the knife. Q = R2 xS1 with local coordinates (x, y, cp) and E given by the zeros of the 1-form w = (sin cp) dx - (cos cp) dy
(free to slip means ds = e =
(E is nonintegrable)
sin
3. Motions of a vertical planar disc of radius r that one allows to roll without slipping on a horizontal plane and that, also, can make pivotations around the vertical line passing through the center. Q = IR2 x Si x Si with local coordinates (x, y, cp, 8), (x, y) E R2, cp is the
angle between the x-axis and the disc, and 8 measures the rotation of the disc when it rolls. E is given by the zeros of two 1-forms, wi = dx - r (cos cp) dO,
W2 = dy - r (sin cp) d8 .
E has dimension 2 and is nonintegrable on Q.
4. Let Q be the Lie group SL(2), set of all 2x2 real matrices with determinant 1. Its Lie algebra is spanned by the matrices
Let X, Y, N be the left invariant vector-fields on Q = SL(2) corresponding to x, y, n, respectively. Consider on SL(2) the left invariant
Riemannian metric defined by (X, X) = (Y, Y) = (N, N) = 1 and (X, Y) = (X, N) = (Y, N) = 0. Let E be the distribution spanned by X and Y. Since [X, YJ = -2N, E is not integrable.
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12.2
77
The total second fundamental form B of E
Let B : TQ x Q EQ - E-'-Q be defined as follows: if . E TQQ, 7 E Eq and z E E9 , let X, Y, Z three germs of vector-fields at q E Q, Y E E, Z E E1 such that X (q) = l;, Y(q) = it and Z(q) = z; one defines the bilinear form B(l;, rl) by 77), z) = (VxY, Z) (q)
and one can show that (Ox Y, Z)(q) depends only on X(q), Y(q), Z(q).
Proposition 12.1. (Gibbs, Maggi, Appell) Given a mechanical system with perfect constraints (Q, (, ), E, F), F E Fk, k > 1, there exists a unique field of reactive forces R E FF such that: i) µ-1 R(vq) E EQ for all vq E EQ;
ii) for each Vq E EQ, the maximal solution t -* q(t) that satisfies it (Lq) = (F + R) (9)
and initial condition q(O) = Vq, is compatible with E. Moreover,
iii)
the motion in (ii) is Ck+2 and is uniquely determined by vq E EQ;
iv)
the reactive field of forces R E FF is given by
R(vq) _ iiB(vq,vq) -µ([µ'-1.r(vq)] 1)
,
Vvq E EQ
Proposition 12.2. (Q, (, ), E, F) defines a 2"d order vector field on EQ given by
E(vq) = S(vq) + Cvv [(µ-1 F +'0- , R) vq]
or
E(vq) = S(vq) + Cv,, [B(vq,vq)+Pif'F(vq)],
Vq E EQ
and the Newton law becomes Dq dt
_ p1-1 'P(4)] + B(),
9(0) E EQ .
Corollary 12.3. Given a mechanical system with constraints, (Q, (, ), E, F), and considering the vector field X.F on TQ defined by (Q, (, ),.F), then
E = TP(X,) holds on EQ.
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78
12.3
Orientability of E and conservation of a volume
A distribution E : q E Q -> Eq C TqQ on a Riemannian manifold (Q, (, )) is orientable if there exists a differentiable exterior (n - m)-form 0 on Q such that, for any q E Q and any sequence (zl,..., zn_,,,) of elements in EQ we have 54 0 if, and only if, (zl,..., zn_r) is a basis of E9 . In fact this is equivalent to saying that EQ is orientable (as a manifold). In the codimension one case (m = n - 1), E orientable is equivalent to the existence of a globally defined unitary vector-field N with Nq E EQ
VgEQ. Proposition 12.4. If E is orientable, there is a volume form on EQ invariant under the flow E = TP(X,) defined by the mechanical system (Q, (, ), E,.F = -d(U r)) if, and only if, the trace of the restriction to ElQ xQ E1Q of Bl (total second fundamental form of El) vanishes. 12.4
Conservative non-holonomic mechanical systems
The system (Q, (, ), E,.F) is conservative if F(vp) = -dU(p) for all vp E TQ, U : Q - Ra C2 potential energy. In this case the generalized Newton law is written as DqT=(P grad U) q(t) + B(4,
and there is the conservation of energy along trajectories on EQ. In fact, dt [Em(4)l = dt
_
(()
+ U(q(t)))
,4)
=(
dt
+ dU(q(t)) 4(t)
\- (P grad U) q(t), 4) + ((grad U) q(t), 4>
=0
(because 4(t) E Ea(t) for all t).
13
Dissipative non-holonomic mechanical systems
Let (Q, (, ), E, F) be a non-holonomic mechanical system such that Q is compact, E is perfect and the field .P: TQ - T`Q is Ck, k > 1, given by
F(vp) = dV(p) + D(vp), such that V : Q - R is Ck+' and D = p'1 D is dissipative with respect to E, that is,
(P D(v), v) < 0
Vv E EQ
.
D strictly dissipative means dissipative plus the condition "(PD(v), v) = 0 v E EQ, implies v = 0". D strongly dissipative means that D satisfies the condition "there exists a continuous function c: Q - R+/(0) such that (P D(vp), vp) < -c(p) lvpj2, vp E EQ" .
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79
The system (Q, (, ), E, fl is said to be strongly dissipative if Q is compact, E is perfect, V is a Morse function and D is strongly dissipative.
Proposition 13.1. Let (Q, (, ), E,.F) be a strictly dissipative mechanical system and A C EQ be the attractor (set of all bounded and globally defined orbits of the associated vector field defined in EQ). Then
i) A is compact, connected, invariant and maximal; ii) A is uniformly asymptotically stable for the flow;
iii) A is an upper semi-continuous function of the pair (V, D); iv) If 4) 1 is the time-one map associated with the flow and 5 = {x E EQ I Ern(x) < a) with a > 0 and sufficiently large, then A = nn>o `I'i (B)J
v) -T(A) = Q; vi) If the system is strongly dissipative and A is a differentiable manifold, then dim A = dim Q.
14
Mechanical systems with semi-holonomic constraints. The dynamics of pseudo-rigid bodies
One can consider a C°° Riemannian manifold (Q, (,)) with a constraint E. If the distribution E is involutive, that is, integrable, we obtain on Q a foliation. If we proceed as Section 12 we need to look for motions compatible
with the foliation. Let us analyze a special case called the dynamics of pseudo-rigid bodies. Physically speaking we want to deal with bodies that can "deform" along the motion but the volume has to be conserved. Assume that Q is the group GL+(3) of all nonsingular 3x3 matrices. If P E GL+(3), let A, B be in the tangent space Tp GL+(3) and consider the Riemannian clef metric (A, B) p trace A A. 'B, where tB is the transpose of the matrix B.
Let V : GL+(3) -, R be a smooth "potential energy" and if f = det P (the determinant of P E GL+(3)) one assumes that f is constant along the motion of P. That is equivalent to considering the foliation defined by the distribution E spanned by df = 0. The corresponding Newton law is
DP
-dV + R(P)
where R is the reaction of the constraint. By d'Alembert's principle, µ-1 R(P) is unique if it is orthogonal to the constraint, relatively to the metric of the trace.
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80
The equations of motion constrained to det P = const are
P=-aP+Aap, f = det P = const > 0 (or df = 0) where A = A(P, P), uniquely determined, is the so called Lagrange multiare 3x3 matrices: (P)t; _ (q23 ), 8V = g v plier. Here P, P, P, and But II( DP) [T s -'a q-.;] dqt;; since K` = 2 trace(P. and _ t LP) =
2
(qii + q12 +
+ q33) we obtain, easily, the local equations
aV Of +A , aqj, aq:j f = det P = const (df = 0) .
In the analyic situation (V E C'° which is the case of the gravitational potential) the motion is given by P(t) = tT(t) A(t) S(t) (bipolar decomposition) where A(t) = diag(al(t),a2(t),a3(t)) and T(t), S(t) are orthogonal paths. Then T(t) tT(t) = S(t) tS(t) = I and by differentiation we get the skew-symmetric matrices W (t) : = 2'(t) tT(t)
A' (t) : = S(t) tS(t) .
The derivative of P = tT (tn'A + A + A A') S gives
P = tT(tn*A+A+AA*)S + tT(tct*A+A+AA')S
+tTAS+tTd(AA'-WA)S aV + A a(det aP Op p) P=tTAS together with the restriction
det P(t) = det A(t) = al(t) a2(t) a3(t) = const From the above equations we have
A+cr(S2'A-A-AA')+(-S2'A+A+AA*)A*+d (AA* -SZ'A) aA
+A A-1 (det A)
where we assume that V(P) = V(A), that is, the potential depends on al, a2, a3, only. If we compare the last conclusions with formula (57),
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81
p. 71, §27 of the famous book on Riemann ellipsoids (see [C] Ellipsoidal Figures of Equilibrium, Dover, 1987) written by Chandrasekhar, we see (see the mentioned book for the meanings of pc and p). that A det A = Moreover A can be explicitly computed and we arrive (after some tedious calculations) at the expression A=
1
(det A)2 trace(A-1)2
[_2traCeA(PA)(P2A)
+ det A trace SZ'2+As2+A-1
A_
where PA = diag(a3, al, (12) and P2A = diag(42, a3, al). The final system has variables (A, A, St', A') plus the condition al a2 a3 = const, and is not, in general, in normal form in Q* and A' due to the presence of the term Tt(A A' - Q* A).
Special solutions of the above system are the ellipsoids discovered by MacLaurin, Jacobi and Riemann, whose stability is nowadays an important field of research.
15
Structurally stable systems in mechanics: Morse-Smale and Anosov examples
Let M be a COO manifold and X a Cr, r > 1, vector field on M. X is Morse-Smale (MS) if the non-wandering set is the union of a finite number of critical elements (points and closed orbits) all hyperbolic with transversal stable and unstable manifolds. The (MS) flows on a compact manifold M are structurally stable (see [P], Palis and Smale when dim 1v1 > 3 and see [Pe], Peixoto when dim M = 2).
Historical remark: For many years the mathematical community believed that structural stability of flows was, generically, related with simple structures; in fact, that is true in two dimensions. But, in 1967, D.V. Anosov studied, extensively, special flows, nowadays called Anosov flows (AF) which
are also structurally stable and constitute a class of nontrivial and complex dynamical systems. Moreover a Holder Cl (AF) which has an invariant volume form is ergodic. Let us consider a mechanical system (Q, (,),.F) without constraints but with a strongly dissipative force, that is, ,F(v,,) = dV(p) + D(vp) such that
V = -U: Q -+ fit is a Ck+1 Morse function, and D = IA-'-D: TQ - TQ, a strongly dissipative force. Then we say, simply, that (V, D) is strongly dissipative.
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Proposition 15.1. Let (V, D) be strongly dissipative. Then i)
The singular points of the vector field on TQ, defined by the generalized Newton law, are hyperbolic, lie on the zero section and the projections on Q are the critical points of grad V;
ii)
The stable manifold W°(Op) and the unstable manifold W"(Op) of a critical point Op are embedded on TQ. Moreover, dim W" (Op) = Morse index of V at p E Q
,
and
dimW"(Op) < dimQ !5 dim W" (0p)
.
Remark 15.2. To be (MS) we need the transversality between stable and unstable manifolds because the dissipation of energy implies that nonwandering points are critical points. In fact,
T [Em(4)] =
Wt
C 1 (4,4) +
= (D(4)
U(q(t))) _
()+du((t)) -4(t)
- dU(q(t)), 4) + dU(q(t)) 4(t)
= (D(4),4) < 0
(when 4(t) 34 0).
The flows on TQ are vector fields, so elements of C"(TQ,TTQ), r > 1; one considers the Cr Whitney topology which has the Baire property. Denote
SDMS(D) = (V, D) strongly dissipative with fixed D} SDMS(V) = { (V, D) strongly dissipative with fixed V)
Theorem 15.3. Let there be given (Q, (,) ), Q compact, dim Q = d > 1 and r > 3 (1 +d). Let G be the subset of SDMS(D) [resp. SDMS(V)] such that there is transversality between stable and unstable manifolds. Then G is open and dense in SDMS(D) [resp. SDMS(V)J.
Remark 15.4. The structural stability of the elements in G can also be proved with respect to the (compact) attractor (A C TQ, the set of all bounded and global orbits).
Definition 15.5. Let M be a C°° compact Riemannian manifold. A nonsingular flow Tt : M -+ M is partially hyperbolic if the (derivative) variational flow DTt : TM -+ TM satisfies: i) for any p E M, where TM = Xp ®yp E) Z., where X, Y, Z are
invariant sub bundles of TM, dim X. = P >- 1, dim yp = k > 1, Zp J [(T tp)i=o];
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83
ii) X and Y are uniformly contracting and expanding, respectively, that is: there exist a, c > 0 such that
a
e-Ci
IDTLiI:5aIile`t,
,
dt > 0, Vl E Xp, pEM,
bt<0, VPEYp, pEM;
iii) Z is neutral (neither uniformly contracting nor uniformly expanding).
If, in particular, Zp = [(Ttp)t-o], then Z is neutral and the flow is called hyperbolic or Anosov flow (AF).
15.1
Sectional curvature and the Anosov geodesic flow
Let Q be a manifold and V an affine connection. The curvature tensor of V is the (1, 3)-type tensor (a, X, Y, Z) E l,1 (Q) xX3(Q) - a(RxyZ) E D(Q)
where
RxyZ = V1x,y1Z - Vx(VyZ) - Vy(VxZ) . Assume now that V is the Levi-Civita connection associated to a Riemannian metric on Q. Let P be a 2-dimensional subspace of TpQ, p E Q. If (v, w) is an orthonormal basis of P, define K(P) feel
(RvwV, W)(p)
where V, W are vector fields on Q such that V (p) = v and W (p) = w. The number K(P) is well defined and is called the sectional curvature of (Q, (,)) at P C Tp(Q). Theorem 15.6. (Hadamard, Anosov) The geodesic flow on the unitary tangent bundle M of a compact Riemannian manifold Q, having strictly negative all their sectional curvatures, is an Anosov flow on M.
Example 15.7. Let us come back to the Lie Group Q = SL(2) with the left invariant vector-fields X, Y, N corresponding to the basis of the Lie Algebra formed by the matrices
x = (1 1)
,
y = (1
and Ol)
n = (Ol 0)
The metric introduced on Q gives IXI = IYI = INI = 1 and
(X, Y) = (X, N) = (Y, N) = 0
.
The distribution E is spanned by X and Y, is nonintegrable because [X, Y] = -2 N, and VNN = 0. That last condition means the conservation
84
W.M.Oliva
of a volume under the flow of (Q, (, ), E,1 = 0), on EQ. The sectional curvatures of the 2-planes of the distribution are negative (equal to -7 in this example). SL(2) is not compact but there exists a discrete subgroup V (uniform subgroup) of SL(2) such that M = SL(2)/D is a dimension 3 compact manifold. The non-holonomic flow is well defined on E1M, the 4dimensional compact manifold of all unitary elements of the 5-dimensional vector bundle EM. The flow on E1M is partially hyperbolic and commutes with a nonvanishing vector field W. The bundle defined by W is invariant under the partially hyperbolic flow that induces on the quotient E1M/W an Anosov flow.
REFERENCES [C]
Chandrasekhar, S., Ellipsoidal Figures of Equilibrium, Dover, New York, 1987.
[O]
[P]
Oliva, W.M., Geometric mechanics, preprint, 2000, Dept. Math., IST-UTL, Lisbon. Palis, J., On Morse-Smale dynamical systems, Topology 8 (1969), 385-405.
[Pe]
Peixoto, M.M., Structural stability on two-dimensional manifolds, Topology 1 (1962).
[PS]
Palis, J. and Smale, S., Structural stability theorems, in: Global Analysis, Proc. Symp. in Pure Math., 14 (1970), A.M.S., Providence, RI.
W.M. Oliva Instituto Superior Tecnico Departamento de MatemStica Avenida Rovisco Pais 1 Lisboa, Portugal
vamolivaOmath.ist.utl.pt
Twist Mappings, Invariant Curves and Periodic Differential Equations R,afael Ortega 1
Introduction
Let us consider the periodic differential system in the plane, x = X (t, x), x E R2, where X satisfies X (t + 27r, x) = X (t, x),
`d(t, x) E R2.
In this course we will study the quasi-periodic solutions of this equation and we will show that these solutions play an important role in the study of the dynamics when the equation has a hamiltonian structure. Before trying to do any theory, we discuss a simple case at an intuitive level.
Example. z + 2x = sin t. This equation has the unique 21r-periodic solution x* (t) = sin t.
The other solutions are not periodic, in fact they are of the form
x(t) = sin t + cl sin Vt + c2 cos f t, cl, c2 E R. These functions are each sums of two periodic functions with nonconmensurable periods (Ti = 27r, T2 = v12--7r). As we shall see, they are quasi-periodic solutions with frequencies wl = = 1 and w2 = 21 = V2.
From the mechanical point of view these solutions are very natural. In the absence of external force the oscillator
1 +2x=0
86
R. Ortega
has periodic solutions with frequency w2 = f. We now add an external force with frequency wl = 1 and the typical solution of the forced equation will combine both frequencies. Next we shall look at the equation (1.1) from a geometrical point of view.
Let us consider the space of three dimensions with coordinates (x,.t, t). First we draw in this space the curve produced by the periodic solution x*. This is the helix with parametric equation (sin t, cos t, t). The other solutions of (1.1) satisfy 2 (x (t) - cos t)2 + (x(t) - sin t)2 = constant, Vt E R.
(1.2)
This identity follows from the conservation of energy for the autonomous equation. The solutions corresponding to the level constant = 0.25 are
x'(t) = sin t + 0.5 sin(vt + c), c E R. The corresponding initial conditions at t = 0 define an ellipse in the plane (x, i), namely 2(.t -1)2+x2=0.25. Eo:
This ellipse is transported by the flow according to (1.2). It becomes an ellipse E(t) with moving center at the helix generated by x*. In this way we produce a cylinder that is invariant by the flow. See the figure below.
The intersection of the cylinder with t = 0, ±27r, ±41r.... is always the initial ellipse. However, the corresponding solutions are not periodic and will not arrive at the same point of the ellipse. To be more precise let us define the Poincare map P : 1R2 --+ R2, (x(0), x(0)) " (x(2ir), x(2ir)),
where x(t) is an arbitrary solution of (1.1). The ellipse Co is invariant by
P, that is
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87
The following exercise gives more details on this curve.
Exercise 1. Define t = (xo, io)t E R2 and prove that the Poincare map can be expressed in the form
P(t) = At + b.
Compute the matrix A and the vector b and prove that, for each t E Co, 0 the orbit {P"(t)}"EZ is dense in Eo.
The periodic solution x' corresponds to the fixed point (0, 1)t of P. In a similar way, the family of quasi-periodic solutions {xC}CER is associated
to the invariant curve CO. As we shall see later, this correspondence between invariant curves and quasi-periodic solutions is typical in the class of periodic equations. We have been able to understand so well the previous equation because it is linear and the solutions have been explicitly computed. We can now ask what happens in the case of nonlinear equations. As an example consider the forced oscillator
+ g(x) = f(t),
(1.3)
where g is nonlinear, g(x) - ±oo as x - ±oo and f is 21r-periodic. Now the dynamics can be complicated but still many things can be said. In 1976 Morris studied this equation when g(x) = 2x3 (see [14]). He proved the existence of many invariant curves for the Poincare map of this equation. Each of these curves produces a family of quasi-periodic solutions with frequencies wl and w2. The frequency wl = 1 is due to the period of f but w2 changes with the invariant curve. This is an important difference with respect to the linear case. These families of quasi-periodic solutions can be visualized as invariant cylinders in the space (t, x, i). These cylinders are 27r-periodic in time and they become the so-called invariant tori after the identification t = t + 2ir. By uniqueness, the solution starting inside a cylinder will never escape. This implies that all the solutions are bounded. In contrast to the linear case the invariant curves probably do not foliate the plane. The work of Morris was motivated by a problem posed by Littlewood, and several authors have extended it to more general equations of the type
(1.3). See [11] for a description of the results obtained in the superlinear case. In all these works the existence of invariant curves is proved using the Twist Theorem. This result was obtained by Moser in the sixties in order to solve a problem in the theory of stability. By now it has become a crucial tool for the understanding of the dynamics of periodic hamiltonian systems in the plane. As we mentioned already, this course is devoted to the study of quasiperiodic solutions. They are interesting by themselves, especially from a mechanical point of view, but they are also useful to obtain interesting properties of the equation. Due to the associated invariant cylinders they
R. Ortega
88
can be employed to prove boundedness (invariant curves around infinity) or stability (invariant curves around the origin). The plan of this course is the following. First we shall study some generalities of quasi-periodic functions and quasi-periodic solutions. This includes the precise connection between invariant curves and quasi-periodic solutions. After this we shall state a simple version of the Twist Theorem that is ready for applications. It follows from the works of Moser in [15] and Herman in [4, 5]. Finally we shall apply all the previous theory to a concrete equation of the type (1.3): the asymmetric oscillator. This section combines [16, 3] with ideas taken from [13].
To conclude this preliminary chapter we introduce some notation. The circle will be identified with the quotient group
T' = R/21rZ. Given a real number 0 E R, the corresponding equivalence class is denoted by 8 = 0 + 2irZ.
The rotation of angle a is defined as
R,, : T' - Ti, 1
(8) = B + U.
Periodic functions f = f (0) will be identified with functions defined on the circle.
The torus is defined as
T2 = T' x T' and, in an analogous way, doubly periodic functions f = f (01, 02) will be identified with functions defined on the torus.
2
Quasi-periodic functions with two frequencies
Let w1,w2 E R - {0} be two real numbers which are not commensurable; that is, (2.1)
W1 VQ.
A function x : R - RN, t H x(t), is said to be quasi-periodic with frequencies (Wi,w2) if there exists another function F E C(T2, RN), F = F(01, 02), such that
x(t) = F(wit,w2t), Vt E R. The condition (2.1) says that the set {(wlt,w2t In consequence, F is uniquely determined by x.
:
t E R} is dense in T2.
Twist Mappings, Invariant Curves and Periodic Differential Equations
89
Example. The function x(t) = sin t + 0.5 sin f t is quasi-periodic with frequencies w1 = 1, w2 = v'. Notice that sup x(t) = 1.5 and inf x(t) _ -1.5 are not reached and therefore x is not periodic of any period. The class of quasi-periodic functions with frequencies (wl,w2) will be denoted by QP(wl,w2). The target space RN will not be specified in this notation because it will be fixed. We have in mind the case N = 2. Exercise 2. Prove that every function in QP(WI, w2) is almost periodic.
0 Exercise 3. Given x E QP(WI, w2), the limit below exists, 1
T
x:= Ti Moreover, 1
X
(27r)2
I I 2a
2,r
F(0i, 02)d91d92.
[Hint: study first the case x(t) = e'(nw'+mW2)t]
A function x in QP(wl,w2) can also belong to other spaces QP(wi,w2). In such a case, the function F representing x will change when we change the frequencies. As an example let us go back to the function x(t) = sin t +
0.5 sin it. It also belongs to QP(3 +
5 + 2v) and the corresponding
function F* representing x is now F*(91, 02) = sin(201-92)+0.5 sin(-501+ 302). The next result clarifies this. Proposition 2.1. Let Wt, w2 and w*, w; be two couples of frequencies satisfying (2.1). Then the following statements are equivalent: (i) (ii)
QP(wl,w2) = QP(wj,w2) There exists a 2 x 2 matrix A with entries in Z and such that
detA = ±1,
(w*,w2)t = A(wi,W2)t.
To prove this result we need an algebraic result on subgroups of (R, +) with two generators. Given wi and w2 satisfying (2.1), the subgroup generated by these two numbers is denoted by (WI, W2) = {nW1 + mw2 : n, m E Z}.
The reader is invited to prove the following result.
Lemma 2.2. The condition (ii) is equivalent to (iii) (WI, W2) = (wi,W2)-
Proof of Proposition 2.1. The implication (ii) = (i) is easy once we notice that the matrix A-1 has also its entries in Z. For instance, given
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R. Ortega
x E QP(w,,w2) with x(t) = F(wjt,w2t), we can also represent this function as x(t) = F*(w*t, wet) where F* = FoA-1. This function is doubly periodic precisely because A-1 has integer coefficients. (iii). Since we In view of the previous Lemma we shall now prove (i) can interchange the roles of the frequencies wi and wi , it is sufficient to prove the implication QP(wl,w2) C QP(w1,w2)
(WI, W2) C (wi,w2)
By assumption, the real and imaginary parts of the function ei`''lt belong to QP(wi , w2 ). Let F be a function in C(T2, CN) with Fourier series
F(0,, 02) > F'n,mei(ne'+m02) n,m
and such that e.w't = F(wlt, w2t).
Since the system ei(" B1+M02) is complete in L2(T2) it is possible to find some integers n and m for which Fn,m 36 0. We now apply Exercise 3 to the function ei"'te-i(n"i+mw2)t to deduce that lim
1
T
ei(w'-nwi-rnw;)tdt
T-oo 2T -T
# 0.
This implies that w, = nwi + mw2. In the same way one can prove that also w2 is a Z-linear combination of wi and w2. This shows that the group generated by the omegas is included in the group generated by the omegas*. For the applications to 21r-periodic differential equations, it is convenient to freeze the first frequency wl = 1 while the second will be simply denoted by w, where w V Q. The previous proposition implies that the identity
QP(l,w) = QP(l,w*) occurs only in the cases
w+w*EZ or w-w*EZ. The minimal period (or maximal frequency) is a very useful concept in the theory of periodic functions. We now define a related concept in the class QP(1,w). A function x E QP(1,w) belongs to the class M(w) if 2ir is the minimal period of F with respect to the second variable 82i that is, F(81, 82 + P) = F(81, 82) V(8,, 82) E R2
P E 21rZ.
As an example, consider the functions xi(t) = sin t+0.5 sin /t and x2(t) _
sin t + 0.5 sin 3 f t. The function x, belongs to M(f) while x2 is in QP(1, v-) - M(vf2-). Exercise 4. Let x : R --+ RN be a continuous function with minimal period
P and assume that w =
Q. Prove x E M(w).
Twist Mappings, Invariant Curves and Periodic Differential Equations
Exercise 5.
91
Prove the equivalence below for a function x E QP(1, w),
n=2,3,....
xEJet(w)
0
There are several nonequivalent definitions of quasi-periodic function, depending on the smoothness imposed on the function F. In the book of Siegel and Moser [21, Section 36) the function F is real analytic. Nevertheless, the previous discussions were inspired by this book.
3
Periodic differential equations. The Poincare map
Let us consider the differential equation ± = X (t, x),
(3.1)
where X : R x R2 -p R2, (t, x) '-* X (t, x) is continuous and 27r-periodic in t. We also assume that there is uniqueness for the associated initial value problem. Given p E R2, tt(p) denotes the solution of (3.1) satisfying x(O) = p. It is defined in the maximal interval Ip. For each t E R we can define the map R2, p'--' ye(p), q5t : Dt C R2 where Dt = {p E R2: t E Ip}. The theorem on continuous dependence implies that Pt is open and ¢t is a homeomorphism from Dt onto Ot(Dt). The family {4t} satisfies the property cbt o 102,, 02,, o It = #t}2,, Vt E R. The map P = q2,r will be called the Poincare map of (3.1). It satisfies
-
P n = 02an, 'm E Z. Exercise 6. Construct examples where the open set V2,r is disconnected. Construct another example with D2,, = 0. 0 Exercise 7. Construct an example of a differential equation (3.1) such that the field X is not locally Lipschitz continuous but there is uniqueness for the initial value problem. 0
The dynamics of (3.1) can be studied through P. Usually the properties of the differential equation are translated to the language of discrete dynamics via P. For example, a 27r-periodic solution corresponds to a fixed point,
a subharmonic solution with minimal period 27rn corresponds to an nperiodic point. Let us now assume that x(t) is a solution of (3.1) in the class QP(1,w). How is this property reflected on the Poincare map? This will be the problem of the next section.
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R. Ortega
92
Quasi-periodic solutions and invariant curves
4 4.1
Families of quasi-periodic solutions
Given an autonomous system ± = X (x) and a nontrivial periodic solution ry(t), we know that also ry(t + c) is a periodic solution for each c E R. In a similar way we shall show that quasi-periodic solutions of a periodic system appear in families depending on one parameter.
Lemma 4.1. Let x(t) be a solution of (3.1) that belongs to the class QP(1, w) and let F E C(T2, R2) be the associated function on the torus. For each c E R define X, (t) = F(t, wt + c). Then xc(t) is also a solution of (3.1).
Proof. The periodicity of (3.1) implies that the translates x(t + 21rn),
n E Z
are also solutions. From the definition of the family {xC } we deduce that x2,rnw (t) = F(t, wt + 21rnw) = F(t + 2irn, w(t + 27rn)) = x(t + 27rn). This implies that the functions x2,r,,,,, are solutions of the equation. Since w is not in Q, the sequence {2irwn}nEZ is dense in T'. Using this density and the theorem on continuous dependence we deduce that all the functions x, are solutions. 0
The next result studies the properties of this family of solutions when we are in the class M(w). The proof is left as an exercise. Lemma 4.2. Assume that x E M(w) is a solution of (3.1) and let {xc} be the associated family. Then, (i)
xc, = xc,
c1 - C2 E 21rZ.
(ii) x,(t + 21r) = xe+2x(,(t)
Vt E R.
An interesting fact about the family {x. } is that it allows us to reconstruct the function F from it. In fact, if we consider the definition
x'(t) = F(t, wt + c) and define the new variables 91 = t, B2 = it + c we are lead to the formula F(91i 92) = xe2-me, (91).
(4.1)
The next result shows that quasi-periodicity can be characterized in terms of properties (i) and (ii).
Twist Mappings, Invariant Curves and Periodic Differential Equations
93
Proposition 4.3. Let {xI}cER be a family of solutions of (3.1), defined in (-oo, +oo) and depending continuously on the parameter; that is, the function R x R --4 R2, (t, c) '-' X, (t)
is continuous. In addition, properties (i) and (ii) of the previous Lemma hold. Then x, is in M(w). Proof. Define F as in (4.1). This function belongs to C(R2, R2) and satisfies
xc(t) = F(t, wt + c).
It remains to prove that it is doubly periodic and that 21r is the minimal second period. The 27r-periodicity in 01 follows from property (ii). The rest 0 is a consequence of (i).
We are now ready to understand the effect of a quasi-periodic solution on the Poincare map.
4.2
From quasi-periodic solutions to invariant curves
Given a solution x(t) of (3.1) that belongs to M(w) and the associated family {x,}, we define the parametrized curve
ti : R -> R2, O(c) = x,(0) = F(0, c). The property (i) and the uniqueness for (3.1) imply that r/' is 21r-periodic and one-to-one on (0, 27r). Thus, r = O(R) is a Jordan curve in the plane. Moreover,
PO(c) = Px.(0) = x.(2ir) =
O (c + 27rw).
This identity implies that r is invariant under P, that is
P(r) = r. This is not the only information given by the previous identity. In fact, if we parametrize r with respect to a circle, the mapping
becomes a homeomorphism and the following diagram is commutative.
p r
r
0
t T1 - T't R2am
0
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R. Ortega
In other words, the restriction of the Poincare map to the invariant curve is conjugate to a rotation of the circle. Moreover, the frequency of the quasi-periodic solutions can be recovered from the rotation number. Exercise 8. Let r be a Jordan curve such that P(r) C r. Then r is invariant under P. [Hint: T' is not homeomorphic to any subset of It).
4.3
From invariant curves to quasi-periodic solutions
Let us now see the converse. We start with a Jordan curve in the plane, r C R2, which is invariant under P and such that the restriction of the Poincare map to r, denoted by Pr, is conjugate to a rotation R2,,,,,, for some w V Q. We shall show that quasi-periodic solutions can be produced from r. First we allow the flow to evolve from r and consider the family of solutions starting at this curve. The invariance of r implies that these solutions are defined in (-oo, +oo) and we can define xc(t) = 1t('Y(C)), c,t E R, where i : R - r is the 2r-periodic parametrization such that Pr 0 'V =
'Vr
0 R2,,.
We shall show that this family satisfies the conditions (i) and (ii) of Lemma 4.2 and so Proposition 4.3 can be applied to deduce that xc E M(w). The property (i) is immediate because, by assumption, t/i is one-toone in [0, 27r). To prove (ii) we use the commutativity of {qSt} with P and obtain xc(t + 27r) = -Ot+2, ()(C)) = ct o 102.(P(C)) = Ot(P o P(C))
= 4t(i(c + 27rw)) = xc+2x,,,(t).
To sum up, we can say that finding a solution in M(w) is equivalent to finding a Jordan curve r which is invariant under P and such that Pr is conjugate to Rte,,,,,.
Exercise 9.
Given w, w*
Q, prove the equivalence
.M(w) = M(w*) b 4.4
is conjugate to IZ2i,,
.
Invariant cylinders
Given a Jordan curve r in the plane, the bounded component of R2 - r will be denoted by R,(r). Let r be a Jordan curve included in Dt. Prove that R,(r) C Dt and ot(Rt(r)) = Rt(ot(r)). Exercise 10.
Twist Mappings, Invariant Curves and Periodic Differential Equations
95
Let us now assume that r is an invariant curve for P. Since ¢t is a homeomorphism also rt = ¢t(r) is a Jordan curve. Moreover, rt+2,, = rt
Define
dt E R.
C={(x,t)ER2xR: tER, xER,(rt)}.
This set is invariant with respect to the differential equation (3.1). That is, given a solution x(t) of (3.1), such that (x(to),to) E C for some to, then it is defined in (-oo, oo) and
(x (t), t) E C Vt E R. This is a consequence of Exercise 10.
4.5
Some examples
We shall now analyze the quasi-periodic solutions of three equations.
Example 1. + w2x = f (t), f E C(T1), w > 0, w V Q.
This is a continuation of the starting example, where w = f and f (t) _ sin t. There is a unique 27r-periodic solution and the other solutions are in the class M(w). The Poincare map has a fixed point that is the center of a family of concentric ellipses which are invariant under P. Moreover, the restriction of the Poincare map to each of these ellipses is conjugate to the rotation of angle 27rw. Exercise 11.
Discuss the case w E Q.
Example 2. Let us assume now that w is a smooth function from [0, oo) into R, with w(P)>0 Vp>0. We consider the system
i i = -w(P)x2, -+2 = w(P)xl,
where p = x1 + x2. This system is autonomous but we shall look at it as a 27r-periodic system. The nontrivial solutions are xi(t) = Po 2 cos(w(Po)t + c), x2(t) = P1112 sin(w(Po)t + c)
with po > 0 and c E R. This system is nonlinear but easy to integrate because the function p is a first integral. When w(po) §E Q the solution belongs to M(w(po)). The origin is a fixed point of the Poincare map and
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R. Ortega
the circles around this point are invariant. The difference with the previous example is that the rotation number of each of these curves depends on p. If we perturb this system and introduce periodic coefficients, in most cases the first integral will disappear. Also, the foliation of the plane by invariant curves will be destroyed but still some curves will remain if the perturbation is not too large. These statements are not rigorous or precise but the reader can be convinced by herself (or himself?) via numerical experiments.
Example 3.
i+ca= f(t,x), c>O. This is the general equation of motion in the presence of friction. The force
f will be smooth and 27r-periodic in t. We are going to prove that this equation cannot have quasi-periodic solutions in M(w) for any w V Q. We
do it by contradiction. We know that such a solution would produce an invariant curve r for the Poincare map. The region Rj(I') would also be invariant. Due to the friction, the mapping P is area contracting. Thus,
meas(P(R!(r))) < meas(R,(r)), and this is not compatible with the invariance of R,(I').
5
Invariant curves of mappings of the annulus
We shall consider a system of polar coordinates in the plane. Every point in R2 - {0} has coordinates (B, r) where 0 E R and r > 0. Given b > a > 0, A is the annulus defined by
A={(9,r):
ET', a
A universal cover of this annulus is the strip
A= {(9,r) : 0ER, a
M: A -R2, (B,r)'-' (Bi,ri) we can find a lift to A, given by M : A -+ R2,
(0, r) '- (01, r1).
Both, the mapping and the lift, are denoted by M. An invariant curve of M is a Jordan curve r C A, which is homotopic in A to the circle r = a and such that
M(r) = r.
Twist Mappings, Invariant Curves and Periodic Differential Equations
97
Intuitively speaking, to assume that the curve is homotopic to r = a just means that the curve goes around the hole. It is easy to construct examples of mappings M and curves 1, such that m(r) = r but r can be deformed to a point in A. They are not considered as invariant curves in the previous definition. Let us now study some simple mappings in the annulus.
Example 1. Rotations. 91 = 9+i3, r1 =r. Here ,l3 E R is the angle of rotation. All circles r = constant are invariant curves. Moreover, the restriction of M to each of these circles is always conjugate to Rp.
Example 2. Twist mappings. 91 = 9 +,Q + a(r), r1 = r. Here a : [a, b) -+ R is a smooth function with
a'(r) > 0
dr E [a, b].
Again the circles r = constant are invariant but now the restriction of M to each of them is conjugate to a different rotation Rs+a(r) The monotonicity of a says that the rotation number is monotone with respect to the radius r. The name "twist mapping" can be justified by the following geometrical observation: given a radial segment 9 = constant the map M transforms it into a twisted arc. Our goal is to obtain a theorem on the existence of invariant curves for small perturbations of the twist mapping. However, the next example will show that we shall have to impose some additional condition.
Example 3. Twisted contractions. 91 = 9 + Q + a(r), r1 = (1 - e)r.
If e > 0 is small this mapping becomes a small perturbation of the twist mapping. However, it is clear that it has no invariant curves because all orbits will escape from the annulus in a finite number of iterations. To exclude mappings like those in Example 3 we shall introduce the following definition. M has the intersection property in A if
m(r) n r # 0, for any Jordan curve r in A which is homotopic to r = a. The mapping of Example 3 does not satisfy the intersection property. In fact, circles r = constant are transformed in circles of smaller radius. On the other hand, rotations and twist mappings have this property in any annulus centered at the origin. To prove this we notice that in these
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R. Ortega
cases M can be seen as an area-preserving homeomorphism of the whole plane with M(O) = 0. Given a Jordan curve r with 0 E Rt(r), the curve
t1 = M(I') also satisfies 0 E R;(r1). If r and I'1 do not intersect then one of the regions R;(r), R,(I'1) should be strictly included in the other. This is not compatible with the area-preserving character of the mapping because
measR,(r) = measR,(I71). Exercise 12. Let M be the twist mapping of Example 2. Find a Jordan curve r in the plane such that M(F) n F = 0.
Exercise 13. Assume that M is a homeomorphism from A onto M(A). For each k = 0, 1, . . . , oo, w, we say that M has the Ck-intersection property if the previous definition is restricted to Jordan curves of class Ck. Prove that all these properties are equivalent. [Hint: an elegant proof using Riemann's theorem on conformal mappings can be seen in [6]]. We are now going to state the Twist Theorem. It guarantees the existence of invariant curves for small perturbations of the twist mapping having the intersection property. The perturbation will be small in class C4.
Theorem 5.1. Let a E C4 [a, b] be a function satisfying
a'(r) > 0 Vr E [a, b]. Then there exists e > 0, depending on b - a and a, such that a mapping M : A - R2 has invariant curves if it satisfies the conditions below, M has the intersection property, the lift of M can be expressed in the form 01
=0+Q+a(r)+cpl(9,r),
r1 =r+co2(9,r),
with
IIv01IIC4(A) + IIW2IIC4(A) < E.
Remarks. 1. This theorem was proved by Moser in (15] assuming that M was of class 333. An analytic version of the theorem was presented in [21] and a version in class C5 can be seen in Russmann [19]. The version we have just stated can be proved using the techniques developed in the two works by Herman [4, 5]. 2. The proof of this theorem gives additional information and implies the existence of many invariant curves in A. In fact, there exist infinitely many numbers v in the interval [/3 + a(a), /3 + a(b)] for which it is possible to find an invariant curve r such that Mr is conjugate to R.V. That is, the
Twist Mappings, Invariant Curves and Periodic Differential Equations
99
diagram below is commutative, r
0
Mr -+
r
T
T
T1
T1
R
0
(As before, Mr denotes the restriction of M to r. The admissible numbers v are not in 21rQ and must satisfy some additional properties. See [4].) Here V is some homeomorphism. Exercise 14.
Consider the annulus A = {1 < r < 2} and the mapping M : A --+ R2,
01 = 9 + 7r, rl = r + e sin 20.
i) Prove that, for small e, this mapping has the intersection property in A. [Hint: apply Stokes Theorem with the differential form w = rd9]. ii) Prove that all orbits with 9o E [0, 2ir) - {0, a , ir, 2 } escape from the annulus. iii) Deduce that M has no invariant curves for small e. This proves that there is no analogue of the Twist Theorem when a = 0. O In many applications the mapping M is a perturbation of the small twist mapping defined by
91 = 9 + Q + ba(r), rl = r,
where E is a small parameter. In these cases the previous theorem is not applicable but one can use the so-called Small Twist Theorem. Theorem 5.2. Let a E C4 [a, b] be a function satisfying
a'(r) > 0 dr E [a, b]. Then there exists e > 0, depending on b - a and a, such that a mapping M : A -+ R2 has invariant curves if it satisfies the conditions below,
M has the intersection property, the lift of M can be expressed in the form. 91 = 9 + A + 6a(r) + acpl (9, r), r1 = r + 6W2 (0, r), for some b E (0,1) and cp1i cp2 E C4 (A), IICv1I1C4(A) + IIV2IIC4(A) < C.
Remarks. 1. The proof of this result is similar. Notice that it does not follow directly from the Twist Theorem because the number a is independent of 6.
2. As b tends to 0 the rotation numbers of the invariant curves will tend to p. Again it is possible to find many invariant curves.
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6 The asymmetric oscillator Let us consider the differential equation
x + ax+ - bx- = 1 + ep(t),
(6.1)
where a and b are positive constants, a is a parameter and the function p is 27r-periodic. This equation was proposed by Lazer and McKenna as a simplified version of their model of a nonlinear suspension bridge (see [10]).
When a = b we go back to the linear oscillator and we can have the classical phenomenon of resonance. As an example consider the equation
x+x=1+esint. In this case it is easy to prove that, for any e jA- 0, all solutions are unbounded. The next result shows that the situation is different for the case
a#b.
Theorem 6.1. Assume that a and b are positive constants with a 54 b. In addition, the function p is of class CS(T'). 'ten there exists e* > 0 such that all the solutions of (6.1) are bounded if jej < e*. Remarks. 1. This theorem was proved in [16] assuming less regularity for p, namely p E C. The extra regularity CS will allow us to obtain a simpler proof. I do not know of an example showing that unbounded solutions can exist when a is still small but p is not smooth. 2. In [3], J.M. Alonso and I constructed many examples showing that unbounded solutions can exist if a is not small and the parameters a, b satisfy 1
f + vrb E Q.
(6.2)
I do not know of an example showing that unbounded solutions can exist when a is large and (6.2) does not hold. In a recent paper [13], Liu Bin has given a new proof of the theorem. In the next pages I shall present a proof that combines ideas from [13] and from [16, 3]. However, the most technical part of the proof (the estimates) will be obtained by a new procedure. I hope that the reader will find this procedure rather simple. The proof will follow after several steps.
Step 1. The asymmetric polar coordinates We start with the autonomous ("homogeneous") equation
x + ax+ - bx- = 0.
(6.3)
The nontrivial solutions of this equation oscillate and satisfy alternatively the linear equations x + ax = 0 if x > 0 and x + bx = 0 if x < 0. They are periodic, with minimal period n 7r
T=
7a=
+ b'
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101
and they look more or less like a sinusoidal function with two asymmetric bumps. It will be convenient to select one particular solution that plays the role of the "asymmetric cosine". Let C(t) be the solution of (6.3) satisfying C(0) = 1, C(0) = 0. Since the equation is autonomous and positively homogeneous, all solutions can be expressed in the form
x(t) = aC(t + Q), a > 0, A E R. Exercise 15.
Prove
rT
J0
C(t)dt = 2 a(
a - b )'
0
Next we define the "asymmetric sine" as S(t) = C(t). The conservation of energy for (6.3) leads to S(t)2 + aC+(t)2 + bC-(t)2 =a dt E R.
(6.4)
It is convenient to notice that, for a = b = 1, the functions C and S are C(t) = cost and S(t) = -sin t. In such a case the identity (6.4) becomes the classical trigonometric identity. We shall now analyze (6.3) from a geometric perspective. If we look at
the phase portrait in the plane (x,±), we find that the nontrivial orbits are closed curves obtained by gluing two ellipses. Namely, 2 y2 + 2x2 = cl, if x > 0 and 2i2 + bx2 = c2i if x < 0, where Cl and C2 are appropriate constants. The energy
E = 1x2 +2
(x+)2+2(x-)2
is preserved along these orbits. Since the minimal period is always T, the origin is an isochronous center. We shall use this phase portrait to define a system of coordinates. Define
x = 7I1/2C(!), y = yJI/2S(!), I > 0, 0 E R, where 'y > 0 is a parameter that will be determined later and ) = 22, . The mapping if : Tl x (0, oo) - R2 - {0}, (B, I) '-+ (x, y) is one-to-one and onto. This can be proved using (6.4). Since C is C2 and S is Cl, we can say that T is Cl. We shall now prove that, for an appropriate value of 7, IF will transport the symplectic forms. This means dx n dy = dO A dI,
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or, in the language of jacobians,
det V(9, I) = 1.
This property and the local inverse function theorem imply that' is a C' symplectic diffeomorphism. A computation and the identity (6.4) lead to dx A dy = SI (S( )2 - C(jj)S(
)jd9 n dI =
2n
ad0 A dI.
We define -y = V 2i2 n Exercise 16.
Prove that the equations
x=-tI°C(?i ), y=ryl°S(ii), I>0, 0 E R, define a Cl-diffeomorphism for any a yl- 0. For which values of a and -y is it symplectic?. 0
A mechanical interpretation. The variables (9, I) are the so-called "actionangle" variables for the oscillator (6.3). In this case the action is, up to a constant, the energy. In fact, using again (6.4),
E= 1 2+2(x+)2 +2 (x-)2=
ry22I
=S2I.
The angle 0 can be interpreted by the formula 0 = T r(x, y), where r is the time employed by a particle to travel from the point (ryJ1/2, 0) to (x, y). Of course this motion follows the law (6.3).
The coordinates that we have constructed are important because they reduce (6.3) to its simplest possible form. To change variables in this equation we use that T is symplectic and so the hamiltonian structure is preserved. The equation (6.3) is equivalent to
x = Hy, y = -He, H(x, y) = 1 y2 + 2 (x+)2 + . (X-)2. 2 2 In the new variables,
0=h1=I2, I=-he=0, h(0,I)=H(T(9,I))=Ill. Let us now consider the nonautonomous equation
i + ax+ - bx- = f (t)
with f E C(T1). It is reasonable to expect that the previous change o variables will simplify it. The hamiltonian
Twist Mappings, Invariant Curves and Periodic Differential Equations
H(t, x, y) = 1 y2 + 2
(x+)2
(X-)2
+
103
- xf (t)
2
is transformed into h(t, 0, I)
= ci - 7I112C(
) f (t)
and we are lead to the system e-
si - 211/2C(c)f(t),
1 = I112S(St)f (t).
(6.6)
It is convenient to notice that this system is not completely equivalent to (6.5). The reason is that T introduces a singularity at the origin and so some of the solutions of (6.6), (9(t), 1(t)), have a maximal interval of definition smaller than (-oo, +oo). They correspond to the solutions x(t) of (6.5) passing through x = i = 0 at some time t. Finally we define p = 7 and (6.6) becomes
ci - -!C( )f (t), p = 2p
s(ii)f(t).
2
(6.7)
This system is not hamiltonian. This is not surprising because (9,1) (9, p) is not symplectic.
Let P be the Poincare map associated to (6.7). It is easy to prove that there is a disk such that P is well defined outside this disk. Let us assume for
a moment that we could prove the existence of a family of Jordan curves that were invariant under P and surround infinity. They would produce invariant cylinders in the space (t; x, ±) and the proof of the Theorem would be essentially complete. In view of this optimistic argument one could try
to apply the Twist Theorem to P. However we shall not follow this idea, due to the lack of regularity of the equation. Since (6.7) is only C' in 9 we cannot guarantee that P is of class C4 as required in the Twist Theorem.
To overcome this difficulty we notice that if f is smooth, then (6.7) is smooth with respect to t and p. This fact will motivate us to interchange the roles of 9 and t. The new independent variable will be 9 while the new unknowns will be t = t(9) and p = p(9). In this way we shall obtain a new (and smooth) Poincare map. This trick was employed by M. Levi in [11] to prove boundedness in a superlinear oscillator. The idea of applying it to the asymmetric oscillator is due to Liu Bin (see [131).
Step 2. The successor mapping Consider the system TO =
F(9, t, p), ae = G(9, t, p),
(6.8)
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with
F(9,t,P)
=
G(9, t,P)
=
-
11)f(t)]-1
2p
fjS(j)f(t)[fZ- 2PC()f(t)]-1
Let p,, > 0 be a positive number such that SZ
-
2pL11C11-11fII. > 0.
The functions F and G are well defined for p > p* and, if f is of class C"(T1), they belong to Co,"(T1 x E*), where
E*={(t,p)ER2: p> p*}. These functions are also 27r-periodic with respect to t and so we can interpret (t, p) as a system of polar coordinates in the plane. Then we can consider that the equation (6.8) is defined in the exterior of the disk p < p* if S* is interpreted as the universal covering of
E*={(t,p)ET1x1R: p>p}. Let (t(9), p(9)) be a solution of (6.8) defined in a certain interval I = (90, 91] is positive and so and such that p(9) > p,, for all 9 in I. The derivative the function t is a diffeomorphism from I onto J = [to, t1], where t(9o) = to
and t(91) = t1. The inverse function will be denoted by 9 = 9(t). It maps J onto I. Let us define
x(t) = -yp(9(t))C(O(t) ), t E J. It is easy to verify that this function is a solution of the original equation (6.5). Of course this is not surprising in view of the way we constructed (6.8). There are some interesting aspects of this solution that we have constructed. The derivative can be expressed in the form
x(t) = ryp(9(t))S(O(t) ), t E J and the zeros of x correspond to the values of 9 such that C(e) = 0. These zeros are nondegenerate because S and C do not vanish simultaneously. The zeros of i correspond to the values of 9 such that S(A) = 0. In particular,
if x(t) reaches a local maximum at t2 E (to,t1), then 9(t2) E 2irZ and x(t2) = 'YP(t2)-
Let (t(9), p(9)) be the solution of (6.8) satisfying the initial conditions t(9o) = to, p(9o) = po. It is not hard to show the existence of a number p* > p* such that if p0 > p*, then (t(9), p(9)) is well defined in [9o, 9o + 2i]
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105
and remains in .6*. The Poincare map associated to (6.8) will be denoted by S. The previous property implies that S is well defined on the set
E* = {(t, p) : p > p*}
and satisfies S(E*) C E. The smoothness of S will not be a big problem because S is of class C" in E* if f belongs to C" (T' ).
Understanding S geometrically. Let us use the notation S : (to, po) (ti, pi). The mapping S can be defined directly from the original equation. We consider the solution of (6.5) satisfying x(to) = 7Po, ±(to) = 0.
If po is large enough the function x reaches a local maximum at this instan
Then tl > to is the next instant where x reaches a maximum and x(ti) = ypl. This observation justifies the name of successor mapping for S. See [18, 1, 7, 161 where variants of the successor mapping were employed in the study of second order scalar differential equations.
A strategy for the proof. To prove the Theorem we will find a family of invariant curves of S that surround infinity. To be precise, we shall look for a sequence of numbers {Rn} and a sequence of Jordan curves in E*, denoted by {rn}, satisfying the conditions
p*
rn is homotopic to p = constant in An,
s(rn) = rn. Lemma 6.2. Assume that f E C(T') and we can find sequences Rn and rn in the previous conditions. Then the solutions of (6.5) are bounded.
Proof. The previous assumptions imply that R,(rn) C R,(rn+l) and R2 = Un>o R, (r"). The invariant curves of S produce invariant cylinders for (6.8) in the space (6; t, p). The solutions lying between two of these cylinders are bounded. From these facts and from the equation itself we can deduce the existence of a new number p** > p*, such that if (t.(9), p(8)) is a solution of (6.8) with p(Oo) > p** for some Bo, then the solution is defined
in the whole line, bounded and such that p(8) > p*, V O E R. From this solution we can construct a bounded solution of (6.5). Now we prove the Lemma. Given a solution x(t) of (6.5) we distinguish two cases. If P(t) :=
?j-1/2
{2x(t)2 +
,a-2x+(t)2
+ 2x_(1)2}
106
R. Ortega
remains always below the number p**, then the solution is obviously bound-
ed. On the other hand, if for some t the function p(t) is above p**, then x will correspond to a solution of (6.8) living between two invariant cylinders and so p will be uniformly bounded in R. Exercise 17. Prove that, in the conditions of the previous lemma, the solutions of (6.5) are equi-bounded; that is, given a constant 7* > 0 we can find another constant 7* > 'y* such that sup Ix(t)I + It(t)I < 7*,
inf Ix(t)I + Ix(t)I < 7*
tER
tER
for any solution x(t). Exercise 18.
Consider the equation (6.5) where f is an arbitrary function
in C(T1). Let x(t) be a solution and let M. denote the set of instants to E R where x reaches a local maximum. Prove that x(t) is bounded if and only if one of the conditions below holds: (i) sup{x(to) : to E M=} < co, (ii)
M. = 0.
[Compare with proposition 4.2 in [16]]. Exercise 19. Let r C E* be a Jordan curve such that 8(r) = r and such that the following diagram is commutative,
r
S
r T
T
T1
-->
T1
R2ra
where a
Q. We want to produce a family of quasi-periodic solutions of
(6.5).
Let r = r(s), p = p(s) be a lift of '0. We can assume T(s + 21r) = T(s) + 27r, p(s + 27r) = p(s).
(If T(s + 21r) = T(s) - 27r we replace a by -a). Let x(t; T, p) denote the solution of (6.5) satisfying x(r) = 7P, i(r) = 0. Prove (i) x(t; T(s), p(s)) = x(t; T(s + 21ra), p(s + 21ra)), (u) x(t; T(s + 2rr), p(s + 2rr)) = x(t - 2ir; T (s), p(s)), (iii) xc(t) = x(t;T(ac),p(ac))) is a family of quasi-periodic solutions in .M (al).
Step 3. Applying the Twist Theorem Intersection property
Twist Mappings, Invariant Curves and Periodic Differential Equations
107
Lemma 6.3. Let r be a Jordan curve in E* that is homotopic to p = p*. Then
s(r) n r # 0. Proof. In R3 we consider a system of cylindrical coordinates defined by
tET', p>O, OER, where the associated cartesian coordinates are X = p cos t, Y = p sin t,
Z=9.
Let A be the vector field in R3 described, in cylindrical coordinates, by the equations
At=j, Ap =
Ae = SZ - 2pC(O)f(t)
Using the standard formula for cylindrical coordinates,
divA =
1
a
pap
(pAp) +
1 aAt + 9Ao pat a9
we conclude that divA = 0 for p 34 0. For p = 0 the field has a singularity but it is rather weak. Around the singularity A satisfies
Ax = 0(1), Ay = 0(1), Az = 0(1).
(6.9)
Define r1 = S(r). Since S is a topological mapping, r1 is a Jordan curve in E* that is homotopic to p = p*. Let us consider the flow in R3 given by t = At, = Ap, 0 = A9. If we start with the curve IF x {0}, lying in the plane Z = 0, and allow the flow to evolve up to Z = 2ir, then we arrive at r1 x {27r} via a smooth cylinder. The domain enclosed by this cylinder in {0 < Z < 27r} will be denoted by D. The boundary of D is composed by the cylinder itself and the two faces R;(r) x {0} and R2(r1) x {27r}. The outward normal vector to aD satisfies A n = 0 on the cylinder. Also, n = (0,0,-l) (resp. n = (0, 0,1)] at R;(r) x {o} [resp. R=(r1) x {21r}]. Given a small e > 0 we apply the Divergence Theorem to the vector field A on the domain
DE={(X,Y,Z)ED: X2+Y2>e, 0
0=JD divA=J
aD
J;R (r ,) x {2A}RAe-f , (r) x {0} AB.
(6.10)
If we assume for a moment that the conclusion of the Lemma does not hold
and r and r1 do not intersect, since they are both homotopic to p = p*
108
R. Ortega
in E, either R=(I'1) C R,(17) or R. (I') C R;(I'1). Assume for example that the first inclusion holds. We reach a contradiction with the identity (6.10) because
Ae-f fR(r)x(o}
L.(r)_R(rI){_f(t)}144(4t > 0.
Ae
2p
R;(r1)x{2n}
Inversion and change of scale. To apply the Twist Theorem we need a fixed annulus and, for this reason, we perform the change of variables
r = p,
r E [0.5, 2.5],
where 6 > 0 is a parameter. As b -' 0 the annulus r E [1, 2] is mapped onto the annulus around infinity p E [ -27 , J. In this way we introduce a small parameter in the system (6.8). It becomes f 114
_ [I - 5
f(t)C(f)]-1
,
i
dr _ -b 2n f (t) S(±) )[cl - b f (t)C(0 )]
(6.11)
The Poincare map M6 is well defined in the annulus r E [1, 2] for small b. Moreover, the previous Lemma implies that M6 satisfies the intersection property. In order to apply the Twist Theorem we also need some estimates. To obtain these estimates we are going to consider that (6.11) is a differential equation depending on a parameter and we shall apply the following consequence of Peano's Theorem. Differentiability with respect to parameters. For the moment we shall consider a general differential equation depending on one parameter of the type dz dB
= F(9, z, b),
(6.12)
where the function
F : [0,21r] x D x [0,A] - RN, (0, z, b)
F(9, z, b)
is of class for some v > 0. Here V is an open and connected subset of RN and A > 0. The solution of (6.12) satisfying z(0) = zo will be denoted by z(9; zo, b). The general theory of differential equations says that z is of class CO-"+1,"+1 in its three arguments whenever it is defined. The following result is a consequence of this fact. C°,"+1,"+1
Proposition 6.4. Let K be a compact subset of V such that for every zo E K and b E [0, 0] the solution z(9; zo, b) is well defined in [0, 2ir]. Then, for each (9; zo, b) E [0, 2ir] x K x [0, A], the expansion below holds, z(9; zo, b) = z(9; zo, 0) + as (9; zo, 0)b + R(9; zo, 5)b
Twist Mappings, Invariant Curves and Periodic Differential Equations
109
where the remainder R satisfies IIR(9;
6 - 0,
0,
6)IIC- (K)
uniformly in 0 E [0, 27r].
This result is a consequence of the regularity of the solution together with the following Lemma.
Lemma 6.5. Let cp be a function in
C°'"+1,"+1([0, 2a]
x K x [0, A]). Then
w(9; z, 6) = w(0; z, 0) + as (B; z, 0)6 + R(9; z, 6)6
with 1Z satisfying the same conditions as R in Proposition 6.4.
Proof. It is a consequence of the identity 1
W(O; z, 6) = 0(O; z, o)+ as (9; z, 0)6+6
Exercise 20.
{
J
T6
(9; z, as)-
ab
Let z(0; zo, 6, e) be the solution of dz dB= F(9, z, b, E),
(O; z, 0)}ds.
z(0) = zo,
where
F : (0,27r] x D x 10,A] x [-1,1]
RN, (0, z, 6, e) H F(0, z, 6, e)
is of class Co,'+1,"+1,o for some v _> 0. Assuming that z(9; zo, 6, E) is well defined in [0, 21r] if zo E K, b E [0, 0], e E [-1,1], then z(9; zo, 6,
E) =
z(O; zo, 0, e)
8z
+
(9; zo, 0, E)6 + R(9; zo, 6, E)5
C7(S
where the remainder R satisfies
IIR(9; ,b,E)IIc-(K) -,0, 5-' 0, uniformly in 0 E [0, 27r], f E [-1, 1].
The estimates. We are going to apply the previous Proposition to the system (6.11). For b = 0 this system becomes = I, = 0. Then t(9;to,ro,0) = to + ?j
,
r(0;to,ro,0) = ro.
The derivatives with respect to the parameter will be denoted by (9) _ (9; to, ro, 0) and 17(9) _
f
(9; to, ro, 0). They satisfy
= nrrof (to + ff)C(n ), 0) = 0,
1 o = - n'srof (to +
17(o)
n)S(0 ),
= 0.
110
R. Ortega
In consequence, e
(O) = 202 ro f A to + e)C(E) )de, n(0) = -
r02
2
1
f(to + e )S(e )de.
If we apply Proposition 6.4 to the system (6.11) with A sufficiently small
and K = {(t, r) E R2 : 0 < t < 27r, 1 < r < 2}, then we are lead to the following result.
Proposition 6.6. Assume that f E C"+1(T1). Then the Poincar6 map M5 of (6.11) satisfies the expansion t1 = to + T + 5e(27r) + o(8),
irl = ro + 8i(27r) + o(8),
and the remainders o(8) are understood in the C" sense. Remark. This proposition is also true if one only assumes f E Cn (T'). See Proposition 6.1 in [17]. It is possible to prove this result using a refinement of Proposition 6.4.
Proof of Theorem 6.1. Let us now assume that f (t) = 1 +Ep(t) with p E C5(T') and IEI < 1. The numbers p* and p* employed in the construction of the Successor mapping can be chosen independent of E. Let Mb be the Poincar6 map of (6.11) when f = 1 + Ep. We can also find another number independent of e, A > 0, such that if IEI < 1 and 8 E [0, A], then Ma is well defined in the annulus A : 1 < r < 2. We want to prove the existence of E* > 0 such that if IEI < E*, then Ma has invariant curves in A for small 8. Once we have proved this, the Theorem will follow from Lemma 6.2. To find the invariant curves of Ma we are going to apply the Small Twist Theorem. The intersection property is an easy consequence of Lemma 6.3. To obtain the estimates we can employ Exercise 20 to obtain a variant of
Proposition 6.6 that is valid for f = 1 + Ep. More precisely, Mb has the expansion
ti = to + T + 8a(ro) + 8{e
rl = ro + 6{-E
212 2
2
f
2 Jo p(to +
r 21r
2J
p(to +
with
a(ro) = S
9
(
)S(
)C(
9
)d9 + Rl (to, ro; 6, E) }, -D
)d9 + R2(to, ro; 8, f)}
T
C(t)dt)ro
and
C)IIca(K) - 0, 8 -+ 0, i = 1,2, IIRi(', uniformly with respect to f E [-1, 1]. Exercise 15 implies that a has a nonvanishing derivative and therefore the Small Twist Theorem is applicable whenever 6 E [0, A] and c is small enough. 0
Twist Mappings, Invariant Curves and Periodic Differential Equations
111
Remark. More results on the boundedness problem for asymmetric oscillators can be found in [16, 3, 13]. Other nonlinear oscillators with linear growth have been studied in 120, 2, 9, 17, 12]. The paper of Markus Kunze in these lectures notes [8] also deals with semilinear oscillators and contains a new application of Proposition 6.4.
Acknowledgements. Liu Bin has kindly allowed me to use his unpublished paper [13] as material for the course. Aureliano Robles has read and
corrected the proofs of this manuscript. Juan Campos and Jose Miguel Alonso have helped me to draw the invariant cylinder on the computer. My thanks also to the people who attended the course; their comments and questions have been useful to clarify and improve the preliminary version of these notes.
REFERENCES [1] V.M. Alekseev, Quasirandom dynamical systems II. One-dimensional
nonlinear oscillations in a field with periodic perturbation, Math. USSR Sb. 6 (1968), 505-560.
[2) J.M. Alonso and R. Ortega, Unbounded solutions of semilinear equations at resonance, Nonlinearity 9 (1996), 1099-1111.
[3] J.M. Alonso and R. Ortega, Roots of unity and unbounded motions of an asymmetric oscillator, J. Dif. Equations 143 (1998), 201-220. [4) M.R. Herman, Sur les courbes invariantes par les diffeomorphismes de l'anneau I, Asterisque 103-104 (1983). [5] M.R. Herman, Sur les courbes invariantes par les diffeomorphismes de l'anneau II, Asterisque 144 (1986).
[6] M.R. Herman, Demonstration du theoreme des courbes invariantes par les difeomorphismes de 1'anneau, unpublished manuscript.
[7] H. Jacobowitz, Periodic solutions of x" + f (t, x) = 0 via the PoincareBirkhoff theorem, J. Differential Equations 20 (1976), 37-52.
[8] M. Kunze, Remarks on boundedness of semilinear oscillators, in this volume.
[9] M. Kunze, T. Kiipper and J. You, On the application of KAM theory to discontinuous dynamical systems, J. Dif. Equations 139 (1997), 1-21.
[10] A.C. Lazer and J.P. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Review 32 (1990), 537-578.
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[11] M. Levi, Quasiperiodic motions in superquadratic time-periodic potentials, Comm. Math. Phys. 143 (1991), 43-83. [12] B. Liu, Boundedness of solutions for semilinear Duffing equations, J. Differential Equations 145 (1998), 119-144. [13] B. Liu, Boundedness in asymmetric oscillations, J. Math. Anal. Apps. 231 (1999), 355-373.
[14] G.R. Morris, A case of boundedness in Littlewood's problem on oscillatory differential equations, Bull. Austral. Math. Soc. 14 (1976), 71-93.
[15] J.K. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Gottingen Math. Phys. 11 (1962), 1-20.
(16] R. Ortega, Asymmetric oscillators and twist mappings, J. London Math. Soc. 53 (1996), 325-342. [17] R. Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theorem, Proc. London Math. Soc. 79 (1999), 381-413. [18] E. Picard, Sur 1'application des methodes d'approximations successives
a 1'etude de certaines equations differentielles ordinaires, Jounal de Liouville 9 (1893), 217-271; reprinted in Oeuvres de Emile Picard, vol. 2, Editions du CNRS (Paris, 1979). [19] H. Riissmann, Kleine Nenner I: Uber invariante Kurven differenzierbarer Abbildungen eines Kreisringes, Nachr. Akad. Wiss. Gottingen Math. Phys. KI II (1970), 67-105. [20] G. Seifert, Resonance in an undamped second-order nonlinear equation with periodic forcing, Quart. Appl. Math. 48 (1990), 527-530.
(21] C.L. Siegel and J. Moser, Lectures on Celestial Mechanics, SpringerVerlag, Berlin, 1971.
Rafael Ortega Departamento de Matematica Aplicada Facultad de Ciencias, Universidad de Granada 18071 Granada Spain rortegac.ugr.es
Variational Inequalities, Bifurcation and Applications Klaus Schmitt 1
Introduction
Although the study of variational inequalities dates back to the origins of the calculus of variations, their systematic development began in the sixties with the work of Fichera ([20]) and Stampacchia ([40], [41]), which was motivated by problems in mechanics (obstacle problems in elasticity
- the Signorini problem) and potential theory (the study of the capacity of sets). After the fundamental work of Lions and Stampacchia ([32]), the study of variational inequalities intensified and became an important subject in nonlinear analysis. The rapid growth of the theory was made possible by the work of Brezis ([4], [5]), Browder ([8], [9]), Kinderlehrer and Stampacchia ([25]), Duvaut and Lions ([18]), .... It brought about important contributions to nonlinear analysis, calculus of variations, optimization theory, optimal control, and to many branches of mechanics, mathematical physics, and engineering. An elementary example of a variational inequality is the following simple deformation problem of a beam constrained by an obstacle. If we consider a homogeneous elastic beam occupying an interval [a, b], subject to a force f, and lying above an obstacle Vi, where 0 is a measurable function, the displacement of the beam is then constrained, and the set of admissible displacements is described by the convex set K = {v : v > t(i, a.e. on [a, b]}.
Using the principle of energy minimization, the deflection u of the beam must satisfy the minimization problem
uEK:E(u)<E(v),VvEK, where
E(v) =
1
(1.1)
jb (ti i)2 fb - fv
d enotes the potential energy. Using the fact that K is a convex set, we must have that
(1-t)u+tvEK,VvEK, VtE[0,1],
114
K. Schmitt
and, hence, the function
i(t) = E((1 - t)u + tv), must have a minimum at t = 0, and i'(0) > 0, i.e.,
i'(0) =Ju"(v-u)"-Jf(v-u)>0, b b VvE K, a
(1.2)
a
which can be viewed as the Euler-Lagrange inequality corresponding to (1.1). On the other hand, if ul and u2 both satisfy (1.2), then, after an elementary calculation, we find that
-
b
f
(U- u)2 > 0
or
u' = u'2, on [a, b], and, because ul and u2 both satisfy the boundary conditions, we conclude
that ul = u2, on [a, b].
Thus, the minimization problem (1.1) is equivalent to the variational inequality (1.2). In a similar vein, minimization problems on a more abstract level lead to variational inequalities. For example, if F is a real convex functional of class C', defined on a Banach space V, and K is a closed convex subset of V, then the minimization problem
uEK:F(u)
(1.3)
is equivalent to the variational inequality
u E K : (F'(u),v - u) > 0, Vv E K,
(1.4)
where stands for the duality pairing between V and its dual space V*; the argument for the equivalence may be found in ([18]) and is very similar to the one given above. Detailed presentations and surveys of the theory of variational inequal-
ities and their applications may be found in [25] and [30] (general theory and applications), [3] and [21] (applications to free boundary problems), [18] and [391 (applications of variational inequalities in physics and mechanics), [29] (bifurcation theory).
Variational Inequalities, Bifurcation and Applications
2 2.1
115
Minimization of functionals Notation and assumptions
Let E be a real reflexive Banach space and let E' be its topological dual. Let us assume that f : E -+ R U {oo} is a weakly lower semicontinuous functional which is coercive, i.e.,
asv -oo, and let K be a weakly closed set.
2.2
A minimization result
Theorem 2.1. Assume the above, then there exists u E K such that (2.1)
f(u) = m n f(v).
Proof. Since f coercive, f is bounded below on K. Hence a = inf f (v) > uEK -oo. Choose a minimizing sequence {vn} C K, i.e., f (vn) - a. Again, since f is coercive, we obtain {vn } is bounded. Further, since E is reflexive, we have that {vn} has a weakly convergent subsequence, say, after relabeling, vn - u. Since K is weakly closed, we find that u E K. Now f being weakly lower semicontinuous implies f (u) < lim inf f (vn), n-oo
and therefore f (u)
= vrninn f (v)
0
proving the theorem. 2.2.1
Consequences
IfK=Eand f EC1,then fl(u) = 0, and u is a critical point for f
.
If K is convex, then
(f '(U), v - u) > 0,
Vv E K,
i.e., u is a solution of a variational inequality.
116
K. Schmitt
2.2.2 On bilinear forms Let
a:ExE -R
be a continuous, symmetric, coercive bilinear form, i.e., Ia(u, v)I <- cl IIuII IIvII, a(u,u) > c21Iu1I2,
where cl and c2 are positive constants. Let b E E' and K be a weakly closed set. Let us consider the functional
f (u) = 2 a(u, u) - (b, u),
then f is coercive, weakly lower semicontinuous and C', hence 3u E K such that f (u) = min f (v). vEK
If K is convex, then
a(u, v - u) > (b, v - u),
Vv E K.
(2.3)
One immediately sees that problem (2.3) has a unique solution and that problems (2.3) and (2.1) are equivalent problems. Let T : E` - E be defined by Tb = u, where u is the unique solution of (2.3), then
IITb, - Tb21IE 5
Therefore, if
Ilbi -
(2.4)
F:E-4E`,
the variational inequality
a(u, v - u) > (F(u), v - u), Vv E K,
2.5)
is equivalent to the fixed point problem
u = TF(u).
(2.6)
2.2.3 Convex functionals
A functional
f:E
is strictly convex if for all u, v E K, u j4 v, 0 < t < 1, f(tu + (1 - t)v) < tf(u) + (1 - t)f(v).
(2.7)
Variational Inequalities, Bifurcation and Applications
117
It is convex if 0<-t<-l.
f(tu + (1 - t)v) < tf(u) + (1 - t)f(v),
(2.8)
If f is convex and C', then f (u) > f (v) + (f '(V), u - v),
Vu, V E K.
(2.9)
From this we deduce that if f is convex and Cl and v E K is such that
(f'(v), u - v) > 0,
Vu E K,
(2.10)
then
f (v) = min f (w).
(2.11)
And therefore if f is convex and C', then (2.10) and (2.11) are equivalent problems, which are uniquely solvable if f is strictly convex and if f is convex, the solution set is convex. 2.2.4
Cones
If K is a subspace of E, then (2.10) becomes (f '(v), u) = 0,
Vu E K,
(2.12)
i.e., f'(v) E Kl = {b E E* I (b, u) = 0 `du E K }. If K is a cone , i.e.,
u+vEK,tuEK,Vu,vEK,t>0, then (2.10) becomes
(f'(v), u) > 0, 1(f'(v),v) = 0.
du E K,
(2.13)
2.2.5 An obstacle problem Let fl be a bounded domain in RN and let E = L2(fl). Let 0 E E be given and let K = {u E E I u(x) > tp(x), a.e. in f2}. Then K is a closed, convex subset of E.
Let 'y E E and define f : E -+ R as f (u) =
1JuII2
2
- (y, u). Then there
exists a unique u E K such that f (u) = min f (v) and furthermore u solves vEK the variational inequality
(f'(u), v - u) > 0,
dv E K,
118
K. Schmitt
J(u - y)(v - u) > 0,
`dv E K,
(2.14)
and the latter must have a unique solution. The natural candidate for this solution is u = max(z[i, y), as one easily verifies by substituting into (2.14). 2.2.6 Another example i
Let E = L2(0,1), K = {u I f u = 1). Then K is closed and convex (hence 0
weakly closed). Let
i
f(u) = f u2
= IIuI!2,
0
then f is weakly lower semicontinuous, coercive and C1, hence there exists
a unique u E K such that f (u) = min f (v), which holds, if and only if, ((,) is the L2 inner product) (f'(u),v - u) > 0,
i.e.,
Vv E K,
(u,v-u)>0, VvEK,
or
(u, v) > (u, u), `iv E K, i.e.,
uv>
u2,
VvEK.
0
0
O n the o th er h an d
I ui < (f u2)j, 0
0
and hence
VvEK. 0
Clearly u = 1 solves the inequality.
2.2.7 Some references For additional and more detailed examples see [5], [11], [13], [18], [21], [25], [31].
Variational Inequalities, Bifurcation and Applications
119
Variational inequalities
3 Let
a:ExE-'R
be a continuous, coercive, bilinear form, b E E*, and K a closed convex set.
The problem
3.1
We pose the following problem: Find (prove the existence of) u E K such
that a(u,v - u) > (b,v - u), dv E K.
(3.1)
In case a is symmetric, this problem has been solved above, thus, what is of interest here is the case that a is not symmetric. The development in this section follows closely the development in [32) and [25]. 3.1.1
Uniqueness of the solution
Using properties of bilinear forms, one concludes that for all b E E', problem (3.1) has at most one solution and if bl, b2 E E' and solutions u1i u2 exist, then IIu1 - U211E <-
1 Ilbl -
where c2 is a coercivity constant of a (cf. (2.4). 3.1.2 Existence Since the existence of a solution has been proved if a is symmetric, we write
a = ae + a0, where a. (u, v) =
2
(a(u, v) + a(v, u)),
ao(u, v) = 2 (a(u, v)- a(v, u)),
then ae is a continuous, symmetric, coercive, bilinear form and ao is continuous and bilinear. Consider the family of problems ae(u,v - u) + tao(u,v - u) > (b,v - u),
dv E K,
0 < t < 1,
(3.2)
`dv E K,
0 < t < 1.
(3.3)
or equivalently ae(u,v - u) > (b,v - u) - tao(u,v - u),
120
K. Schmitt
For w E K consider
ae (u, v - u) > (b, v - u) - tao(w, v - u),
dv E K,
0,
t < 1.
(3.4)
Note that for fixed w E K,
bw = b - ta,,(w, .) E E', hence there
exists a unique u = Tw solving (3.4) and IITwI - Tw2IIE 5
2 Ilbw, - b,,,, 11
On the other hand Ilbw, - b,,,2 IIE = sup tl ao(wi, u) - a,,(w2, u)l 5 tc1llw1- w2IIE, IIuII=1
and hence
IITwI-Tw2IIE:5
tzlllwl-w2IIE,
t'
< 1. We hence have and T : K -+ K is a contraction mapping provided a unique solution of (3.3) as long as t < a. Let at, = ae + toad, where to = -, then at, is coercive, continuous and bilinear, and the problem
ato (u, v - u) > (d, v - u),
t1v E K
(3.5)
has a unique solution for all d E E*. Note that the coercivity and continuity constants of ato may be chosen the same as those of ao. Hence by the uniqueness result in 3.1.1 above, we have for d1, d2 E E' and U1, u2 solutions of (3.5) that Ilul-u2I1E< 2Ild1-d2IIE
For fixed w E K consider
ato (u, v - u) > (d, v - u) - ra,, (w, v - u),
yv E K,
(3.6)
and apply earlier reasoning to conclude that (3.6) has a unique solution
TWEK,and, as long as,0
cC'
Ilw1 - w2IIE
Thus, we may apply the contraction mapping principle to get a unique solution of (3.4) for 0 t < 2to. Continuing this way we arrive at the unique solvability for t = 1.
Variational Inequalities, Bifurcation and Applications
121
3.1.3 A second order boundary value problem
Let 11 be a bounded open set in RN, let {atj(x)} 1,3 =1 C L°°(fl) be such
that
E
co
E = Ho (St) with IIuIIHo(0) _ f IDuI2, and let a(u,v) be given by a(u,v) _
J a,j(x)O,u81v = /VvT{azj)Vu.
Then Ia(u,v)I < c1IIVvIIL2IIVuIIL-,
cl = maxllaijIIL'O(n)
and
Ia(u, u)I > collVujI2L2.
For b E L2 (g) C Ho(0)` we obtain the existence of a unique u E HH(1) such that a(u, v - u) > b(v - u), Vv E Ha (11),
J
hence
a(u, v) =
by E HO '(P),
Jbv,
or in a distributional sense the partial differential equation
- 1: Oj (atj atu) = b t.j
will have a unique solution u E HH(1l) (see also [71, [19], [241, [311).
3.1.4
A unilateral problem
Let bo, co E L°O(0,1), h E L2(0,1), t1'o, 01 E R. Consider the unilateral problem
-u" + bou'+ cou = h, (U)
u(0) > rGo, u'(0) < 0, u(1) > 01, u'(1) > 0, (u(0) - 1Go)u'(0) = 0 = (u(1) - V1)u'(1). 1
1
We let E = H1(0,1), Ilull2 = f u2 + f (u')2, 0
0
K={uEEI u(0)>iIio,u(1)>tio}.
122
K. Schmitt
On E x E define the continuous bilinear form it
it
a(u, v) =
J0
u'v' +
J bou'v + J couv. 0
0
Then (U) is equivalent to 1
a(u, v - u) > J h(v - u), VV E K.
(3.7)
0
If u solves (U), then u" E L2(0,1), hence the boundary conditions make sense. Thus multiplying the differential equation in (U) by v - u and integrating by parts one obtains 1
1
1
1
f u'(u - v)' + f bou'(v - u) + f cou(v - u) = f h(v - u) + (v - u)u'101 0 0 0 0
VvEK, or
a(u,v - u) -
f
h(v - u) = (v(1) - u(1))u'(1) - (v(0) - u(0))u'(0).
0
If u(O) > 1Go and u(1) > 01, then the right-hand side equals 0. If, on the other hand, u(0) = 1Go, then u'(0) < 0 and v(0) > u(0) or u(1) = Vi1i then
u'(1)>0andv(1)>u(1).Thus, inanycase, 1r
a(u, v - u) >
J0
h(v - u), VV E K.
Conversely, if u E H'(0, 1) is a solution of (3.6), we may choose v = u + 0 E Co (0, 1), and obtain a(u, ¢) > J hO,
V0 E Co (0, 1),
and hence the differential equation (U) holds in a weak sense implying that
u" E L2(0,1), and thus u' has a trace. Since U E K, we automatically have u(0) > i'o, u(1) > ik1. Again, since the differential equation is satisfied, we may multiply it by (v - u) and integrate by parts and obtain that (v(1) - u(1))u'(1) - (v(0) - u(0))u'(0) > 0,
Variational Inequalities, Bifurcation and Applications
123
Vv E K. Choosing v(O) = u(O) and v(1) > u(1), we obtain u'(1) > 0 and similarly we obtain u'(0) < 0. Further, choosing v(1) = 01, v(0) = tlo, we obtain 0 > (u(0) - Oo)u'(0) > (u(1) - Ol)u'(1) > 0,
and hence
(u(0) - tho)u'(0) = (u(1) - ?Pi)u'(1) = 0. The partial differential equations analogue of this problem is N
N
- r'a; (aiic%u) + E a;8,u + aou = h,
in St,
i=1
+,i=1
subject to the unilateral constraints
0,
E a=?n38,u,
on 852,
=,i
where v = (n I, n2i , nN) is the unit outward normal vector field to 52. For further material on unilateral problems see [16], [17], [28], [27], [29].
3.2 3.2.1
Quasilinear inequalities Set-up
We now consider the case of more general inequalities of the form (A(u) - f,v - u) + j(v) - j(u) > 0,
here
WEE,
(3.8)
A:E-+E*, fEE`
and
j: is a convex lower semicontinuous functional such that
D(j)={vEEI j(v)
j(0) = 0, j : E - [0, oo]. Concerning A we shall assume:
124
K. Schmitt
A is monotone, i.e.,
(Au - Av, v - u) > 0 dv, u E E. A is continuous on finite dimensional subspaces, i.e., for all finite dimensional subspaces M of E, the mapping u
,-
M -->
(Au, x), R
is continuous Vx E E.
A is a bounded mapping.
3uo E D(j) such that lim
(A(u), u - uo) + j(u)
IIUII-+ao
= 00
Hull
(a coercivity assumption). We have the following theorem (see [6], [8], [35]):
Theorem 3.1. V f , Yj (3.8) has a solution. If A is strictly monotone, i.e., (A(u) - A(v), u - v) > 0,
v# u,
only one solution exists. This will allow us to define
TA,i:E'-'E as
TAJf = u, where u is the unique solution of (3.8), and, if we are given F: E -' E`, to find the solutions of the inequality
(A(u) - F(u), v - u) + j(v) - j(u) > 0,
Vv E E,
(3.9)
is equivalent to finding solutions to
u = TA,,F(u).
(3.10)
We sketch a proof of the existence of a solution to the equation (3.8) in the
case j = IK, where Ik(v)
l
0o,
v ¢ K,
0, VEK
is the indicator functional of a closed, convex set K in E.
Variational Inequalities, Bifurcation and Applications
125
Let KR denote the set KR = {u E K I IIulI < R}, and consider the problems
(A(u), v - u) > 0,
Vv E K,
(3.11)
Vv E KR.
(3.12)
and
(A(u), v - u) > 0, We have the following lemma:
Lemma 3.2. A necessary and sufficient condition that (3.11) have a solution is that there exists R > 0 such that a solution uR of (3.12) exists with IIuRII < R.
Proof. If a solution of (3.11) exists with Dull < R, then u solves (3.12). Conversely, if UR solves (3.12) with 11UR1I < R, then given y E K,
w = uR + E(y - uR) E KR, for e > 0, small. Consequently
UREKRCK, and
0 < (A(uR), w - uR) = E(A(uR), Y - UR),
Vy E K.
0
Hence, since e > 0, the lemma is proved. 3.2.2 Finite dimensional considerations
We next consider the case that the problem is finite dimensional and K is a bounded convex set. Then the problem
(A(u), v - u) > 0,
Vv E K,
(3.13)
is equivalent to the problem
(-A(u), v - u) < 0, Vv E K, or
(u, v - u) > (u - A(u), v - u), Vv E K, and hence by earlier considerations, it is equivalent to the fixed point problem (here we have identified pairing with the inner product)
u = T(I - A)(u),
126
K. Schmitt
where T is the solution operator defined by the bilinear form a(u, v) = (u, v) (the nearest point projection, cf. Section 2.2.2). On the other hand, since
T(1-A): K -K, it has a fixed point. If K is not bounded, we consider the problem
(A(u), v - u) > 0,
` V E KR.
This problem has a solution UR E KR. We now let R -
(3.14) oo and use the
coercivity condition, which reads that for some uo E K, lim IiisO-+oo
(A (u), u - uo) - 00.
(3.15)
Hull
Let us then consider IIuII >> 1, u E K. We then conclude from (3.15) that
(A(u), u - uo) > 0. Choosing R so that uo E KR, we have
(A(u), uo - u) < 0 for IIuII >> R, hence IIuRII < R and we obtain the result. To complete the proof we shall need the following result (cf. [33]).
Theorem 3.3. (Minty) Let A: E - E` be monotone, then u E K satisfies (A (u), v - u) > 0,
Vv E K,
(A (v), v - u) > 0,
Vv E K.
if and only if
Proof. If 0 < (A(v) - A(u), v - u) = (A(v), v - u) - (A(u), v - u), then
0 < (A (u), v - u) < (A(v), v - u). Conversely, let w E K and set v = u + t(w - u), 0 < t < 1. Then
0 < (A(u + t(w - u)), v - u), and thus
0<(A(u),w-u), VwEK.
O
Variational Inequalities, Bifurcation and Applications
127
Let E be infinite dimensional and K be bounded. Then for v E K,
S(v) = {u E K (A(v), v - u) > 0} is weakly closed, hence compact. Therefore n S(v) vEK
is a weakly compact set. That this set is nonempty will follow from the finite intersection property. Thus let {vl,..., vm} C K. Then we claim
S(v,)nS(V2)n...nS(vm) o0. Let M be the finite dimensional subspace spanned by {vl, ..., vm} and let KM = K n M. Then, as before, there exists um E KM such that
(A(uM), v - um) ? 0,
Vv E KMT,
and by Theorem 3.3,
(A(v), v - um) > 0,
dv E KM,
in particular
(A(v,),v1-u1)>0, Vi=1,...,m, so
uM E S(V,),
Vi=1,...,m.
Hence, there exists u E S(v) such that
(A(v), v - u) > 0,
Vv E KM
and using Theorem 3.3 once more, we obtain the result. If K is unbounded, we employ the coercivity condition imposed on A and argue as before. 3.2.3 Fixed points for non-expansive operators
Let E be a Hilbert space and let K be a bounded, closed, convex subset of E.
A mapping
F:K -K
is called non-expansive (see [101), whenever
IIFu - Fv11 < flu - vfl, `du,v E K. Using the above result we establish the following theorem.
K. Schmitt
128
Theorem 3.4. Let E be a Hilbert space, K a closed, bounded, convex subset and
F:K-'K
be a non-expansive mapping. Then the set Fix F = {u E K I F(u) = u} is a nonempty, closed, and convex subset of K.
Proof. The following calculation shows that the mapping I - F is monotone:
(u - v - F(u) + F(v), u - v) = (u - v, u - v) - (F(u) + F(v),u - v) = Ilu - vII2 - IIF(u) - F(v)IIIIu - vII > 0.
Let P be the projection operator associated with K, then
I - FP is monotone also, and for any uo, lim
(u - FP(u), u - uo)
OUR-
--oo.
Hull
Using the results in the previous section with Au = u - FP(u), one concludes that there exists u E K such that
(u - FP(u), v - u) > 0,
dv E K,
and hence
(u - F(u), v - u) > 0, Vv E K. Letting v = F(u), we have
(u - F(u), F(u) - u) > 0, or
u = F(u). That Fix F is closed and convex follows easily.
O
Continuity of the solution operator
3.2.4
We return to inequality (3.8) and impose the following additional conditions
on A: A is strictly monotone.
3c > 0, 3p > 1 such that (Au, u) >- cIIull".
Variational Inequalities, Bifurcation and Applications
129
e A belongs to class (S), i.e., V{vn} E E such that vn -s v and lim(A(vn), vn - v) = 0, it follows that vn -4 V.
Define the solution operator
P : E` -- E by
Pf=u
where u is the unique solution of
(A(u) - f, v - u) + j(v) - j(u) > 0,
WEE.
(3.16)
We have the following theorem (see [29]):
Theorem 3.5. P is a continuous operator. Proof. Thus, let {fn} be such that
fn -+f in E` and let un = P f, Then it follows from (3.16) and the properties of A that Cillun11P 5 (fn,un)
and hence (since p > 1) that the sequence {un} is a bounded sequence. It therefore has a weakly convergent subsequence, say, after relabeling, un
A straightforward calculation shows (using Minty's theorem and the fact that solutions are unique) that w = u must solve (3.16). I.e., all such subsequences must have the same limit and thus the whole sequence converges weakly to u. Again, using the form of (3.16), we obtain
(f - fn, u - un) > (A(un) - A(u), un - U) and therefore by the monotonicity of A that
lim(A(un) - A(u), un - u) = 0. And therefore
lim(A(un), un - u) = 0. Since A belongs to class (S) we deduce that
un "4u, proving the continuity.
0
130
K. Schmitt
3.2.5
On the p-Laplacian
Let 11 be a bounded open set, with smooth boundary 81t, E = W"(1), p E (1, oo), E' = W-1,9(fl), where the dual space is given by (-1)1aJOa9a 9. E L9(11)},
W-1'Q(fZ) = If I f = lal<1
and P + v = 1. The pairing (f, u) is given by
(f, u) = E f9a8U. Ial51 n
Let A : E - E* be defined by
r (A(u), v) =
J n
IVulp-2Vu Vv.
Then for p > 2, N > 1, (Au - Av, v - u) > cllu - vllp Whereas for N > 1 and all p E (1, oo),
(Au - Av,v - u) > c(Ilullp-l- IIvIIp-1)(IIuli - IIvII) These calculations show that A is a strictly monotone operator. We may therefore conclude that for all f E W-1"9(0) (e.g., f E L9(SZ) ), 3!u E E such that -div(IVulp-2Vu) = f.
On the other hand, if vn - v and lim(Avn, vn - v) = 0, then (Avn - Av,vn - v) > c(Ilvnllp-l- IIvIIp-1)(IIvniI - IIvII)
implies that (note p > 1) Ilvnll
Ilvll'
and since W0" (1t) is a uniformly convex space,
vn -'V
Ilvnll -IIvII
implies vn -4 Vi
and therefore A belongs to class (S) which has as a consequence that the solution operator is a continuous mapping.
Variational Inequalities, Bifurcation and Applications
4 4.1
131
Bifurcation for variational inequalities The bifurcation problem
Let us now assume that we have the following situation. The mapping
j: E-+RU{oo} is such that j(O) = 0 and j > 0. Further we have a mapping
which is completely continuous and satisfies F(1i, 0) = 0
and A is as before with A(0) = 0. Let us consider the variational inequality
(A(u) - F(a, u), v - u) + j(v) - j(u) > 0,
VEE
(4.1)
(see (3.8)). Then (4.1) is equivalent to the problem
u = TA,,F(,\, u),
(4.2)
where the operator TAj is the solution operator defined earlier. We henceforth drop the subscript A, j and consider the fixed point problem
u = TF(A, u).
(4.3)
Note that this fixed point equation involves a completely continuous op-
erator TF, since F is completely continuous (by assumption) and T is continuous. On the other hand, this operator, in general, is not smooth. Thus in order to obtain bifurcation results for (4.3), hence for (4.1), results using smoothness of operators cannot be used. We shall show below that in many situations, however, the global bifurcation theorem of Krasnosel'skii-Rabinowitz (cf. [12], [14], [26], [29], [38]) may be applied..
4.2
The Krasnosel'skii-Rabinowitz theorem
Let us write (4.3) as u = Q(A, u).
(4.4)
A point (Ao, 0) is called a bifurcation point for (4.4), if every neighborhood U of (Ao, 0) in R x E, contains a solution (A, u) of (4.4) with u 34 0. We note that if (a, 0), a E R, is not a bifurcation point of (4.4), then, if a
132
K. Schmitt
varies over a compact interval, containing no bifurcation values, then for 0 < r << 1, the Leray-Schauder degree d(1- Q(a, ), B,.(0), 0)
is defined (here B,.(0) = {u E E I Dull < r}) and constant. The global bifurcation result, applicable in our context, is the following:
Theorem 4.1. (Krasnosel'skii-Rabinowitz) Assume there exist a, b E R, a < b, such that (a, 0) and (b, 0) are not bifurcation points and that d(I - Q(a, ), B,.(0), 0) # d(I - Q(b, ), B,.(0), 0).
(4.5)
Let
S = {(A, u) I (A, u) solves (4.4), u 0 0) U [a, b] x {0}
and C be the maximal connected subset of S containing [a, b] x {0}. Then either i) C is unbounded in R x E or ii) C fl (R \ [a, b] x {0}) # 0.
4.3
Homogenization procedures
In order to apply the above theorem, real numbers a and b must be found
so that the above degree calculations may be performed and a change of degree will take place. This is usually accomplished by applying the homotopy invariance property of Leray-Schauder degree. Here we provide conditions, in terms of related variational inequalities, which will allow for such calculations and procedures. Assume that A, j, F are as above and also that
3a:E-'F', 3f:RxE-'E', satisfying the following conditions:
a satisfies the same hypotheses as A, and for all sequences {vn} C V,
Ian) CR+,an>0,vn-'v,an--,0+, an
1
A(an, vn) -' a(v),
in V'.
`d{an} C R+, an > 0, {An} C R, vn - v, an -, 0+, A. -' A, ap i F(An, anon) - f (A, v)
Variational Inequalities, Bifurcation and Applications
If vn
133
v, a,, - 0+, then J(v) < liminf v j(anvn),
and for all v E E, each sequence {an}, U. -+ 0+ , 3{vn} C E, such
that v, - v, and o 3(Onvn) - J(v). Under those assumptions one verifies the following:
f : R x E - E' is completely continuous. J is a proper, lower semicontinuous, convex functional, and J(0) = 0,
J(v) > 0.
a(av) = ap-la(v), o > 0, f(A,av) = ap-1f(A,v), a > 0,
J(av) = o J(v) , a > 0. Associated with these operators is the variational inequality
(a(u) - f(A,u),v - u) + J(v) - J(u) > 0,
WEE,
(4.6)
which has 0 as a solution and is equivalent to the fixed point equation u = TQ,j f (A, u).
(4.7)
It follows from the above properties that if u is a solution of (4.6), then Cu
is also for all a>0. Values of A for which (4.6) has a nontrivial solution will be called eigenvalues. We now have the following theorem:
Theorem 4.2. If (a,0) is a bifurcation point for (4.1), then (4.6) has for A = a, a nontrivial solution. If a, b E R, a < b, are not eigenvalues for (4.6)
and if
d(I - T.,jf (a, ) , Br(0), 0)
d(I - Ta,.1f (b, .), Br(0), 0),
then if S and C are as in Theorem 4.1, with Q(A, u) = TAj F(,\, u), we have the conclusion of this theorem. For the details concerning this result and its applications we refer to [29].
134
K. Schmitt
Applications
5
Let A, F, j, or, f, J be as in the previous section, we shall now give several examples.
Application 1 Assume K is a closed subset of E and j = IK, then since j(O) = 0, it 5.1
follows that 0 E K.
Let A be defined by a quadratic form a: E x E -, R which satisfies earlier hypotheses. Let F(A, u) = MAu+B(u) where A is compact linear and B is completely continuous, B(u) = o(IIulI) as h uHH -, 0. We consider the variational inequality
a(u, v - u) - (F(A, u), v - u) + IK (v) - IK (u) ? 0, WEE,
(5.1)
i.e.,
(A (u), v) = a(u, v).
In this case p = 2 and (a(u), v) = a(u, v), and f (J1, u) = M u.
Further J = IKo, where Ka = U tK, the so-called support cone of K. t>o
Hence the homogenized variational inequality is
a(u, v - u) - (AAu, v - u) + IK, (v) - IKo (u) > 0, WEE.
(5.2)
If we assume, for example, that Ko is a closed subspace El of E, then (5.2) becomes
a(u, v) - (,\Au, v) = 0,
VV E El,
(5.3)
which is equivalent to u = Ta,IKo (,\Au),
in El,
(5.4)
where Ta,IK0 is the solution operator, which is Lipschitz continuous. Note that Ta,IK0 is a linear operator, hence (5.4) becomes
u = .\T.,I,,. Au,
in El,
and Ta,IK0 A is a compact linear operator. Hence possible bifurcation points are to be found among the characteristic values of the compact linear operator TO,IK0 A: El -, El
and those of odd multiplicity are bifurcation values.
Variational Inequalities, Bifurcation and Applications
135
Application 2 Note that K,, = E, whenever 0 E int K. On the other hand, it may be 5.2
the case that Ko = E without 0 E int K, e.g., if i E Ho (0, 1), O(x) > 0, x E (0, 1) and
K={uEH1
'(x),0<x<1},
u(x)
then 0 is not interior to K, yet K,, = Ho (0, 1).
5.3
Application 3
Let E = Ho (Sl), Sl be a bounded open subset of RN with smooth boundary. Let S11 be an open subset of Sl with S21 C Sl with Ofl, smooth also.
Let 01, 02: Sl --+ R be continuous functions with 01,02 # 0 on Sl \ Sll and
01(x)=0=02(X),xESll, further 01(x) < 0 < 02(x), Vx E U.
Let K = {u 0i(x) < u(x) < 42(x) , Vx E ill, then K, may be identified with Ho '(!Q \ SZ1).
Hence if a : Ho x Ho is a continuous, coercive bilinear form and
(F(a, u), v) =
,
then the problem a(u, v) =
J
Vt E Ho (S2 \ S21),
Auv,
(5.5)
is the homogenization of the problem
a(u, v - u) - (F(.1, u), v - u) + IK (v) - IK (u) > 0,
Vv E Ho (S?), (5.6)
and therefore the real eigenvalues of odd multiplicity of (5.5) will yield bifurcation points of (5.6).
5.4
Application 4
In this application we give some functionals j and their homogenizations. 5.4.1
A global constraint
Let E = Ho(Il), a: E x E -+ R a continuous, coercive, bilinear form and Sl C R2. We let
j(u) =
f'I(u), n
136
K. Schmitt
where T : R [0, oo], is a proper, convex, lower semicontinuous functional with %1(0) = 0, W(t) > 0, t 34 0, W(ts) = tW(s), t > 0. In this case (assuming p = 2)
J = I{o} and hence no bifurcation will occur. To check that j is lower semicontinuous (it clearly is convex), let vn then, since (cf. [1])
v,
H2(fl) --+ C(S!)
is a compact embedding vn(x) -+ v(x), x E SZ, hence
%I(v(x)) < liminf 'P(vn(x)) by the lower semicontinuity of T. If v(x) 0, then f 'I (v(x))dx > 0, hence, using Fatou's lemma
0 < j(v) < rliminfT(vn) < liminf j(vn). If, on the other hand, v(x) J 0, the result is trivially true. Hence
j(anvn) = liminf n J 'P(vn(x))dx
llminf n
if v = 0,
0,
oo, ifv0. 5.4.2 A boundary constraint
The set-up is similar to the above, but E = H2(Sl) n HO (11) and
j(u) =
f
W(anu)
an
In this case we may compute J = IHo (n)'
The lower semicontinuity follows from the compactness of the embedding
H2(S2) - L2(O1), u
.-+
f3nu
(cf. [1]) and Fatou's lemma.
Note that
j(v) _
0,
v E Ho (S?),
> 0, v ' H02 (!Q), and an easy calculation shows then that ( as vn v, an -+ 0+), lim
1
f
i(anvn) = 1
0O0,
,
V E HO2
v V H02(0).
Variational Inequalities, Bifurcation and Applications
5.5
137
Application 5
With p > 1, consider the variational inequality
fQ IVuIP-'VU V(v - u) - nf [AIuIP-2u
+ g(x, u, -\))(v - u) + j(v) - j(u) > 0,
VV E E
where
E={uEW"P(11)Iv=0,onr} and r is a relatively open subset of 8Sl with positive measure. The norm of E is given by Hull = IIuHIw.", =
if luiP+ f lVulP]P n
0
Assume that
g:0xRxR-+R
is a mapping which is such that g(x, u, A) =
o(IuIP-1)
as u - 0 uniformly in x, A and satisfies the growth conditions I g(x, u, A)l 5 c(A)[m(x) +
Let
MIuIP-1],
m E LI (fl).
j:E-RUfool
be given by
j(u) = f W(u), an\r where WY : R -+ [0, oo], is a proper, convex, lower semicontinuous functional, 41(0) = 0,11(t) > 0, t 0, W(ts) = t'(S), t > 0. Here
(Au,v) = f lVulP-2Vu Vv n
with IlAull < IIuIIP-1, U E E. Also p-1 P
II Aun - Aull <
f IlVurelP-2Vun
- lVulJ-2Vul
,
138
K. Schmitt
hence, if un - u in E, then IVunlp-20un
-' IVulp-2Vu
in [LtT ]N,
and A is a continuous mapping from E to E. One next computes that &(u) = A(u) and
f
(f (u, A), V) = A
lulp-2uv.
We claim that luo.PW). To see this, let on --+ 0+, v -k v. Then, since (cf. [1]) Wl.p(1l)
-* L°(M),
u'-' Ulan is compact for 1 < q < p, N-1 N-p
p-
,
N>p,
00, N < p,
hence for such q
a.e. on Oft.
vlan
vnlan
If v=0,a.e.on Oft,vEW"(f),then J(v)=0.If v 34 0 on a subset of Oft of a positive measure, then
0<
f
W(v) =
f
W(v).
aw
an
Hence, again, by Fatou's lemma and the lower semicontinuity of 'Y,
0<
f
f
W (v)
lim inf 1 (vn )
an\r <
liminf j(vn),
and also hm inf &.V.) ap.
= liminf = 00.
on
1 j (vn )
Variational Inequalities, Bifurcation and Applications
139
Hence
J(v) < lim inf
j(onvn.
0+, we choose vn = v, and since
On the other hand, if v E E, Qn
j()>0,
0,
v
Qp n
v E 4Vo1P(Q),
vVWo,P(f),
we get lim
0, V E Wo'P(1Z), / cc, v ¢ Wo', P(c'),
_ = lim 1(v) Qn-1
i (Qny) Qn
= J(v) = Thus the limiting variational inequality (the homogenization) is given by the boundary problem
J
IVuVP-2Ou Vv
+ A J Iu,P-2uv = 0 `/v E W0
"P(SZ).
This problem has a "first" (i.e., smallest) eigenvalue Al = inf{J IVutP I u E Wo,P(Sl)
IuIP = 1}
,J
0
11
which is isolated (see 12)) and
d(I - Ta,Jf (A,'), Br(0), 0)
=
1,
A < A1,
-1, Al < A < A2,
where A2 = inf{A > Al I A an eigenvalue}.
See [15], [22] and [23] for other bifurcation results for quasilinear equations involving p-Laplacian like operators.
5.6 A unilateral problem Consider the differential equation subject to unilateral constraints -u" + u = Au cos u,
on (0,1),
u(0),u(1) > 0, u'(0) '< 0:5 u'(1),
u(0)u'(0) = 0 = u(1)u'(1).
The variational inequality formulation (as observed earlier) of this problem is 1
1
r
fu'(v' - u ')+u(v-u)>A J ucosu(v-u), VvEK,uEK, 0
0
K. Schmitt
140
where
u(1)>0}. Here p = 2 and the homogenization becomes i
ri
t
u(v - u) + J(v) - J(u) > 0,
u'(v' - u') + u(v - u) - A
J
J
0
0
`/v E H'(0,1), where J = IKo and Ko is the support cone of K, which equals K.
5.7 A beam with elastic obstacles As a final example we consider a beam subject to elastic obstacles. We refer to [34], [36], and [37] for more details. Let us consider the cases that E=H 2(0, 1) n Ho(0,1) which is the natural space for a simply supported beam and E = H2 (0,1) for the clamped beam. Let Il , I2 be subintervals of (0, 1) of positive measure. We consider the variational inequality I
fuh'(v -u )"_A 0
+
f kl(v
Vv E E,
rl+U
u,2(v-u),
0
)y + f k2(v+)o
0,
-
where i(U) =
and
fki(u
)y +
fk2(u+)v
u+=I u, if u > 0, 0, if u < 0, u_ _
f -u, 0,
if u < 0,
ifu>0.
Here again p = 2 and the homogenization becomes
Ju"v - u)" - A f u'(v - u)' + J1(v) - J2(v) - Jj(u) - J2(u) > 0, 0
Vv E E.
One computes
Ji (u) =
IK1(u), if 1 < 7 < 2, (u- )2, if 7 = 2,
f ki
r,
0,
if -y > 2,
Variational Inequalities, Bifurcation and Applications
J2(u) =
141
IK2(u), if 1 < Q < 2, f k2(u+)2, if Q = 2, 12
0,
if $ > 2,
where
K1 = {u I u > 0, a.e. Ill and
K2={uIu
R. Adams, Sobolev Spaces, Academic, New York, 1975.
[2]
A. Anane, Simplicite et isolation de la premiere valeur propre du pLaplacien, C. R. Acad. Sci Paris 305 (1987), 2725-728.
[3]
C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Flee Boundary Problems, John Wiley & Sons, New York, 1984.
[4] H. Brezis, Equations et inequations non lineaires dans les espaces vectoriels en dualite, Ann. Inst. Fourier 18 (1968), 115-175. [5]
H. Brezis, Problemes unilatt raux, J. Math. Pures Appl. 51 (1972), 1-168.
[6]
H. Brezis, Operateurs Maximaux Monotones, North-Holland, 1973.
[7]
H. Brezis, Analyse Fonctionnelle, Masson, Paris, 1983.
[8]
F. Browder, Nonlinear monotone operators and convex sets in Banach spaces, Bull. Amer. Math. Soc. 71 (1965), 780-785.
[9] F. Browder, Existence and approximation of solutions of nonlinear varational inequalities, Proc. Nat. Acad. Sci. USA 56 (1966), 10801086.
[10] F. Browder, Fixed point theory and nonlinear problems, Bull. Amer. Math. Soc. 9 (1983), 1-39. [11] M. Chipot, Variational Inequalities and Flow in Porous Media, no. 52 in Applied Mathematical Sciences, Springer-Verlag, New York, 1984. [12] S. Chow and J. Hale, Methods of Bifurcation Theory, Springer-Verlag, Berlin, 1982.
[13] P. G. Ciarlet and P. Rabier, Les Equations de von Karman, SpringerVerlag, Berlin, 1980.
142
K. Schmitt
[14] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.
[15] M. A. del Pino and R. Manasevich, Global bifurcation from the eigenvalues of the p-Laplacian, J. Differential Equations 92 (1991), 226251.
[16] C. Do, The buckling of a thin elastic plate subjected to unilateral conditions, Lecture Notes Math. 503 (1976), 307-316.
[17] C. Do, Bifurcation theory for elastic plates subjected to unilateral conditions, J. Math. Anal. Appl. 60 (1977), 435-448.
[18] G. Duvaut and J. L. Lions, Les Inequations en M6canique et en Physique, Dunod, Paris, 1972.
[19] L. C. Evans, Partial Differential Equations, Berkeley Lecture Notes, Berkeley, CA, 1994.
[20] G. Fichera, Problemi elastostatici con vincoli unilaterali: it problema di signorini con ambigue condizionial contorno, Mem. Accad. Naz. Lincei
Ser. VII7 (1964), 613-679. [21] A. Friedman, Variational Principles and Free Boundary Value Problems, Wiley-Interscience, New York, 1983.
[22] M. Garcia-Huidobro, R. Manasevich and K. Schmitt, On principal eigenvalues of p-Laplacian like operators, J. Differential Equations 130 (1996), 235-246. [23] M. Garcia-Huidobro, R. Manasevich and K. Schmitt, Some bifurcation results for a class of p-Laplacian like operators, Differential Integral Equations 10 (1997), 51-66.
[24] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.
[25] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities, Academic, New York, 1980.
[26] M. A. Krasnosels'kii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon, Oxford, 1963.
[27] R. S. Kubrusly, Variational methods for nonlinear eigenvalue inequalities, Differential Integral Equations 3 (1990), 923-932.
[28] R. S. Kubrusly and J. T. Oden, Nonlinear eigenvalue problems char-
acterized by variational inequalities with applications to the postbuckling analysis of unilaterally supported plates, Nonlinear Anal. TMA 5 (1981), 1265-1284.
Variational Inequalities, Bifurcation and Applications
143
[29] V. K. Le and K. Schmitt, Global Bifurcation for Variational Inequalities: Applications to Obstacle and Unilateral Problems, vol. 123, Applied Math. Sciences, Springer-Verlag, New York, 1997. [30] J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Nonlincaires, Dunod, Paris, 1969.
[31] J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Springer-Verlag, Berlin, 1972.
[32] J. L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math. 20 (1967), 493-519.
[33] G. S. Minty, Monotone (nonlinear) operators in Hilbert spaces, Duke Math. J. 29 (1962), 341-346.
[34] J. T. Oden and J. A. C. Martins, Models and computational methods for dynamic friction phenomena, Comp. Meth. Appl. Mech. Eng. 52 (1985), 527-634.
[35] D. Pascali and S. Sburlan, Nonlinear Mappings of Monotone Type, Noordhoff, Alpen aan den Rijn, 1978. [36] P. J. Rabier and J. T. Oden, Solution to Signorini-like contact problems through interface models, I, preliminaries and formulation of a variational equality, Nonlinear Anal. TMA 11 (1987), 1325-1350. [37] P. J. Rabier and J. T. Oden, Solution to Signorini-like contact problems through interface models, II, existence and uniqueness theorems, Nonlinear Anal. TMA 12 (1988), 1-17. [38] P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Fhnct. Anal. 7 (1971), 487-513.
[39] J. F. Rodrigues, Obstacle Problems in Mathematical Physics, NorthHolland, Amsterdam, 1987. [40] G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris 258 (1964), 4413-4416.
[41] G. Stampacchia, Le problbme de Dirichlet pour les equations elliptiques du second ordre h coefficients discontinus, Ann. Inst. Fourier 258 (1965), 189-258.
Klaus Schmitt Department of Mathematics University of Utah 155 South 1400 East Salt Lake City, UT 84112, USA
schmitttmath.utah.edu
Part 2
SEMINAR PAPERS
Complex Dynamics in a Class of Reversible Equations Francesca Alessio
Marta Calanchi Enrico Serra 1
Introduction
We consider the pendulum type equation
ii + W'(t, u) = h(t) t E R, u E R,
(E)
where the functions W and h satisfy
(Hl) WE C2(R x R; R) is 1-periodic in t and T-periodic in u, (H2) h E C(R; R) is 1-periodic and fo h(t) dt = 0, (H3) V (t, u) = W (t, u) - h(t)u is even in t.
The purpose of this note is to establish the existence of a large class of homoclinic and heteroclinic solutions to periodic orbits of (E) whose presence displays some chaotic aspects of the associated dynamics. We are referring to the so-called multibump type solutions, which have been studied extensively in the last few years in the context of Lagrangian and Hamiltonian systems starting with the pioneering papers [3], [6], [14], [15].
Variational methods have been proved to be successful in the study of this type of problem. Indeed, they often allow one to show the presence of complex dynamics under nondegeneracy conditions on the set of homoclinic solutions weaker than the classical transversality of the intersection between the stable and unstable manifold (see e.g. (17]). We refer the reader to the papers (2], [5], [12], [8], [16] and the references therein. The main novelty here is that the search for homoclinic and heteroclinic
solutions to periodic motions is complicated by the fact that one has to work with nonintegrable functions, so that the problem does not look fit to the variational approach. In spite of this, P. H. Rabinowitz showed in [10] that a global approach could be constructed, provided the equation is reversible in time (condition (H3)). He proved (for systems) the existence The first author is supported by CNR, Consiglio Nazionale dells Ricerche.
148
F. Alessio, M. Calanchi, E. Serra
of heteroclinic solutions which connect periodic minimizers of the natural action functional associated to (E). Further results were given in [11], and a multiplicity result was given in (9] for the case of the forced pendulum equation. In all these papers periodic minimizers are assumed to be isolated
or nondegenerate in a suitable sense. In a recent paper, [4], the authors proved that in the one dimensional case these assumptions can be relaxed and in fact the existence of heteroclinics is obtained under the much weaker assumption (m') (see below). In [4], also the existence of homoclinic solutions, i.e., heteroclinic solutions to the same periodic orbit, is investigated. It is proved that if the set of the heteroclinic orbits is discrete in a suitable sense (assumption (*) below), then the equation also admits homoclinic solutions. These solutions can be viewed as a special case of multibump type solutions and precisely, two-bump solutions. Here, following the technique developed in (1), we prove that the same discreteness assumption on the set of heteroclinic motions introduced in [4] is sufficient to have multibump dynamics. More precisely, in Theorem 3.4 we prove the existence of solutions q asymptotic to given periodic states
at infinity and such that q oscillates N times (N arbitrary) between these states. Next, in Theorem 3.6, we prove the existence of bounded solutions q which oscillate infinitely many times between two periodic states. Moreover, points where these solutions are close to a given state can be prescribed in an arbitrary fashion, provided they are far enough from each other. The existence of this type of solutions is particularly meaningful since
it gives some evidence of the chaotic nature of the dynamics associated to equation (E), showing in particular that the system exhibits sensible dependence on the initial conditions. In Section 2 we describe the variational formulation associated to the problem and we list some preliminary properties obtained in earlier papers. Section 3 is devoted to the statement and proof of the main results. After the present work was completed the authors were informed that P. H. Rabinowitz has recently obtained, in [13], results related to ours for the case of systems.
Notation. If uo and u1 are 1-periodic (continuous) functions such that uo(t) < u1(t) for all t, we say that u E [uo,u1] if uo(t) < u(t) < u1(t), Vt E dom(u). Therefore we will write for example u E Hio, fl [uo, u1] to mean that u E H1oc(R; It) and uo(t) < u(t) < u1 (t) Vt E R. Finally, we will abbreviate expressions like limt.+,,. (q(t) -u1(t)) = 0 as q(+oo) = u1.
Complex Dynamics in a Class of Reversible Equations
2
149
Preliminary properties
In this section we describe precisely the setting in which we are going to work and we recall some properties that we will frequently use. These properties were obtained in (10] and [4]. We refer the reader to the latter paper for their proofs.
Let E = Hl',,JR;R) and El = {u E E I u(t + 1) = u(t) a.e.}. Writing L(u) for a u2 - V (t, u), we define the usual action functional over the space El as I(u) = fo L(u) dt. It is well known that, under assumptions (H1) and (H2), the functional I possesses an ordered family of 1-periodic minimizers at level co = min I, E1
which are classical solutions to equation (E). In the sequel we will assume that there exist two global minimizers u0, ul E El satisfying the discreteness condition
Juo(t) < ul(t) for all t E (0, 1], if u E El n [uo, ul] and I(u) = co, then u E {up, ul }. The most important consequence of (H3) for our purposes is the equality (see [10] for its proof)
Nmo )I = EnI = co.
(2.1)
This equality allowed P. H. Rabinowitz in [10] to build a global approach
to heteroclinic solutions in spite of the fact that one has to work with nonintegrable functions. Indeed, for all q E E n [uo, ul] and for all k E Z, let Jk(q) =
f
k+1
L(q) dt - co;
k
then Jk(q) > 0 by (2.1) and equality holds if and only if q = uo or q = u1 in [k, k + 1], since uo and u1 satisfy (m*). Define now a functional J J(q) = > Jk(q) kEZ
The main features of the functional J can be collected in the following proposition (see (4], Lemma 2.6).
Proposition 2.1. The functional J has the following properties. 1. The sublevels of J in E n [rco, u1] are bounded in E.
2. If qn E E n [uo, u1] and qn - q weakly in E, then q E [uo, ul] and
J(q) 5 limn J(qn)
150
F. Alessio, M. Calanchi, E. Serra
3. If q E E n [uo, ul] and J(q) < +oo, then q(±oo) E {uo, ul }. The simplest type of heteroclinic solutions connecting uo to ul and vice-
versa can be obtained by minimizing J over appropriate subsets of E n [uo, u1]. Set
ro = {q E E n [uo, ul] I q(-oo) = Uo, q(+oo) = ul } and
r1 = {q E E n [uo, u1] I q(-oo) = u1, q(+oo) = uo},
and note that, because of (H3), if q E ro, then q*(t) := q(-t) E r1 and J(q) = J(q' ); this implies that c:= inf J = inf J. ro
r,
This number is readily seen to be positive. One of the main results in [4], which will be our starting point, is the following theorem.
Theorem 2.2. If (m*) holds, the infimum c is achieved in ro (and in p1). Moreover there exist qo E ro n C2(R; R) and q1 E r1 n C2(R; R) solutions of (E) such that J(qo) = J(q1) = c. Remark 2.3. i) The fact that minimizers in rQ, a E 10, 1}, solve the equation is not obvious because of the constraint q E [uo, u1], see [4], Proposition 3.7.
ii) Of course, by reversibility, one can assume that q, (t) = qo(-t). The main purpose of the present work is to show that, under discreteness assumptions on the set of heteroclinic solutions in r, o E {0, 1}, there is a very rich structure of solutions. Indeed, the solutions provided by Theorem 2.2 will be used as building blocks for the construction of more complicated
types of solutions, in the spirit of the classical multibump construction. Assuming, as in [4], that S :_ {q(0) I q E ro, J(q) = c} 0 (uo(0), u1(0)),
(*)
we will apply the "gluing" procedure developed in [1], proving the existence of multibump type solutions to (E).
We remark that in [4] the authors proved the existence of solutions asymptotic to the same periodic orbit as t - ±oo; these can be seen as a special case - 2-bump solutions - of multibump type solutions. The problem here is the construction of solutions with arbitrarily or infinitely many bumps and to this aim we will need a refined analysis of the problem. As far as condition (*) is concerned, one can readily see that in the autonomous case the set S coincides with the interval (uo(0), u1(0)) because of the invariance of the problem under translations and that in such case, modulo translations and reflections, the equation admits a unique heteroclinic solution. We refer to [8], [2], [16] (for Duffing-like equations), [5] for
Complex Dynamics in a Class of Reversible Equations
151
Hamiltonian systems and [4], where it is proved that analogous conditions are necessary to obtain multiplicity of connecting orbits. From now on we will assume that also condition (*) is satisfied. The important consequence of assumption (*) for our purposes can be summarized in the following estimate ([4], Lemma 4.3). Roughly, it expresses the fact that there exist points through which no minimizer in 1'a (a E {0,1 }) can pass.
Proposition 2.4. For all 5 > 0 there exist bo, bl E (0, b) and A > 0 such that inf {J(q) I q E ro u I'1, q(0) = uo(0) + bo or q(0) = ul (0) - bl } > c + A. To proceed towards the construction of multibump type solutions we will need a few technical results. To state the first one, let d
2
emir lui(t) - uo(t)I.
(2.2)
Lemma 2.5. There exist S E (0, (1) and AO > 0 such that for every b E (0, S) and a E {0,1 } there exist wa a E H'(0, 1) fl [uo, ul ] such that 1wb,o(0) = uo(0), Iw6,o(1) - uo(1)I = b,
- uo(0)I = b, wda(1) = u,,(1),
1I wd o(0)
and
JO (W,+5,,) :5 Ao6. Jo(wia) Aob, Proof. The straightforward proof is contained in [4]. For instance, if a = 0, then wa,o can be taken to be uo(t) + bt. 0
The preceding Lemma will be often used below to compare the value of the functional J on different functions. To this aim we define the following "cut-off" operators. Let s E Z, a E {0,1 } and q E E be such that Iq(s) - ua (s) I = b for some b E (0, S) (where S is provided by Lemma 2.5). We define
Xsaq(t) =
ua(t)
if t < s - 1,
q(t)
if t > s,
w6a(t-s+1) ifs-1
and
if t < s, XLq(t) = wa (t - s) if s < t < s + 1, ua(t) ift>s+1. The property of this cut-off procedure that we will use in the following section is expressed in the next statement. The idea is that whenever q E E is close to ua (a E {0,1}), it can be modified in an interval of length one and extended as ua without affecting too much its level when evaluated by J. The proof of the following statement is a direct computation based on the estimates given in Lemma 2.5. q(t)
152
F. Alessio, M. Calanchi, E. Serra
Lemma 2.6. Let b E (0, b) and let p, s E Z U {±oo}, t E Z be such that p + 1 < t < s - 1. If q E E verifies (q(t) - uo(t)I = b for some a E 10, 1), then
s
a
t Jk(Xt oq) :)L Jk(q) + AoS. k=p
k=p
Multibump solutions
3
This section contains the main results in this paper, namely the construction of multibump type solutions. In what follows AO is the number provided by Lemma 2.5, d is defined in
(2.2) and a is the function defined by
a(r) = inf{I(u) - col u E H1(0,1) n [uo,u1], distLo(o,1)(u, {uo,ul}) > r}(3.1)
Note that by (2.1) we have a(r) > 0 for all r > 0. ); then, by PropoWe begin by fixing some constants. Let b = min(8, sition 2.4, for all 6 E (0, b) there exist positive numbers 5o, bl E (0, 6) and
A > 0 such that
inf{J(q) (q E ro u rl, q(0) = uo(0) + bo or q(0) = ul(0) - b1} > c +A. (3.2)
We fix A E (0, A) such that 2 < a(d) - 2Xo max(bo, 61),
(3.3)
an d we set e
= n-d n
A 12A0'
(3 . 4)
Finally we fix an integer m > 4 so large that
2c+A
m > 2a(e)
(3 . 5)
and such that there exist minimizing heteroclinics in the sets
Xo = {q E E n [uo, ul] ( q(-m) < u*(-m) + ao, q(m) ? ul (m) - 61} and
X1 = {q E E n [uo, ul] I q(-m) ? ul (-m) - 5i, q(m) : uo(m) + bo}; this is always possible because of Theorem 2.2 and the invariance of the problem under Z-translations. The next proposition provides the first main estimate to be used later.
Complex Dynamics in a Class of Reversible Equations
153
Lemma 3.1. If q E Xo satisfies 2m-1
A
E Jk(q)
k=-2m
then there exist two integers k- E [-2m+1, -m-1] and k+ E [m+l, 2m-11 such that
q(k) < uo(k-) +E and q(k+) > u1(k+) - E. Proof. We prove the existence of k+, the other case being symmetric. First of all we can assume that for all t E [m, 2m] we have
q(t) > uo(t) +2
u1(t)
this can be proved as in [4], Lemma 4.9, and we omit the details. Therefore we obtain that IN - UOIIL-(k,k+1) >- d for all k E [m, 2m - 1]. To complete
the proof we argue indirectly, by assuming that q(k) < u1(k) - e for all integers k E [m + 1, 2m -1]. In this case we see that IIq - u1IIL-(k,k+l) > E for all k E [m, 2m - 1] fl Z; therefore putting this estimate together with the previous one, we obtain that diet L-(k,k+1) (q, {uo, ul }) > e
for all integers k E [m, 2m - 1].
We deduce from this that 2m-1
Jk (q) 2: ma(E) > c + k=m
2
,
by the choice of m. This violates the assumption.
0
Of course the same type of result, namely with k- and k+ interchanged, holds for functions in X1 (just replace q(t) by q(-t) in the proof). We now show that all functions in X, below a suitable level are "free from constraints" at t = fm.
Lemma 3.2. If q E Xo satisfies A
2m-1
E Jk (q) < c +
,
k=-2m
then
q(-m) < uo(-m) + ao
q(m) > u1(m) - d1.
and
Proof. Let k- and k+ be the numbers given by the preceding Lemma and consider the function k-
k+
154
F. Alessio, M. Calanchi, E. Serra
Then q- E ro and, if the thesis is false, by (3.2) and the invariance under Z-translations, we obtain J(q) > c + A.
(3.6)
However, by Lemma 2.6 and the choice of e, k+-1
J(4) <
Jk(q) + 2Aoe < C + 2 + 2Aoe < c + A,
k=k-
0
which contradicts (3.6).
Once again, the same type of property holds for functions in X1. We now define the classes of functions in which we look for multibump solutions. Let us fix N E N, p:= (p1, ..., pN) E ZN such that pi -pi- I > 4m and or :_ (al, ...vN) E {0,1 }N with vi 0 Qi_ 1 for all i = 2, .., N. We set rN,p,o = {q E E n (uo, u1] I q(-oo) = u4,, q(+oo) = u1-ON a q(' - pi) E X,;, i = 1, ..., N}.
From this definition we see that each function in the class rN,p,, looks like a (piece of) heteroclinic solution in the interval [pi - 2m, pi + 2m]. In particular, functions in rN,p,, are alternately close to uo and u1 at the points pi - m and pi + m. We are now ready to prove the main property. To state it we set
CN,p,, = inf J rN.p.o
and we let Ai = {pi - 2m, , pi + 2m - 1} be the set of indices relative to the i-th "block" (pi - 2m, pi + 2m]. The next proposition says that any minimizing element in rN,p,, has a uniformly low level on every block [pi - 2m, pi + 2m], independently of the number of blocks.
Proposition 3.3. If q E rN,p,, satisfies J(q) = CN,p,,, then
AM < c +
A ,
for all i = 1,
kEA;
Proof. Let P = {i E {1, ... , N} I EkEA; Jk(q) > c + A/2},
P+={iE {1,...,N}\PIi+1EP},
P- ={iE {1,...,N}\PIi-1EP};
,
N.
(3.7)
Complex Dynamics in a Class of Reversible Equations
155
to prove the proposition we must show that P is empty. Let qi be a minimizing heteroclinic in X,; and note that, since J(qi) = c, Lemma 3.1 applies to give the existence of two points ki E [pi - 2m + 1, pi -
m - 1] and k; E [pi + m + 1,pi + 2m - 1) such that qi(k;) <
e
and gi(ld) > ul(kt) - e if ai = 0 and the same with "k; " and "k-" interchanged if ai = 1. Finally we set (Xki
qi =
o
o Xk 1)qi
(Xki 1 o Xk+ o)qi
if ai = 0, if ai = 1.
Note now that Lemma 3.1 applies for every i V P. Assuming for instance that ai = 0, we replace q in [pi - 2m, pi + 2m) with the function (Xk. p o Xk 1)q
ifiEP, ifiEP- nP+,
XkS ,oq
if i E P- \ P+,
qi
q=
rxk;,1q
ifiEP+\P-,
q
otherwise.
If ai = 1 an analogous definition holds. Having modified q as 4 for all i = 1, ... , N, we extend it to R by continuity as uo, u1, or q, according to its behavior at the endpoints of [pi - 2m, p, + 2m]. We obtain therefore a function q E rN,p,, and we now estimate its level.
Note first that if i E P, then Jk (q) < c + 2aoE < E Jk (q) + 2Aoe - A/2, kEA;
kEA1
while, if i V P,
E Jk(q) S c + A/2. kEA;
To proceed in the estimate, let
l Z(P) = {k E Z I k V UiEPUP-UP+A,}; then
J(4) = > Jk(q) + > Jk(q) + > > Jk(q) iEPkEA; kEZ(P) iEP+UP- kEAi
< > Jk(q) + >(E Jk(q) + 2Aoc - A/2) kEZ(P)
iEP kEA;
+ > (> Jk(q) + 2Aoe ) iEP+UP- kEA;
F. Alessio, M. Calanchi, E. Serra
156
Jk(q) + >2 +
A/2)IPI
Jk(q) +
iEPkEAi
kEZ(P)
>2
>2 Jk(q) +
U P-I
iEP+uP- kEAi
= J(q) + (2Aoe - A/2) IPI + 2AoeIP+ U P-I where I I denotes the cardinality of a set. Now since w e clearly h a v e I P+ U P I < 21 PI, we deduce that P must be
empty, for otherwise (taking into account the choice of E), J(9) 5 J(q) + IPI(6Aoe - A/2) < J(q),
which is impossible because q is a global minimizer. This completes the proof.
Using the previous notation, we are now able to prove the existence of multibump solutions to (E).
Theorem 3.4. Assume that conditions (m') and (*) hold. Then, for every
N E N, p = (pi, ...,PN) E ZN and a = (a1i ...aN) E {0,1}N such that pi - pi-1 > 4m and ai 0 ai_ 1 for all i = 2,.., N, there exists q E rN,p,, such that J(q) = cN,p,,. Moreover, q E C2(R; R) and is a classical solution of (E).
Proof. Let (qn) C rN,p,, be such that J(qn) -+ cN,p,,. By Lemma 2.1 there exists a subsequence (still denoted qn) such that qn -+ q weakly in E and J(q) < cN,p,,. Since qn -+ q also in LOO (R; R) and since qn is minimizing, we obtain that q E rN,p,, and therefore that J(q) = cN,p,,. Now, as in [41, Proposition 3.7, we can prove that
uo(t) < q(t) < ul(t) Vt E R. Moreover, by Proposition 3.3 and Lemma 3.2, we obtain
q(pi - m) < uo(pi - m) + So and
q(pi + m) > ul (pi + m) - bi
for all i such that ai = 0, and a corresponding property for those i's for which ai = 1. Therefore, using standard regularity arguments, we can conclude that q belongs to C2(R; R) and that it is a classical solution of (E).
0 The next result concerns the existence of orbits with infinitely many bumps, which give, as we have anticipated, some evidence of the chaotic character of the system. Let pi E Z and ai E {0, 1} be bi-infinite sequences such that pi - pi_1 > 4m and ai ai_ 1 for all i E Z. For every N E N, set pN = (P_ N, ..., pN ), aN = (a_N, ..., aN) and, for simplicity, rN = r2N+1,pN,aN
and
CN = inf J. 1'N
Complex Dynamics in a Class of Reversible Equations
157
As a corollary of Theorem 3.4 we have the following lemma, which contains the basic information for the construction of solutions with infinitely many bumps.
Lemma 3.5. For every N E N there exists a solution qN E C2(R; R) of (E) such that qN E rN, J(qN) = cN. Moreover there exists M > 0 such that IIgNIIL-(R;R) <- M,
IIgNIILO°(R;R) <- M,
IIgNIIL-(R;R) s M dN E N. (3.8)
Proof. The first part is a direct consequence of Theorem 3.4. For the second part just notice that it follows from the fact that the sequence qN is bounded in LOO (R; R) independently of N and solves qN = -V'(t, qN) + h, an equation with bounded right-hand side. The next result establishes the existence of solutions with infinitely many bumps.
Theorem 3.6. Assume conditions (m`) and (*) hold. Then for all 5 > 0 there exists m = m(b) E N such that for every bi-infinite sequences pi E Z
and of E {0,1 } with pi - pi_ 1 > 4m and ai # o'i_ 1 for all i E Z, there exists a solution q E C2(R; R) of (E) such that q(. - pi) E X; for all i E Z. In particular, for all i E Z, I q(pi - m) - uo: (pi - m) I < b
and
Iq(pi + m) - u,i,, (pi + m) I < b. (3.9)
Proof. Let b E (0, b) and m = m(b) E N be fixed as at the beginning of this section. For every N E N, let qN be the solution found by means of Lemma 3.5. Each qN satisfies qN( - pi) E X,; and in particular (3.9) for every i = -N, ... , N. Moreover, by the estimates (3.8), the sequence qN is uniformly bounded in C2(R; R); therefore, by the Ascoli-Arzelh Theorem,
it has a subsequence which converges in Ci ,(R; R) to some q. The fact that each qN solves (E) shows that q E C2(R; R) and solves (E) as well. Properties (3.9) are clearly preserved by local uniform convergence, so that q is the required solution.
Theorem 3.6 can be refined and made more precise in several ways. We list below some of them, without giving the proofs, which, in most cases, consist in variations of arguments already described. First, other types of solutions could be found. In particular we refer to
orbits which are asymptotic to uo or ul as t -' +oo or t - -oo, and that exhibit infinitely many oscillations between uO and ul only before or after a certain time to. Moreover, one could give a much more precise description of all these solutions (and of those found in Theorem 3.6), in the spirit of the shadowing
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F. Alessio, M. Calanchi, E. Serra
Lemma. For example one can prove that the orbits found are "closer than 5" to uo or ul in the intervals [pi + 3m, pi+i - 3m], i E Z. Finally, if the points pi are distributed periodically, one could look for periodic orbits of large period that "shadow" heteroclinics in every period. This can be done with the same type of arguments as above, by working directly in a space of periodic functions (see e.g. [7] for analogous results). REFERENCES [1] F. Alessio, M. L. Bertotti and P. Montecchiari, Multibump solutions to possibly degenerate equilibria for almost periodic Lagrangian systems, Z. Angew. Math. Phys., to appear.
[2] U. Bessi, A Variational Proof of a Sitnikov-like Theorem, Nonlinear Anal. TMA 20 (1993), 1303-1318. [3] S. Bolotin, Existence of homoclinic motions, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 6 (1980), 98-103. [4] M. Calanchi and E. Serra, Homoclinic solutions to periodic motions in a class of reversible equations, Calc. Var. and PDEs 9 (1999), 157-184.
[5] K. Cieliebak and E. Sere, Pseudo-holomorphic curves and the shadowing Lemma, Duke Math. J. 77 (1995), 483-518. [6] V. Coti Zelati, I. Ekeland and E. Sere, A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann. 288 (1990), 133-160.
[7] V. Coti Zelati and P. H. Rabinowitz, Multibump periodic solutions of a family of Hamiltonian systems, Top. Meth. in Nonlinear Analysis 4 (1994), 31-57.
[8] P. Montecchiari, M. Nolasco and S. Terracini, A global condition for periodic Duffing-like equations, Trans. Amer. Math. Soc. 351 (1999), 3713-3724.
[9] D. C. Offin and H-F. Yu, Homoclinic orbits in the forced pendulum system, Fields Inst. Comm. 8 (1996), 113-126. [10] P. H. Rabinowitz, Heteroclinics for a reversible Hamiltonian system, Ergod. Th. and Dyn. Sys. 14 (1994), 817-829. [11] P. H. Rabinowitz, Heteroclinics for a reversible Hamiltonian system, 2, Differential Integral Equations 7 (1994), 1557-1572.
[12] P.H. Rabinowitz, A multibump construction in a degenerate setting, Calc. Var. and PDEs 5 (1997), 159-182.
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159
[131 P. H. Rabinowitz, Connecting orbits for a reversible Hamiltonian system, preprint, 1998. [141 E. Sere, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Zeit. 209 (1991), 27-42.
[151 E. Sere, Looking for the Bernoulli shift, Ann. Inst. H. Poincare, Anal. Non Lineaire 10 (1993), 561-590. [161 S. Terracini, Non degeneracy and chaotic motions for a class of almost-
periodic Lagrangian systems, Nonlinear Anal. TMA 37 (1999), 337361.
[17] S. Wiggins, Global Bifurcation and Chaos, Applied Mathematical Sciences 73, Springer-Verlag, New York, 1988.
F. Alessio, M. Calanchi and E. Serra Dipartimento di Matematica Politecnico di Torino Corso Duca degli Abruzzi, 24 10129 Torino Italy
SERRAQpolito.it
Symmetry and Monotonicity Results for Solutions of Certain Elliptic PDEs on Manifolds Luis Almeida Yuxin Ge ABSTRACT In this work we use an analogue of the method of moving planes to obtain symmetry results for positive solutions of some elliptic partial differential equations on manifolds, under certain symmetry requirements for the problem considered. We also obtain some monotonicity results in this setting by using the sliding method.
1
Introduction
The method of moving planes is based on ideas first introduced by Alexandrov (see [1]) for studying embedded hypersurfaces of constant mean cur-
vature. Later on, Serrin (see [13]) reintroduced this method in order to study elliptic equations and Gidas, Ni and Nirenberg ([7] and [8]) used it to obtain symmetry results for positive solutions of elliptic equations in R". Subsequently, there have been many extensions of these results and, in particular, Berestycki and Nirenberg (see [3]) improved the proof given in [7], by using a maximum principle in narrow domains due to Varadhan and based on Alexandrov-Bakelman-Pucci's inequality (for similar results on manifolds see the work of Cabre [4]). Their method enables them to
assume no boundary regularity and to deal with fully nonlinear elliptic equations. Here, we follow an approach that is close to the method of Berestycki and Nirenberg, using the maximum principle for operators in divergence form in general domains on manifolds, to obtain symmetry results under some symmetry hypothesis described below. The presentation given is inspired by the work of Damascelli for domains in R' (see [5]).
For the special case of spaces of constant curvature, analogous results have also been obtained very recently by Padilla [12) for symmetric domains
in S", and by Kumaresan and Prajapat ([10] and [11]) for balls in H". In our work, we present a rather abstract approach that includes these spaces as special cases. We also give a few new examples, including a case (see
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L. Almeida, Y. Ge
Section 4.3) where we use Theorem 1.1 to obtain symmetry results for a domain in a non-commutative Lie group. Moreover, we indicate a procedure where, taking advantage of the product structure of the spaces considered,
one may produce many examples of domains where our theorem yields symmetry results (see Section 4.4). We consider a domain M of a manifold N (i.e., M is an open submanifold of N) and denote by g the metric on N. We assume the following conditions:
i) Isometrics : There exists a family of isometries It : N - N, t E]0, 2[, which is C' in t, and such that there is a family of hypersurfaces
Ut C N such that It(x) = x a x E Ut, i.e., Ut is the invariant hypersurface under the action of It. ii) Domain decomposition: There exist pairwise disjoint sets Vt, t E]O, 2[, such that a) Vt C Ut, for all t EJO, 2[.
b) For all t1, t2 E [0, 2],
U Vt is an open subset of N. t,
c) M = U Vt, and A
is compact.
O
iii) Inclusion in increasing t: Let Qt, = U Vt and Qt' = U Vt, then O
tl
a) It(Qt) C Qt, for all t E]O,1[. b) For all t EJO,1[ and for every component C of Qt, there exists a point xEOCf18M : It (x) EQt. Moreover, we say that M is symmetric if we also have
iv) Inclusion in decreasing t:
a) It(Qt) C Qt, for all t E]1,2[ and Q, = b) For all t E] 1, 2[ and for every component C of Qt, there exists a I1(Q').
point xEOCf18M : It(x)EQt. For 0 < t < 1 and x E Qt we define xt = It(x) and ut(x) = u(xt). For simplicity, in this work we will concentrate on functionals of the form
E(u) = fM a(IVU12)dvol ,
(1.1)
but the results can be extended to more general cases (a depending also on x E M or on Vu in a more general way) - in particular, we can work in the setting studied in [5], as long as we make further assumptions analogous to those made by Damascelli in his work.
Symmetry and Monotonicity for Elliptic PDEs on Manifolds
163
The Euler-Lagrange operator associated with (1.1) is Lu = b(a'(I VuI2)du)
where bw = (-1)'(P-1)+i * d * w, for any p-form w, and * is the Hodge star operator. We will study the equation Lu = f (u)
(1.2)
,
which, in local coordinates xl,... , x", on M, can be written as
Lu = -
etg ax'
(aI(I Vu12)
f(u) .
det99`j axe
(1.3)
We suppose the following conditions are satisfied a E W2'OO(]0, +oo[) f1 C°([0, +oo[) ,
(1.4)
3a>0 : a'(t) - 2 (a" (t))- t>a>0, VtER+,
(1.5)
SQ > 0 : a'(t) +2Ia"(t)It :5,6, dt E R+
,
f is locally Lipschitz.
(1.6)
(1.7)
where (p' denotes, as usual, the negative part of the function W. Let u, v E Hl (.M), and A be the section of T*M ® TM given by
Ay = A(x) = +2
J0
J0
1
1
a'(IDpt(x)I2)dt IdT=mot
Vpt(x)I2)(Vpt(x))* ® (Vpt(x))dt ,
(1.8)
where pt(x) = tv(x) + (1 - t)u(x). From condition (1.6) it follows that for all
TIM, f 1 a'(Iopt(x)12)dt 0
2 fo l a"(IDpt(x)12)(Opc(x),E)2dt
< f (a'(IVpt(x)I2) + 2Ia"(IVpt(x)I2)1 IVpt(x)12) Ie12dt < (1.9)
where we used Cauchy-Schwarz for the first inequality. Likewise, using condition (1.5) and the Cauchy-Schwarz inequality, we obtain
(AX,)
J
((a(IVpt(x)12)
- 2(a"(IVp(x)I2))
The main result of this paper is
IVpt(x)I2) II2dt > aII2.
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L. Almeida, Y. Ge
Theorem 1.1. Suppose M satisfies conditions i), ii), iii), and u E R) is a positive (i.e., u > 0 in M) weak solution to Ho (.M, R) fl L(u) = f (u) in M,
u = 0 on 8.M,
and conditions (1.4) to (1.7) are satisfied. Then,
u(x) < u(xa),
if x E QA
,
0
(1.11)
Moreover, if M is symmetric we have
u(x) = u(xl), dx E Q1 .
If Ii(Ql)
(1.12)
Q1, we still have the inequality
u(xl) > u(x), dx E Q1
(1.13)
.
The proof of this theorem will be given in Section 3 after having established the necessary comparison principles in Section 2. In Section 4 we will give a few examples where we apply Theorem 1.1 to obtain symmetry results. The results obtained also give an illustration of a general principle indicated at the end of Padilla's work [12]. We end our paper with a monotonicity result (Theorem 5.1) of the same nature of the one obtained by Berestycki and Nirenberg using the sliding method (see also [6] theorem 1.7). Roughly, it says that when our domain M is convex along the action of a group of isometries, if the boundary data is increasing along this action, then so is our solution of (1.2). In collaboration with L. Damascelli, we are now working on extending our results to a class of degenerate elliptic operators which includes the
p-laplacian for 1 < p < 2, and also on obtaining symmetry results in unbounded domains in manifolds with non positive sectional curvature.
2
Comparison principles
We assume that u E Hl(M) fl L°°(M), A E R and g : R -+ R is a continuous nonincreasing function. We say that u is a weak solution (subsolution <, supersolution >) of Lu + Au = g(u) if VW E CC (M), W
(2.1)
0,
JM [a'(IDul2)(Vu, O(p) + Aucp]dvol = im g(u)cpdvol
.
(<, >)
Symmetry and Monotonicity for Elliptic PDEs on Manifolds
165
Let M' be an open subset of M. We have the following Poincare-type inequality: there exist two constants ry, C E R+, such that if vol (.M') < ry, then
f
'I2dvol < Cvol(M')2""
I
f
IV iI2dvol
J
,
VO E Ho(M')
.
(2.2)
(This follows using a partition of unity for M, and Poincart's inequality in the local charts associated with the partition.)
Theorem 2.1. (Weak Comparison Principle) Letu,v E H1(M)f1L°O(.M) and M' be an open subset of M, and assume that u < v on W. Then, if u (v) is a subsolution (supersolution) of (2.1) in M', i) if A > 0, then u < v in M'; ii) if A < 0, vol(M') < ry and CIAI(vol(.M'))2/" < a (ry and C being as in (2.2) and a as in (1.5)), then u < v in .M'.
Proof. Taking as test function (u - v)+ E Ho (M'), and denoting {u >
v}= {xEM':u>v}, we have
0>
f
(a'(IVuI2)(Vu, V(u - v)) u>v}
- a'(IVvJ2)(Vv, 0(u. - v))
+A(u - v)2) dvol 1
o
J(U>v)
d (a'(IVptI2)VPt, V(u - v))dt dvol dt
+
A(u - v )(u - v)dvol
f(u>v}
f
u>v}
(Az0(u - v), 0(u - v))dvol + f
A(u - v)(u - v)dvol , u>v}
where pt and Ax are as in (1.8). Consequently, using (1.10),
-A
J
u>v}
(u - v))dvol
>_ fu>v} (Ax0(u - v), V (u - v))dvol >a IV(u - v)I2 > 0. (2.3) {u>v}
If A > 0, it follows from (2.3) that II (u - v) + II Ho (M') = 0, and thus u < v
in M', as desired. If A < 0, equations (2.2) and (2.3) imply
afM? IV(u - v)+I2dvol < Al IfM' I(u - v)+I2dvo1 <
IAIC(vol(M'))2,1"
JM' IV (u - v)I2dvol
,
and hence if IAIC(vol(M'))2/" < a, it follows that II(u - v)+II HO,(M') = 0,
andthusu
0
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L. Almeida, Y. Ge
Theorem 2.2. (Strong Maximum Principle) Let A E R and suppose that u, v E H' (AM) fl C(AM) satisfy
u < v, Lu + Au < Lv + Av, in M'.
If 3xo E M'
u(xo) = v(xo), then u
:
(2.4)
v in the component of M'
containing xo.
Proof. Let 0 E Ho (AM),
> 0. From (2.4) it follows that
0 > I (a'(IVuI2)(Vu, V) - a'(IVvI2)(Vv, V) + A(u - v)¢) dvol - v), VO)dvol + AJM'
fM'
- v)qS dvol ,
w here A= is as in (1.8). In local coordinates x', ... , x", we can write this
variational inequality as
"
8 (gui
1
i.=1
dt
(
)J
Au - v)) + 8xr
-
(2.5
)
If u(xo) = v(xo) for some xo E M', then supB(u-v) = supM,(u-v) = 0 for every geodesic ball B C M' containing xo, since u and v are continuous in M'. Therefore, our result follows from the strong maximum principle (see Theorem 8.19 of [9]).
3
Proof of Theorem 1
Since la is an isometry, uA satisfies the equation Lua = f (uA),
in Qa
.
From (1.7) (i.e., the fact that f is locally Lipschitz) it follows that 3A1iA2 E P+ s.t. L(uA) + A1ua = f (ua) + A1ua = g1(ua)
L(u) + A1u = f (u) + Aiu = g1(u)
,
L(ua) - A2uA = f (ua) - A2uA = g2(ua)
, (3.2)
L(u) - A2u = f(u) - A2u = 92(u),
where gi is nondecreasing and 92 is nonincreasing in the range of values of
uand uA.
Symmetry and Monotonicity for Elliptic PDEs on Manifolds
Let G :_ {µ E (0,1)
:
167
ua > u in Qx , VA E (0, µ]}. By Theorem
2.1, if vol(QA) < 'y and CA2(vol(QA))2/N < a (which is always true for A sufficiently small since vol(QA) - 0 when A - 0), then ua > u in Q.\ since
ua>uon 9Qa(recall that ua=uon VAand ua>0=uon 8Qaf OM). Thus, G is nonempty. It is clear that G is closed in (0, 1) (continuity of u). By the connectivity
of the unit interval, to show that G = (0, 1) it suffices to prove that G is also open. Suppose that p E G. This implies that u,, > u in Q,,. We claim that then u,, > u in Q. If this were not so, there would 3xo E Q,, : u(xo) = u,,(xo). But then, since u,, > u, using (3.1) and the fact that g1 is nondecreasing, the strong maximum principle (Theorem 2.2) would yield that u = u,, in C, where C would be the connected component of Q,, containing xo. Since u is continuous on M and C C M, it would follow that u = uµ in C. However, this is impossible since by iii) b) there exists x E 8Cn 9M such that x,, = I,,(x) E Qµ, and hence u(x) = 0 < u(x,,) = u,(x) (recall that by assumption u > 0 in M, and x,, E M by construction). This proves our claim. Let K CC Q,, be a compact subset of M such that IQ ,, \ KI < 2
,
and CIA2Ivo1(Q,, \ K)2"" < 2
Since u is continuous in M, It is continuous in (0, 2) and K is compact, using the fact that mmin(ua - u) depends continuously on A (uniform convergence in K of ua to u,,) and that min(u,, - u) > 0 (since p E G), we see that there exists e > 0 such that for A E [µ, p + e),
ua(x)>u(x), VxEK.
(3.3)
Moreover, from the continuity of It it follows that for e sufficiently small, IQ,, \ KI < -y
,
and CIA2Ivol(QA \ K)2I' < a,
VA E [p, s< + e)
.
Thus, applying Theorem 2.1 to u, ua in Q,, \ K, it follows that uA > u in Qa \ K. Together with (3.3) this yields ua > u in QA, and consequently A E G. Since [0, µ] C G by definition, (p - e, p + e) fl (0,1) C G, hence G is open and thus G = (0,1). The fact that u(xi) > u(x), Vx E Qi follows by continuity (of u and It). Under the assumption that M is symmetric, the same argument with A
decreasing from 2 to 1 yields that u(x) > u(xi), dx E Q' which implies that u(xi) < u(x), Vx E Qi. Together with the inequality obtained above, this yields u = ui when M is symmetric, as desired. 0
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L. Almeida, Y. Ge
Examples
4
The examples we will give below are of the form It = At o I o At 1, where I is a reflection, i.e., an isometry of N that leaves a hypersurface U C N invariant, and moves all other points of N. We suppose 8U C eN. The At are a family of isometrics of N such that
A : (t,x) H At (x), is C1([0,2] x N, N).
Ao=Idn(. A sufficient condition for M = U Vt, to satisfy conditions i) and ii) o
of our assumptions, is to require that 8At I
X
T-V, = TTN
,
`dx E V
,
Vt E (0 , 2)
,
(4 . 1)
provided that we also demand that Vt, n Vt, = 0 for t 1 # t2. In all the examples below u will denote a positive solution of equation (1.2). For a few more details on these examples see [2].
4.1
Annular domain in R2
We consider a domain M which can be written in polar coordinates as
M:={(p,9)
:
po
oo, 0 < a < Tr. In this case, we let I be the reflection relative to the line y/x = tan(-a), and At be the rotation of angle at around the origin, i.e., At(z) = e'atz. Our domain will then satisfy all the required conditions and thus, from Theorem 1.1, it follows that u(p, 9) = u(p, -9) ,
in
M
.
4.2 M is a convex geodesic ball in S' or H' Let M be a convex geodesic ball in S' of radius a < 2 , and y be its center - without loss of generality we may suppose that y = N, the north pole of S", i.e., y = (0, ... , 0, 1) E R"+1 In this case, we can write M as M = {x E S" : x'+1 > 1 - cos a}. We choose any direction in the x1, ... , x" hyperplane - without loss of generality we may suppose it is the x" direction - and we want to show that u is symmetric relative to the hyperplane orthogonal to the direction we chose (in our case it will be the x" = 0 hyperplane). We write R"+1 = R"-1 x R2 and define At := IdR..- ® At , where At is the rotation of angle at in R2 _ C, i.e.,
At(xn +ix"+1) = et t(x"
+ix"-F1)
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169
The reflection I is just the symmetry with respect to the hyperplane x +l = tan a, of Rn+l It is easy to check that M satisfies the assumptions of The-
orem 1.1 and thus we obtain the desired symmetry of u relative to the xn = 0 hyperplane. Since we could have chosen any direction in the x1, ... , xn hyperplane, we conclude that u is symmetric relative to any hyperplane containing the xn+1 axis, and therefore u is radially symmetric in M. A similar result was also obtained by Padilla (see (121) using related ideas. Our method also yields unidirectional symmetry results along a band in a sphere as in Padilla's work. An analogous approach, making use of hyperbolic rotations, also yields symmetry results for symmetric domains in Hn - in particular, for balls (in this case, since Hn is a constant negative curvature space, we no longer need an upper-bound on the radius of the ball in order to apply the method). After our work was completed, Prajapat told us that, in collaboration with Kumaresan, she had also obtained the same result for balls in Hn and S" using a more specific approach (see (111).
4.3 A subgroup of the polarized Heisenberg group The non-abelian group H"01 is defined by taking R" x R" x R and defining the group operation to be
(x,y,t) * (u,v,s) _ (x+u,y+v,t+s+u y)
,
where u y denotes the usual scalar product in R". It can be realized as the subgroup of upper triangular matrices (see (14]),
m(x, y, t) = Id +
10 y
t
0
0x
0
0
0
Consider the subgroup of H AI, G:= {(x, y, t) E H;,, : xl = yl}. Since the condition xl = y1, is stable for the group operation, it is clear that G is a subgroup. We consider the left-invariant metric h on G, induced by the standard metric on its Lie algebra n
n
2(dxl)2 + E(dx4)2 + >(dy')2 + (dt)2 i=2
i=2
G is a homogeneous space. We let n I: ((x ,x2 ,... x ), (x ,y 2,... ,y n), t) 1
1
1
2
n (_X1'
Y
2
n
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L. Almeida, Y. Ge
It is easy to see that I is a reflection in (G, h). The At are defined by 2
where xt = (t, 0, ... , 0) E Rn. We can see that the one-parameter family Consider the set M C G given by at is a subgroup of
and let I= A_1 o I o Al
M = {(x, y, t) E G : Ix12 + IiI2 + t2 < 2},
We can check that M has all the properties required for applying Theorem 1.1 and thus obtain the desired symmetry results.
Remark. In the three examples given above we considered homogeneous spaces, thus their isometry group was a Lie group g. All we did was to consider an element a in 0, the Lie algebra of C, and to let At = exp(ta). We always had a transversality condition like (4.1). More generally, we can also consider cases where instead of a fixed a E 6, we take a C1 curve a(t)
in 0. 4.4
Product of manifolds
We consider spaces of the form M = M 1 x M2 C N1 X N2 = N, where M 1 satisfies the conditions of Theorem 1.1 and M2 is a connected Riemannian manifold with compact closure (with or without boundary). This is a simple example of a situation where we have symmetry only along some directions
(those of ,M1), but not necessarily all directions (we made no symmetry hypothesis for M2). We remark that since we made no special assumptions
on the topology of the ambient manifolds Nl and N2, we may consider subdomains M = M1 X M2 in, for instance, Rm x R" or Sm x Hn, having the properties mentioned above. Theorem 1.1 remains valid in this setting. In fact, if I,, A' , and VV1 are defined as before for the manifold M I, for M we just set
Vt=Vtl x.M2, At=A' ®Idg2
,
It=It ®IdM
.
For instance, we may obtain directional symmetry results for B x [-1, 1J"
or B x Sn, where B is a convex geodesic ball in S" or H", or is as in Example 4.3. Moreover, we may consider certain subdomains of N1 xN2i as for instance
in Sn x Sm, denoting its points by X = (x1, ... , xn, x"+1, ... xn+m where (x1, ... , xn) are stereographic coordinates on S" and (xn+l, xn+m) are stereographic coordinates on Sm, we may consider M = {X
IXI < 1}. In this case, applying Theorem 1.1, we obtain that if u is a positive solution of (1.2), i.e., Lu = f (u), on M, then it is of the form
u = f(rl,r2) , where rl = I(x1,... ,x")I , r2 =
,xn+m)I
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Monotonicity result
5
Let M C H be a submanifold such that I8M I = 0 and M is compact and can be decomposed as the union of a family of hypersurfaces as follows:
there exists a hypersurface without boundary U C H such that M = UtE(o,11 At(Ut) where the Ut C U are compact hypersurfaces with boundary
for 0 < t < 1, depending C' on t, and At
,
t E R, is a C' group of
translations on H, transversal to U at t = 0. More precisely we demand the following
Folliation conditions:
8At(x)
TT U = T= M ,
It = o
3e > 0 : Vx E U, dt E (0,1 + e),
dx E U
(5 . 1)
,
At(x) V U
(5.2)
.
Directional convexity conditions:
At(x) E M} is a closed interval which we denote [tI(x),t2(x)], and It E (0,1) : At(x) E .M} is an open interval which we denote (r, (x), T2 (x))
Vx E U, it E [0,1]
:
(5.3)
As a consequence of these assumptions, we may use t as first coordinate in M. A case where these conditions are satisfied is the one described in the remark at the end of Section 4.3, where N is a homogeneous space.
The main result of this section is
Theorem 5.1. Suppose that u E HI(M) fl CO ()R), is a solution of (1.2) with boundary condition ulaM =.0 such that dx1i x2 E OM, if 3t(xl, x2) > 0 such that x2 = Atxl, then q(x2) > qb(xl), (5.4)
VT E (O,t(xl,x2))
,
46(xl) < u(A,.(x1)) < 4(x2)
(5.5)
.
Then, the function u is increasing along the trajectories of the group action, i. e.,
Vx E M, Vt > 0 s.t. At(x) E M, we have u(At(x)) > u(x)
.
(5.6)
The proof of this result is quite similar to that of Theorem 1.1 and can be found in [2].
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L. Almeida, Y. Ge
REFERENCES [1] A. Alexandrov, Uniqueness theorem for surfaces in the large, Vestnik Leningrad Univ. Math. 11 (1956), 5-17. [2) L. Almeida and Y. Ge, Symmetry results for positive solutions of some elliptic equations on manifolds, Global Anal. Geom 18 (2000), 153-170.
[3] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Bras. Mat. 22 (1991), 1-39. [4] X. Cabre, Contribuitions h 1'etude des equations aux derivees partielles elliptiques et paraboliques, These d'habilitation de l'universite de Paris VI, 1998.
[5] L. Damascelli, Some remarks on the method of moving planes, Differential Integral Equations 11 (1998), 493-501.
[6] L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. I.H.P. Anal. Nonliniaire 15 (1998), 493-516. [7] B. Gidas, W.M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209-243.
[8] B. Gidas, W.M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in RN, Adv. Math., Suppl. Stud. 7A (1981), 369-402.
[9] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd Edition. Grundlehren Math. Wiss. 224, SpringerVerlag, Berlin and New York, 1983. [10] S. Kumaresan and J. Prajapat, Serrin's result for hyperbolic space and sphere, preprint. [11] S. Kumaresan and J. Prajapat, Analogue of Gidas-Ni-Nirenberg result for domains in hyperbolic space and sphere, to appear in Rendiconti Dell'Instituto Di Matematica, Dell'Universitd Di Trieste, Nuova Serie.
(12) P. Padilla, Symmetry properties of positive solutions of elliptic equations on symmetric domains, Applicable Analysis 64 (1997), 153-169.
[13] J. Serrin, A symmetry problem in potential theory, Arch. Rat. Mech. Anal. 43 (1971), 304-318.
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[14] S. Thangavelu, Harmonic Analysis on the Heisenberg Group, Progress in Mathematics 159, Birkhauser, Boston, 1998.
Lufs Almeida and Yuxin Ge Centre de Mathematiques et de Leurs Applications Unite associee au CNRS URA-1611 Ecole Normale Superieure de Cachan 61 Avenue du President Wilson, 94235 CACHAN Cedex France
Fax: 33 1 47 40 59 01 almeida@cmla.ens-cachan.fr
Yuxin Ge Departement de Mathematiques, Faculte de Sciences et Technologie Universite Paris XII-Val de Marne 61, avenue du General de Gaulle, 94010 Creteil Cedex France ge@cmla.ens-cachan.fr
Nielsen Number and Multiplicity Results for Multivalued Boundary Value Problems Jan Andres ABSTRACT The generalized Nielsen number is defined for compact admissible (multivalued) self-maps on connected ANR-spaces. This number provides a lower estimate of the number of coincidences rather than of fixed points. Nevertheless, the multiplicity results to corresponding solutions can be obtained in this way for a rather general class of multivalued boundary value problems. Two types of concrete applications are presented.
1
Introduction
For the sake of simplicity, suppose that X is a Banach space, D is an open
bounded subset of X, f : D --+ X is a continuous map such that f (D) is compact and the frontier 8D = D\D of D is fixed-point free. Then it is well-known that we can associate to the triple (f , D, X) an integer ix (f, D) = deg(id - f, D, 0), called the fixed point index of f : D --i X, where deg stands for the LeraySchauder degree of (id - f) on D with respect to 0. Unlike the degree, the index can be also defined when e.g., X is a closed convex set with empty interior in a Banach space. Roughly speaking, the index can be regarded as an algebraic count of the number of fixed points of f in D. An analytically easiest case appears for f E C1(D) n C(D), when X =
R", 0 E R"\(id - f)(OD) and the set {x E (id - f)-1(0)} consists only of regular points, i.e., Sid- f (x) # 0. In such a simple situation, the Leray1991 Mathematics Subject Classification: 34B15, 34C25, 47H04, 47H10, 54H25, 55M20, 58006. Key words and phrases: Nielsen number, number of coincidences, admissible pairs, multiplicity results, differential inclusions.
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Schauder degree reduces to the Brouwer one, and so we have
deg(id - f, D, 0) _
E
sgn 17id_ f(x).
xE(id-f)-1(O)
In Figure 1 below, one can observe an interesting phenomenon related to a homotopy deformation, namely although
i(fi, D) = i(fz, D) = i(f3, D) = 1, each mapping has a different number of fixed points.
Figure 1. So, in spite of the fixed-point index being a topological invariant, some fixed points may disappear under the influence of homotopy. Therefore, we can only deduce that a nontrivial (nonzero) index implies the sole existence of a fixed point. On the other hand, it was an ingenious idea of Jakob Nielsen in 1927 to initiate the fixed-point theory allowing us to make a lower estimate of the number of fixed points preserved under a homotopy deformation. The most popular definition of its central notion, the Nielsen number, is due to another founder of this theory, Franz Wecken, in a series of papers in 1941-2 (for more details and appropriate references see e.g. [6]). A more recent version, suitable for applications, deals with compact self-
maps W on metric ANR-spaces X. We say that x, y E Fix(V) = {x E X : V(s) = x} are Nielsen-related (written x y) if there exists a path u : [0, 1] --+ X so that u(0) = x, u(1) = y and u, W(u) are homotopic
-
keeping endpoints fixed. Two nonrelated fixed points x, y are sketched in Figure 2.
Nielsen Number and Multiplicity Results for Multivalued BVP
177
Figure 2. One can readily check that the relation ".
." is an equivalence. Furthermore, it is known that each fixed-point class is open in Fix(V) and hence
is by the hypothesis compact) the number of such classes is (the set finite. If, for a Nielsen class C C Fix(cp), the associated fixed-point index i(cp,C) (which is well-defined-see e.g. [11]) is nontrivial, then C is called essential. The Nielsen number N(V) is thus the number of essential Nielsen classes.
Its most important properties are that N(ip) < a Fix(cp) and that N(V) is invariant under an appropriate sort of homotopy (for more details see e.g., [6]). The multivalued analogy for lower-semi-continuous maps has been given
in [13], but for the majority of interesting applications to differential systems (which is our interest here), we need the one for so-called admissible mappings (in the sense of 111]), which are in addition upper-semicontinuous. This is, however, a much more delicate case, treated therefore in the next two sections. The last two parts are devoted to applications. Let us also note that the related historical remarks, jointly with the appropriate references, can be found e.g., in [7].
2
Topological preliminaries
Let H be the tech homology functor with compact carriers and coefficients
in the field of rational numbers Q on the category of metric spaces. A nonempty space X is called acyclic if we have
HE(X)=
Q fori=0, 0
for i#0.
We recall that a compact space X is said to be an R6-set if it is an intersection of a decreasing sequence of compact contractible spaces. It follows from
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the continuity property of the functor H that every Ra-space is acyclic; in particular, every compact contractible space is acyclic. A single-valued continuous map p : X -+ Y is called a Vietoris map if p is proper and, for every x E X, the set p-1(x) is acyclic. An upper-semi-continuous map cp : X Y+ Y is called acyclic if p(x) is acyclic for every x E X. For a multivalued map V : X -+ Y, we shall consider the graph r,D {(x, y) y E cp(x)} of cp and two natural projections:
X
rV IVY,
where p. (x, y) = x, q. (x, y) = y. X is acyclic, then p,, : r -+ X is a Vietoris Observe that if cp : X map, and we have W(x) = q,(p;'(x)) for every x E X. The above observation allows us to give the following (cf. [11])
Definition 2.1. A multivalued map cp : X M+ Y is called admissible if there exist a compact metric space r and two (single-valued) continuous maps p:r-+Xand that (i) p is a Vietoris map,
(ii) V(x) = q(p-1(x)) for every x E X; then (p, q) is called a selected pair for cp and we write (p, q) C W.
Recall (see e.g. [111) that any acyclic map W: X w X is admissible, and so we have (p., q,p) C V.
Lemma 2.2. ([111) If cp : X -+ Y and
Y M+ Z are admissible maps, then the composition 0 o cp : X -+ Z of cp and 0 is also admissible.
Remark 2.3. In what follows, by a multivalued map we shall mean a pair (p, q), x r --!+ Y, of single-valued continuous maps with p to be a Vietoris map. Since any admissible map can be represented by an associated pair (p, q), for such a pair x r q+ Y we let: C(p,q) _ {y E r I p(y) = q(y)} and
Fix(p,q) = {x E X X E q(p-1(x))}. Of course, C(p, q) # 0 if and only if Fix(p, q) # 0. Roughly speaking, the set C(p, q) is bigger than Fix(p, q) in general. We recommend [10] for the notion of the generalized Lefschetz number and its properties. Note that for
a pair (p, q), X - r q X, its generalized Lefschetz number is denoted by A(p, q) and we let A(p, q) = A(q+ op.-'), provided A(q, o p.-1) is well-defined; in that case (p, q) is called a Lefschetz pair.
Nielsen Number and Multiplicity Results for Multivalued BVP
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Now, we recall some elementary facts concerning absolute neighbourhood
retracts. Definition 2.4. A space Y is called an absolute retract (an absolute neighbourhood retract) whenever, for any metrizable X and closed A C X, each f : A -+ Y is extendable over X (over an open neighbourhood U of A in X). We use then the notation: Y E AR (Y E ANR). The following theorem characterizes ARs (ANRs) in terms of retraction property (up to a homeomorphism):
Theorem 2.5. A metrizable space is an AR (an ANR) if and only if it is a retract of (some open subset of) some normed space.
Proposition 2.6. If X E ANR and U is an open subset of X, then U E ANR.
Until the end of this section, a pair (p, q), X
r
Y, representing
a multivalued map will be denoted by cQ, i.e., by a multivalued map cp : X Y we always understand the pair (p, q) of the above type (cp(x) = q(p-' (x)). Since p is Vietoris and q is continuous, our multivalued map cp is always upper-semi-continuous with compact values (see [11]). A multivalued map cp : X M+ Y is called compact if W(X)
U WW xEX
is a compact subset of Y. In what follows, cp E K(X) denotes cp : X Y+ X being compact. We recall a particular case of the result proved in [101.
Theorem 2.7. Let X E ANR and cp E K(X). Then cp is a Lefschetz map and if A(W) # 0, then Fix(V) # 0. In particular, if X E AR and cw E K(X), then A(W) = 1, and so Fix(W) # 0.
3
Nielsen number for ]K-maps
It has been shown in [101 (cf. Theorem 2.7 above) that, for any multivalued K-map on an ANR-space, the Lefschetz number A(p, q) E Z is defined and A(p, q) # 0 implies the existence of a coincidence point z E IF (p(z) = q(z)) of the pair (p, q).
Moreover, we have constructed in [7] the Nielsen number N(p, q) for a class of multivalued self-maps on a compact ANR. N(p, q) is a nonnegative integer, a homotopy invariant and OC(p, q) > N(p, q), where pC stands for the number of coincidence points.
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In this section, we generalize (as in (8]) this construction: we drop out the compactness assumption on X by replacing (p, q) to be a 1K-mapping. As in the single-valued case, the definition of a Nielsen number is done in two stages: At first, C(p, q) is split into disjoint classes (Nielsen classes) and then we define essential classes.
Fix a universal covering px : X - X. We define I' = {(a, z) E X x r f -' X by p(i, z) = x.
p. (Y) D p(z)} (pullback) and the map p :
Property A. For any x E X, the restriction ql
: ql P- 1 (x)
: p-1(x) -+ X
admits a lift q, making the diagram
commutative.
Remark 3.1. Note that a sufficient condition of guaranteeing property A is, for example, that p-1(x) is an oo-proximally connected set, for every x E X. It is well-known (see [12]) that any oo-proximally connected subset of an ANR-space is an Ra-set. Lemma 3.2. ([8]) If (p, q) satisfies (1K+A), then there is a lift q : r -> X making the diagram
X_*-I'X P
9
Px! X-6 P
I Pr
[Px
r- 9'X
commutative.
Property B. There is a normal subgroup H C Ox = {a : X -+ X Ipxa = px } of finite index (8x /H-finite), invariant under the homomorphism q!p (q!p'(H) C H), where p! : ex - Or, p!(a)(x, z) = (ax, z), and q! : Or Ox,
Remark 3.3. In particular, if X is a connected space such that the fundamental group ir1(X) of X is abelian and finitely generated, then X satisfies property B (see [14]). Note also that if (p, q) is admissibly homotopic to a single-valued map f, then property B holds true (see [7]). Let us notice that (K + A + B) makes the diagram
XH 9H rH PH XH PxHj
jPrH
Pxrf
Nielsen Number and Multiplicity Results for Multivalued BVP
181
commutative, where pXH : XH - X is a covering corresponding to the normal subgroup H a 9X c n1X and rH is a pullback. As above, we can define homomorphisms pH : 9XH-i erH, qH! : OrH : 9XH, where 8XH = (a: XH -' XH I PXHa = PXH} Lemma 3.4. ([8], Lemma 5.1) We have: (i) C(P, q) = U«EOX,, PryC(PH, aqH),
(ii) if PrHC(PH,agH) n prHC(PH,QgH) is not empty, then there exists
ayE8XH such that (iii) the sets prHC(pH, aqH) are either disjoint or equal.
Thus, C(p, q) splits into disjoint subsets PrHC(pH, a qH) called Nielsen classes modulo a subgroup H. Now, we shall define essential classes. We consider the diagram XH
X0
IPXH
IPrH
PXH 1 P
r
q
>X
Lemma 3.5. ([8]) The multivalued map (j5H,gH) is a 1K-mapping.
Definition 3.6. A Nielsen class mod H of the form prHC(pH,a'H) is called essential if A(pH, agH) 36 0.
By Lemma 6.5 in [7], this definition is correct, i.e., if PrHC(PH, aqH) = PrHC(PH,QgH), then
A(pH, aqH) = A(PH, #gH)
Definition 3.7. The number of essential classes of (p, q) mod a subgroup H is called the H-Nielsen number and is denoted by NH (p, q). Now, we can give two main theorems of this section.
Theorem 3.8. ([8]) A multivalued map (p, q) satisfing (1K + A + B) has at least NH (p, q) coincidence points.
Theorem 3.9. ([8]) NH(p, q) is a homotopy invariant (urith respect to the homotopies satisfying (K + A + B) ).
Remark 3.10. In [8], we have a generalization of this theory to not necessarily compact maps, so-called compact absorbing contractions.
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182
4
Application to differential inclusions
Now, we will apply the Nielsen theory developed in the foregoing section for obtaining the multiplicity results to differential inclusions
X' E F(t,X),
(4.1)
where F : J x RI µ RI is a set-valued (upper) Caratheodory mapping (for the definition and more details see e.g., [12]) and J is an arbitrary (possibly infinite) real interval. By a solution X(t) of (4.1), we always mean a locally absolutely continuous function X(t) satisfying (4.1) for a.e. t E J. For the sake of simplicity, the H-Nielsen number defined above will be denoted without the index H. Considering (4.1) with the constraint, namely
X E S C C(J, R"),
(4.2)
where S is a nonempty subset, we start with the following essential result (see [4], Theorem 2). Let us recall that the appropriate topology in C(J, R') is the one of the uniform convergence on compact subintervals of J.
Lemma 4.1. Let G : J x R" x R'a + R" be a Caratheodory mapping and assume that: (i) there exists a subset Q of C(J, R") such that, for any q E Q, the set T(q) of all solutions of the problem fX' E G(t, X, q(t)),
1 XES, on J E R is nonempty,
(ii) T(Q) is bounded in C(J,R' ), i.e., there exists a positive (singlevalued) function.0 : J -- Rn such that Irr(t)I < qS(t) for all t E J, T E T(q) and q E Q,
(iii) there exists a locally Lebesgue integrable function a : J --> Rn such that (G(t, X(t), q(t)) 1 < a(t) a.e. in J,
for any pair (q, X) E rT, where rT denotes the graph of T. Then T(Q) is a relatively compact subset of C(J, Rn). Moreover, the multivalued operator T : Q M+ S is upper-semi-continuous with compact values
if still
(iv) T(Q) C S. It will be also convenient to use the following definition.
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183
Definition 4.2. We say that the mapping T : Q M+ U is retractible onto Q, where U is an open subset of C(J, R") containing Q, if there is a retraction r : U -+ Q and p E U \ Q, r(p) = q implies that p ¢ T(q). Its advantage consists in the fact that, for a retractible mapping T : Q U onto Q with a retraction r in the sense of Definition 4.2, its composition with r, rIT(Q) o T : Q -' Q, has a coincidence point q E Q if and only if is a coincidence point of T. The following theorem characterizes the matter (see [8]).
Theorem 4.3. Let the assumptions of Lemma 4.1 be satisfied, where Q is a closed connected subset of C(J, R") with a finitely generated abelian fundamental group. Assume, furthermore, that the operator T : Q M+ U, related to problem (4.3), is retractible onto Q with a retraction r in the sense of Definition 4.2 and with R5-values. At last, let G(t, c, c) C F(t, c)
take place a.e. in J, for any c E R. Then the original problem (4.1)-(4.2) admits at least N(rIT(Q) oT( )) solutions belonging to Q. The following statement, which has been also proved in [8], is already suitable for applications.
Theorem 4.4. Consider the boundary value problem
J X' + A(t)X E F(t, X),
L(X)=9, and the associated linear homogeneous problem
fX' + A(t)X = 0, L(X) = 0,
on a compact interval J. Assume that A : J - R"2 is a single-valued continuous (n x n) -matrix and F : J x R" -+ R" is a Carathe odory productmeasurable mapping satisfying I F(t, X )j 5 p(t)(IX I + 1),
for all (t, X) E J x R", where p : J - [0, oo) is a suitable Lebesgueintegrable bounded function. Furthermore, let L : C(J, R") R" be a linear operator such that problem (4.5) has only the trivial solution on J. Then the original problem (4.4) has N(rlT(Q) o T(-)) solutions, provided there exists a closed connected subset Q of C(J, R") with a finitely generated abelian fundamental group such that
(i) T(Q) is bounded,
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(ii) T(q) is retractible onto Q with a retraction r in the sense of Definition 4.2,
(iii) T(Q) C {X E C(J, R") : L(X) = 9), where T(q) denotes the set of (existing) solutions to the linearized system
J X' + A(t)X E F(t, q(t)),
1L(X) = e. Remark 4.5. If Q is additionally compact and T(Q) C Q, then it need not have a finitely generated abelian fundamental group (see [5]). In the single-valued case, the situation becomes still easier (cf. [6], [9)).
As a nontrivial example, consider the system of 2k inclusions,
f xii+aixi E +gi(t,Xi,Y), 1ys' +biyi E fi(t,Xi,Yi)xil/") +hi(t,Xi,Yi), i = 1,...,k,
(4.6)
where Xi = (x1,. .. , xi), Yi = (yl,... , yi), ai, bi are nonzero numbers and mi, ni are odd integers with min(mi, ni) > 3. Let suitable positive constants E0,i, F0,i, Gi, Hi exist such that
Ifi(t,Xi,Yi) < Fo,i, Ihi (t,Xi,Y)I 5 Hi hold for a.a. t E (-oo, oo) and all (Xi, Yi) E R2i, i = 1, ... , k. lei (t,Xi,Yi)I < Eo,i, Igi(t,Xi,Yi)1 <_ Gi,
Futhermore, assume the existence of positive constants eo,i, fo,i, 51,i, b2,i
(i = 1, ... , k) such that 0 < eo,i < ei (t, Xi, Y )
(4.7)
for xi ? -bl,i, yi >- 52,i and a.a. t as well as for xi < 5i,i, yi S -52,i and a.a. t, jointly with 0 < fo,i <- fi (t, Xi, Yi)
(4.8)
for xi > 51,i, yi <- b2,i and a.a. t as well as for xi < -bl,i, yi > -52,i and a.a. t. Another possibility is that (4.7) holds for xi < Si,,, yi > 62,i and a.a. t as well as for xi > -61,i, yi < -52,i and a.a. t and that (4.8) holds at the same time for xi > 51,i, yi >- -b2,i and a.a. t as well as for xi < -61,i, Yi
52,i and a.a. t.
Theorem 4.6. ([8]) Let suitable positive constants 51,1,52,, exist such that the inequalities (i = 1, ... , k) I
I
leO,% 2 "t - Gi1
1
fo,ib21/inc 1
bi I
- HiI
al,i 62,i >
Q' (e'o'i)
)ttf,
,"
(4.9)
Nielsen Number and Multiplicity Results for Multivalued BVP
185
are satisfied for constants eo,i, fo,i, Gi, Hi estimating the product-measurable,
semicontinuous in (Xi, Yi) E R2i, for a.a. t E (-oo, oo), multifunction ei, fi, gi, hi as above, for constants ai, bi with aibi > 0 and for odd integers mi, ni with min(mi, ni) > 3. Then system (4.6) admits at least 2k entirely bounded solutions. In particular, if the multifunctions ei, fi, gi, hi are still w-periodic in t, then system (4.6) admits at least 3k w-periodic solutions, provided the sharp inequalities appear in (4.9).
5
Multivalued Poincare operators approach
The main advantage of this approach consists in a possibility to study the solutions of (4.1) with constraints as the coincidence points of the Poincar6 self-maps on finite(!) dimensional tori. In [3], we have proved that, under the natural assumptions used below, the associated translation operators as well as the first-return maps (i.e., the Poincar6 ones) are admissible in the sense of Definition 2.1 and admissibly homotopic to identity. Thus, we can employ
Theorem 5.1. ([71) Let T'
r Q , F' be an admissible pair. Then there exists a single-valued continuous map p : Tn - Tn (more precisely, a pair representing this map) to which (p, q) is admissibly homotopic and (p, q) has at least N(p) = IA(p)I coincidences. Working with the related translation operators, we can arrive at
Theorem 5.2. ([7)) Assume that
F(t,...,x, +1,...) = F(t,...,xj,...)
for j = 1,...,n,
(5.1)
where X = (x1, ... , xn) and consider system (4.1) on the set Ro x Tn, where Tn = Rn/Zn. Then (4.1) admits for every positive constant w at least N(7-1) solutions X(t) such that
X(0) = 7i(X(w)) (mod1), where 7-l is a continuous self-map on Tn and N(1-1) denotes the associated Nielsen number.
As a consequence, we easily obtain for 71 = -id
Corollary 5.3. If additionally to assumptions of Theorem 5.2,
F(t + w, -X) _ -F(t, X), then system (4.1) admits at least 2n anti-w-periodic (or 2w-periodic) solutions X (t) on Tn, namely X (t + T) _ -X (t) (mod 1).
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J. Andres
Remark 5.4. Unfortunately, we cannot put f = id, because (cf. [7]) N(id) = IA(id)I = IX('><")I = 0,
where X denotes the Euler-Poincare characteristic.
The application of first-return maps leads to
Theorem 5.5. ([2]) Let (5.1) be satisfied and assume still n
n
E f,(t, X) > 0 or
inf R+x1o,11"
i-1
sup
E fi(t, X) < 0,
(5.2)
R: xto,']" i_1
where F = (fl,... , fn)T is bounded. Then system (4.1), considered on the set Rj x T", admits at least N(7-l) solutions X (t) such that
X(0) = 7{(X(r)) (modl)
for some r > 0
and, moreover, X(0) E T"-1 as well as 1f(X(r)) E T"-1, where If is a continuous self-map on Tn-1.
Corollary 5.6. ([1]) Let an autonomous system (4.1), where n > 1, determine a a-equivariantflow on T", i.e.,
fi(xl,...exn)= fi(xi,...,xn)xl,...,xi-1)
for i = 1, ... , n.
Assume, furthermore, (5.1) and (5.2) forF(t, X) - F(X). Then autonomous system (4.1), where n > 1, admits at least n periodic solutions X(t) such that X(t+n-r) __ X(t) (modl) for some r > 0. REFERENCES [1]
J. Andres, Splay-phase and anti-splay-phase orbits for equivariant setvalued flows on tori, Dif. Eqns Dynam. Sys. 4 (1996), 89-98.
[2] J. Andres, A target problem for differential inclusions with state-space constraints, Demonstr. Math. 30 (1997), 783-790.
[3] J. Andres, On the multivalued Poincare operators, Topol. Meth. Nonlin. Anal. 10 (1997), 171-182. [4] J. Andres, Almost periodic and bounded solutions of Caratheodory differential inclusions, Differential Integral Equations, 12:6 (1999), 887912. [5]
J. Andres, Multiple bounded solutions of differential inclusions: the Nielsen theory approach, J. Differential Equations 155 (1999), 285310.
Nielsen Number and Multiplicity Results for Multivalued BVP
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[6] J. Andres, A nontrivial example of application of the Nielsen fixed point theory to differential systems: problem of Jean Leray, Proceed. Amer. Math. Soc., 128 (2000), 2921-2931. [7] J. Andres, L. G6rniewicz and J. Jezierski, A generalized Nielsen number and multiplicity results for differential inclusions, Topology Appl., 100:2-3 (2000), 193-209.
[8] J. Andres, L. G6rniewicz and J. Jezierski, Noncompact version of the multivalued Nielsen theory and its application to differential inclusions, LN of the Schauder Center 2: Differential Inclusions and Optimal Control, Proceedings of the Banach Center Workshop held in Warsaw, September 27-October 3, 1997 (J. Andres, L. G6rniewicz and P. Nistri eds), 25-42. [9] A. Yu. Borisovich, Z. Kucharski and W. Marzantowicz, A multiplicity
result for a system of real integral equations by use of the Nielsen number, preprint, 1997. [10] G. Fournier and L. G6rniewicz, The Lefschetz fixed point theorem for some non-compact multi-valued maps, Pund. Math. 94 (1977), 245254.
[11] L. G6rniewicz, Homological method in fixed point theory of multivalued maps, Dissertationes Math. 129 (1976), 1-71. [12] L. G6rniewicz, Topological approach to differential inclusions. In Topological Methods in Differential Equations and Inclusions, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 472 (A. Granas and M. Frigon eds.), Kluwer Acad. Publ., Dordrecht, 1995.
[13] J. Jezierski, The Nielsen relation for multivalued maps. In Topological Methods in Differential Equations and Inclusions, SERDICA Bulg. Math. Publ. 13 (1987), 174-181. [14] E. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.
Jan Andres Department of Mathematical Analysis, Faculty of Science Palacky University Tomkova 40, 779 00 Olomouc-Hej6in Czech Republic andres@risc.upol.cz
Bifurcation Theory and Application to Semilinear Problems near the Resonance Parameter David Arcoya Jose L. Gamez 1
Introduction
In this note, we present without proof some of the main results obtained in [6] about Bifurcation Theory and related problems: resonance and the antimaximum principle, and we prove also some new applications about existence of solutions for semilinear elliptic problems near the resonance parameter. Specifically, we consider here the semilinear elliptic boundary value problem
-Du(x) = \m(x)u + g(A, x, u) u(x) = 0
if x E SZ,
if x E 852,
for a bounded domain SZ C RN with sufficiently smooth boundary 90 and A E R, under the hypothesis: (H) There exists r E (N, +oo] such that m E L''(S2) with m+ = max{m, 0} 0- 0 and g: R x fl x R -- R is a continuous function. Consider the Sobolev space X = W02"(11) endowed with the usual norm 11. Since r > N, X is compactly embedded into C1(S2). Problem (1.1)
can be written as
FA(u)-u-ALu-N(A,u)=0, uEX, where Lu = (-0)-1(mu), N(A,u) = (-0)-1(g(A,x,u)), for every u E X. Hence, if g satisfies the condition Supported by Acci6n Integrada Spain-Italy H11997-0049, by D.G.E.S. Ministerio de Educaci6n y Ciencia (Spain) PB98-1283 and by E.E.C. contract n. ER.BCHRXCT940494.
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D. Arcoya, J. L. Garnez
(go) liml,l.o g(A, x, s) Is = 0 uniformly for x E 12 and A in bounded sets of R,
then the operator N satisfies lim N(A, u) = 0 IIuII-.0
uniformly for A in bounded sets of R.
Hull
Similarly, if the condition
(g.) liml,l-+. g(A, x, s)/s = 0 uniformly for x E fl and A in bounded sets of R
holds, then NH Il++lllm oo
ullu)
-0
uniformly for A in bounded sets of R.
Consequently, by the results of Krasnoselskii [141, hypotheses (H) and (go) imply that the first positive eigenvalue Al (m) of the eigenvalue problem [11] with weight m associated to the Laplacian operator is a bifurcation point from zero, that is, there exist a sequence {An} of real numbers converging to Al (m) and a sequence {un} converging to zero in X, with un # 0 such that (An,un) is a solution of (1.1). In addition, by the Global Bifurcation Theorem of Rabinowitz [19], we deduce that the bifurcation is global in the sense that the connected component of the closure of E = {(A, u) E R x X / Fa(u) = 0, u # 0} containing (Al (m), 0) is either unbounded or it contains another bifurcation point (At (m), 0), 10 1. Similarly, under hypotheses (H) and (gam), Rabinowitz [20] proved that at Ai(m) there is a Global Bifurcation from infinity.
In both cases, notice that for any sequences {(an, un)} C R x X converging to the bifurcation point, such that Fa (un) = 0, the normalized sequence wn = un/IIunII satisfies
wn = -AnO" 1(mwn) +
A-1
(9(Anxufl)) = 0IIuJI
Using that the inverse of the laplacian operator is compact, we deduce that there exists a subsequence {wn,, } converging to some function 40 E X with 0 = Ai(m)0-'(m4,). Therefore, 0 is a normalized eigenfunction of the laplacian operator associated to the eigenvalue A, (m). Since this eigenvalue is simple, we deduce that 0 = f(&1, where ¢1 denotes a positive associated eigenfunction with norm one in X. Thus, if 0 = +01, then the convergence
wn,, - 0 in C' (U) implies that unk > 0, in CI for nk large enough. On the contrary, if 0 = -41, we deduce un,< < 0, in CI for nk large enough. We distinguish the cases by calling the first one a bifurcation from +oo and the second one a bifurcation from -oo. Actually (see [6]), it is possible to prove that if we consider the conditions
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191
(g+oo) Via-.+oo g(A, x, s)/s = 0 uniformly for x E SZ and A in bounded sets of R,
(g_<,) lim,._. g(A, x, s)/s = 0 uniformly for x E Q and A in bounded sets of R, then
(g+,,,,)
.
bifurcation from +oo at A, (m)
and
(g_a,) = bifurcation from -oo at A1(m). Notice that the above bifurcation results are not complete. Indeed, we know that, for instance, there exists a continuum emanating from (A 1(m), 0) and this means an existence result for (PA). However, the question is: What are the A's for which we have existence of nontrivial solutions? Is it a right interval? A left interval? Both? A careful study of the behavior of the continuum of nontrivial solutions emanating from the bifurcation point is needed to complete the description of the set of A's for which problem (PA) has nontrivial solutions. This study is perhaps easier for bifurcations from zero. Indeed, in this case, if g(A, x, s) is positive in a right neighborhood of s = 0, then the bifurcation of positive solutions is to the left of the bifurcation point. Similarly, if g is negative in a right neighborhood of s = 0, we obtain existence of positive solutions to the right. Therefore, the behavior of the positive solutions near the bifurcation point is decided by the sign of g(A, x, s) near s = 0+. The behavior of bifurcations from infinity at A, (m) has been studied by A. Ambrosetti and P. Hess [3]. The authors prove that the bifurcation from
+oo is to the left provided that lim inf,-+oo g(A, x, s) > E > 0. (Similarly, if lim sup.-+. g(A, x, s) _< -e < 0, then the bifurcation from +oo is to the right.) One can ask: Is it sufficient to know the sign of g near s = +oo to decide if the bifurcation is to the left/right? The following result gives a partial answer to this question for some kind of nonlinearities g:
Theorem 1.1. ([6]) Assume (H), (g+oo) and that there exist B E L'(Q), c < 2 - .1 and so > 0 such that jg(A, x, s)sa I < B(x) for a. e. x E Sl, for all s > so and for A in a neighborhood of A1(m). If there exists the pointwise limit Aa(x) := lira g(A, x, s)sa with (A,a)-'(Ai (m),+oo)
Aa0i_a
fn
34 0,
then the bifurcation of positive solutions at A1(m) from infinity is to the left if such integral is positive, and to the right if negative. In other words, for positive solutions (A, u) near the bifurcation point at infinity, one has sgn [A1(m) - A] = sgn [frl
a,0'-aJ A
.
D. Arcoya, J. L. Gamez
192
Remarks 1.2. 1) This first theorem is only a simple case of a more general result given in [6]. We point out that bifurcations from infinity or from zero at higher eigenvalues are also studied in that paper. In addition, the
extension of the results to the case of quasilinear operators like the plaplacian improves the previous results in [2]. 2) Observe that we have treated here essentially the case g(A1, x, +oo) = 0, In [6], we consider also the case in which g(A1 i x, +oo) is orthogonal to 01.
3) Observe that if, for instance, r = +oo, roughly speaking, the above theorem says that if g(A, x, s) is like C/sa for large s, then the sign of C decides the direction of the bifurcation from infinity provided that a < 2. A counterexample can be seen in [6], showing that the above assertion is false, in general, for a > 2. In the limit case a = 2, the assertion is true. Indeed, for this case we prove:
Theorem 1.3. (The case a = 2 in the bifurcation from infinity) Assume (H) with r = +oo, (g+.) and that there exist co > 0, so > 0 and b E Li B(SI) such that g(,\, x, s)s2 < b(x) (respectively g(A, x, s)s2 > b(x))
for A in a neighborhood of A, (m), and s > so. If, in addition one has that b(x) < -e < 0 in SIo, a neighborhood of on in SZ (respectively b(x) > co > 0 in SIo), then the bifurcation of positive solutions from infinity at A = al (m) is to the right (respectively to the left).
2
Applications to resonant problems and the antimaximum principle
As a consequence of the study of the bifurcations from infinity we can obtain
several results about resonant problems and the antimaximum principle. Let us begin by recalling the Antimaximum Principle [10, Theorem 2] of Clement and Peletier. It states that "given m E Lr(SI), if r > N, then for every positive function h E L''(SI) there exists e = e(h) > 0 such that every solution (A, u) of
-Au = Am(x)u + h(x),
u=0,
if x E SI,
ifxEBSI,
with Ai(m) < A < A1(m) + e, satisfies u < 0 in SI".
The bifurcation point of view of (2.1) helps to understand this fact. Indeed, as a consequence of our bifurcation results, we can recover either an Antimaximum type result or a local Maximum Principle type result:
Theorem 2.1. Let r > N and m E L''(SI). For every h E L''(fl), there exists e =re(h) > 0 such that
1. If
n fn
h41 < 0, then every solution (A, u) of (2.1) satisfies
Bifurcation Theory and Application to Semilinear Problems
193
(a) u > 0 in ci provided that AI (m) < A < A 1(m) + e, (b)r u < 0 in S2 provided that A1(m) - e < A < A1(m).
2. If
h¢1 > 0, then every solution (A, u) of (2.1) satisfies n in
(a) u<0 in f2 provided that Ai(m) 0 in It provided that A1(m) - e < A < A 1(m). (b)jhcii
3. If
= 0 and m(x) > 0 in S2, then every solution (A, u) of (2.1)
with A 0 Al (m) changes its sign in It.
Remark 2.2. Using similar ideas, we can obtain the Local Antimaximum Principle given in [121 for the p-laplacian operator, and also a Local Maximum Principle. With respect to a resonant b.v.p., we consider specifically the problem -Du = Al (M)M(X)u + g(x, u), u = 0,
if x E It, if x E BSl,
where m and g(A, x, s) := g(x, s) satisfy hypothesis (H) with g bounded, and 3g(x, +oo) = limo g(x, s) (pointwise limit), a
3g(x, -oo) = lim g(x, s) (pointwise limit). 8
-oo
Recall the classical result by Landesman and Lazer [15] related to resonance at the principal eigenvalue A1(m), which states that problem (2.2) (with k = 1) admits at least one solution, provided that one of the following two conditions holds: J 9(x, +oo)¢1(x) dx < 0 <
Jn
9(x, -oo)s61(x) dx,
or
0 > j9(x_oo)cbi(x)dx.
Note that the above sufficient conditions fail if for instance g(x, ±oo) 0. Starting from the work [8] (see also [4]), resonant problems have been studied depending on different "degrees of resonance". Roughly speaking, "the smaller g is at infinity, the stronger resonance is". In this way, Bartolo, Benci and Fortunato defined "strong resonance" if g(x, s) = g(s), g(±oo) _ 0 and f ±. g(s)ds = 0. We remark explicitly that the hypothesis on the previous integral is a global condition on g. In contrast, we understand by strong resonance at ±oo the case that g(x, ±oo) is orthogonal to 01.
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D. Arcoya, J. L. G&mez
Different degrees of strong resonance can be measured according to how fast is the convergence of g(x, s) to the function g(x, +oo) (respectively g(x,-oo)). With this purpose we now establish some useful notation. For 0 < /3 < 2 - 1/r, we denote a-,j(x) = , lim
0[9(x,
s) - 9(x, +00)1-90
if the above limit is pointwise in St and in a dominated way, in the sense +oo)Js'a are dominated by a function in that the functions [g(., s) L'' (SZ) for all s E R. We consider the conditions
f or
n
r
Jn
9(x, +00)01 < 0
g(x, +oo)q5i = 0 and
3,3 E (0, 2 - 1/r) such that
(G+OO
or n
f
0
10
g(x, +oo)oi = 0, r = +oo and [g(x, s) - g(x, +oo)[s2 < -co < 0 for s > so.
9(x, +00)01 > 0
0
or
f g(x,+oo)O1
= 0 and
n
2 3 E (0, 2 - 1/r) such that
(G+00
In> 0
or
f 9(x, +oo)¢1 = 0, r = +oo and [9(x, s) - 9(x, +00)182 > co > 0 for s > so.
In the same manner, for 0 < 0 < 2 - 1/r, we define
bp(x) = lim [9(x, s) - 9(x, -oo)]sA D-'-00
pointwise in ft and L'-dominated. Similarly,
Bifurcation Theory and Application to Semilinear Problems
195
f
of 9(x, -00)01 < 0 or
f g(x, -oo)!¢1 = 0 and n
(G=00
30 E (0, 2 - 1/r) such that
fb(x)<
0
or
f off
n
g(x, -oo)¢i = 0, r = +oo and [g(x, s) - g(x, -oo)]s2 < -Eo < 0 for s < -SO.
9(x, -00)01 > 0
or
9(x, -00)01 = 0 and (G±oo
30 E (0,2 - 1/r) such that f bp(x)0i-0 > 0
n
f
or
g(x, -oo)¢1 = 0, r = +oo and
sl
[g(x, s) - g(x, -oo)]s2 > co > 0 for s < -so.
Th approach this type of resonant problems we follow some ideas close to [17] and we embed the problem (2.2) into a one parameter family of problems as follows:
-Du = Am(x)u + g(x, u),
u=0,
if x E 11,
ifxEcl,
(2.3)
with A E R. Observe that the boundedness of the function g ensures that (gam) holds (and then the bifurcation from infinity occurs for problem (2.3) at every eigenvalue of odd algebraic multiplicity). As we will see, the behavior of the possible bifurcations from infinity at Ak(m) for problem (2.3) determines the existence of a solution for the resonant problem (2.2). The key to relate these two problems is to interpretate the concepts of bifurcations to the left and to the right in the sense of a priori bounds for the norms of the solutions. From this point of view, observe that the bifurcation from oo at A1(m) is to the left (respectively to the right) if there exist c > 0 and M > 0 such that for every solution (A, u) of (2.3) one has A E [Al (m), Al (m) + E]
Ijull < M
(respectively X E [AI(M) - e, A1(m)] = huh < M).
With this characterization and based on our results in the previous section to decide the behavior of the bifurcations, we can deduce the following result:
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D. Arcoya, J. L. GAmez
Theorem 2.3. Suppose that condition (H) holds with g bounded and r = +oo. Assume one of the following two possibilities:
1. function g satisfies (G+,) and (G±.), or 2. function g satisfies (G+00) and (G=0). Then problem (2.2) admits at least one solution.
Remarks 2.4. 1) An interesting example of strongly resonant b.v.p. which is covered by the previous result is the case in which 3eo, so > 0 such that Vs > so, g(x, -s) -eos 2, g(x, s) > eos-2. In such conditions, the resonant problem (2.2) admits at least one solution. Observe that in this case the integrals f ±* g(x, s)ds can take any real value (if they exist), and are not constrained to be zero as in the results in [8].
2) The above technique can also be used to handle the case of resonance at higher eigenvalues. In fact, we can recover the extension of [4], which generalizes the conditions by Landesman and Lazer (15] to the case of higher eigenvalues. Also, we can handle the case of quasilinear operators improving the results in [5, 1, 7, 9].
3
Existence near the resonance parameter
In the last section, we devoted our attention to the existence of solutions for b.v.p. which are resonant (at infinity) at the first eigenvalue Al (m) of the laplacian operator. In order to do it, we have studied the problem (2.3) for A = Ai(m). The aim of this section is to complete the study of (2.3) for A near \I(m). In this way, our following result is related to the previous ones in [1, 16, 18].
Theorem 3.1. Suppose that condition (H) holds with g bounded. Assume
that the function g satisfies (G'.) and (G=.). Then there exists b > 0 such that the problem (2.3) has 1. at least one solution for A E (A I (M), A2 (M)), and
2. at least three solutions for \ E (Ai(m) - b, A1(m)).
Proof. We begin by observing that for every A E (A1(m.),A2(m)) there exists R,\ > 0 such that if r E (0, 1] and u is a solution of
-Au = Arn(x)u + rg(x, u),
u=0,
if x E 11,
ifxEBSl,
Bifurcation Theory and Application to Semilinear Problems
197
then IIull < RA. Indeed, this is easily deduced since A is not a bifurcation point from infinity of (2.3). Thus, by homotopy of the degree,
deg (Fa, BR (0), 0) = deg (I - AL, BR,, (0), 0) = -1 0 0.
This means that, for every A E (Ai (m), .12(m)), (2.3) has at least one solution u with norm lull < Ra. In addition, hypotheses (G+.) and (G: ,) imply (see the previous section) that for a fixed ry E (A1(m),1\2(m)) there exists R > 0 such that, for all A E [A, (m), -y], every solution u of (2.3) satisfies IIuII < R. In particular, by excision and homotopy properties of the degree,
deg (Fa,BR (0),0) = -1 # 0, ba E [Aj(m),7]
and by the continuation property (see [18, Theorem 1 and Corollary 2]), we deduce the existence of 6 > 0 such that the problem (2.3) has, at least, one solution u with IIuII < 2R for A E (,\, (m) - 6,A,(m)). Moreover, the existence of two more solutions can be proved by noting that our conditions (G+.), and (G=00) provide two bifurcations, one from +oo and other from
-oo, both "to the left" of \1(m). Remarks 3.2. 1) An analogous (reversed) result can also be deduced for the case in which the function g satisfies the conditions (G+,) and (G±,)
instead of (G+0) and (G=,). 2) In [18] the authors obtain a similar result for functions g satisfying the condition
ug(x,u)>0 for every uiA 0andxE1l.
(3.1)
We point out that our method applies also in such cases (see formula (7) in [6]), but in the case under consideration, our hypotheses involves only the local behavior at infinity of the nonlinearity g. We now fix our attention on the global behavior of the branches of solutions bifurcating from infinity at A, (m). We consider here the case m - 1, and g(x, 0) > 0 (# 0). Indeed, by applying the comparison results given in [13] (zero is always a sub-solution which is not a solution), it can be proved that the branch bifurcating from +oo contains only positive solutions, while the one bifurcating from -oo does not intersect the closed cone of positive functions in X. Hence, the two branches do not meet. Since no positive solutions exist for large positive values of A, we deduce that globally, the branch of positive solutions bifurcating from +oo "goes to" the left. From the uniqueness of the solution for A << 0 follows that the branch bifurcating from -oo "goes to" the right. The knowledge of this global behavior together with the local analysis near the bifurcation points provide the following result:
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D. Arcoya, J. L. GAmez
Theorem 3.3. Assume condition (H) with m =_ 1 and g bounded and Lipschitz, satisfying g(x, 0) > 0, ($ 0). Then
1. If (G+,) and (G=.) hold, there exists A. < Al such that the problem (2.3) has (a) at least one positive solution for ,\ < A., (b) at least two solutions (one of them positive, and the other one not), for ,\ = A., (c) at least three solutions (one of them positive, and the other two
not), for A.
2. If (G+,) and (G±.) hold, there exists A' > Al such that the problem (2.3) has (a) at least one positive solution for A < A1, (b) at least three solutions (two of them positive, and the other one not), for Al < A < A*, (c) at least two solutions (one of them positive, and the other one
not), for ,\ = A', (d) no positive solutions for ,\ > A*.
Remark 3.4. The case g(x, 0) < 0, (0 0) is covered by applying the previous theorem to the equation satisfied by v = -u. Consequently, some of the found solutions are then constrained to be negative instead of positive. With similar techniques, the case g(x, 0) __ 0 can also be studied. Note that this covers the Mawhin-Schmitt situation (3.1). Now, u =_ 0 is a trivial solution, so we are interested in looking for nontrivial solutions of (2.3). The branches bifurcating from ±oo contain solutions which do not change sign in S I. Moreover, the two branches meet at the zero solution, when the parameter A attains the principal eigenvalue associated to the problem linearized at zero. There are many possibilities for the study of the multiplicity of nontrivial solutions in this case, and we consider here only the following example.
Theorem 3.5. Assume condition (H) with g bounded and regular with in L'-(Q). respect to s, g(x,O) 0, and there exists go(x) := lim 9(x's) e-o s We consider the eigenvalue Al = A1(-0 - 9o) (A1 = Al - 9'(0) in the autonomous case). Then, if (G+.) and (G-.) hold, and Al > Al there exist two constants A+, A- < Al such that the problem (2.3) has 1. at least one positive solution for A = A+,
2. at least one negative solution for A = A-,
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3. at least two positive solutions for A+ < A < A1,
4. at least two negative solutions for A- < \ < Al (in particular, four non-trivial solutions in a left-sided open neighborhood of A1), 5. at least two solutions (one of them positive and the other one negative)
for Al < \ < A1, 6. no positive solutions for A < A+, and no negative solutions for \ <
A-. REFERENCES
[1] A. Ambrosetti and D. Arcoya, On a Quasilinear Problem at Strong Resonance, Topological Methods Nonlinear Anal. 6 (1995), 255-264.
[2] A. Ambrosetti, J. Garcia Azorero and I. Peral, Multiplicity results for some semilinear and quasilinear elliptic equations, J. Funct. Anal. 10 (1996), 219-242. [3] A. Ambrosetti and P. Hess, Positive solutions of asymptotically linear elliptic eigenvalue problems, J. Math. Anal. Appl. 73 (1980), 411-422. [4] A. Ambrosetti and G. Mancini, Theorems of existence and multiplicity
for nonlinear elliptic problems with noninvertible linear part, Ann. Scuola Norm. Sup. Pisa 5 (1978), 15-28. [5] A. Anane and J.P. Gossez, Strongly nonlinear elliptic problems near resonance: a variational approach, Comm. Partial Differential Equations 15 (1990), 1141-1159. [6] D. Arcoya and J.L. Gamez, Bifurcation theory and related problems: Anti-maximum principle and resonance, to appear in Comm. on Partial Diff. Equat. [7] D. Arcoya and L. Orsina, Landesman-Lazer conditions and quasilinear elliptic equations, Nonlinear Anal. TMA 28 (1997), 1623-1632.
[8] P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear Anal. TMA 7 (1983), 981-1012. [9] L. Boccardo, P. Drabek and M. Kueera, Landesman-Lazer conditions for strongly nonlinear boundary value problems, Comment. Math. Univ. Carolinae 30 (1989), 411-427. [10] P. Clement and L.A. Peletier, An anti-maximum principle for secondorder elliptic operators, J. Differential Equations 34 (1979), 218-229.
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[11] D.G. De Figueiredo, Positive solutions of semilinear elliptic problems. In Proc. of the First Latin American School of Differential Equations, Lectures Notes n4 957, 1982.
(12] J. Fleckinger, J.P. Gossez, P. Takac and F. de Thelin, Existence, nonexistence et principe de l'antimaximum pour le p-Laplacien, C.R. Acad. Sci. Paris 321 (1995), 731-734. [13] J.L. GiLmez, Sub- and super- solutions in bifurcation problems, Nonlinear Anal. TMA 28 (1997), 625-632. [14] M.A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, McMillan, New York, 1964.
[15] E.M. Landesman and A.C. Lazer, Nonlinear perturbations of linear elliptic problems at resonance, J. Math. Mech. 19 (1970), 609-623. [16] D. Lupo and M. Ramos, Some multiplicity results for two-point boundary value problems near resonance, Rend. Sem. Mat. Univ. Pol. Torino 48 (1990), 125-135.
[17] J. Mawhin and K. Schmitt, Landesman-Lazer type problems at an eigenvalue of odd multiplicity, Results in Math. 14 (1988), 138-146.
[18] J. Mawhin and K. Schmitt, Nonlinear eigenvalue problems with the parameter near resonance, Annales Polonici Mathematici LI (1990), 241-248.
[19] P.H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487-513.
[20] P.H. Rabinowitz, On bifurcation from infinity, J. Differential Equations 14 (1973), 462-475.
David Arcoya and Jose L. Gamez Department of Mathematical Analysis, University of Granada Granada, 18071, Spain darcoyacugr.as,jlgamezougr.es
Orientation and Degree for Fredholm Maps of Index Zero Between Banach Spaces Pierluigi Benevieri ABSTRACT We define a notion of topological degree for a class of maps (called orientable), defined between real Banach spaces, which are ftedholm of index zero. We introduce first a notion of orientation for any linear ftedholm operator of index zero between two real vector spaces. This notion (which does not require any topological structure) allows us to define a concept of orientability for nonlinear
ftedholm maps between real Banach spaces. The degree which we present verifies the most important properties usually taken into account in other degree theories, and it is invariant with respect to continuous homotopies of ftedholm maps.
1
Introduction
We define an integer valued topological degree for a class of Fredholm maps
of index zero between real Banach spaces (i.e., C1 maps such that at any point of the domain the derivative is Fredholm of index zero). This is a revised and simplified version of a recent study in collaboration with M. Furi (see [2] and [3]). The starting point is a definition of orientation for Fredholm linear operators of index zero between real vector spaces. Such a notion is completely algebraic (no topological framework is needed). Any operator admits exactly two orientations. Furthermore, in the domain of real Banach spaces, given a (bounded) Fredholm operator of index zero L, an orientation of L induces by a sort of continuity an orientation on any operator L' sufficiently close to L. This "local stability" of the orientation allows us to define a notion of orientation for (nonlinear) Fredholm maps of index zero between Banach spaces. More precisely, given two real Banach spaces E and F and an open subset fl of E, a Fredholm map of index zero, f : Il - F, is called orientable if one can assign in a "continuous way" (we will make precise the sense of such a continuity) an orientation to the derivative of f at any point of its domain. The degree is consequently defined for the class of oriented maps. Such a degree verifies the most classical and important properties usually
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taken into account in degree theory. Our construction extends and simplifies the theories given by Elworthy and Tromba and, later, by Fitzpatrick, Pejsachowicz, and Rabier. The first two authors introduced the notion of Fredholm structure in order to extend to infinite dimensional manifolds the classical concept of orientation of finite dimensional manifolds (on which is based the Brouwer degree) (see [4] and [5]). By a different approach, Fitzpatrick, Pejsachowicz and Rabier define a notion of orientation for Fredholm maps of index zero between real Banach spaces and then a degree for the class of oriented maps (see [7] and references therein). The concept of orientation introduced by Fitzpatrick, Pejsachowicz and
Rabier has interesting similarities and differences with our one and the reader can find a comparison (also with the work of Elworthy and Tromba) in [2] and [3]. We conclude this paper with an application to bifurcation theory. Precisely we present a global bifurcation theorem in the domain of Fredholm maps which are oriented according to our construction. More precise results in bifurcation theory and a comparison with analogous results obtained by Fitzpatrick, Pejsachowicz and Rabier will appear in a forthcoming paper in collaboration with M. Furi.
2
Orientable maps
We introduce here a completely algebraic notion of orientation for Fredholm linear operators of index zero between real vector spaces. Therefore, we will
show that, in the context of Banach spaces, an oriented bounded operator induces, by a sort of continuity, an orientation to any sufficiently close operator. This "continuous transport of orientation" allows us to define a concept of an oriented C' Fredholm map of index zero between open subsets of Banach spaces (and, more generally, between Banach manifolds). Let E be a vector space and consider a linear map T : E --, E of the form
T = I - K, where I denotes the identity of E and K has finite dimensional range. Given any finite dimensional subspace E0 of E containing Range K, T maps E0 into itself. Then, consider the restriction TI& : E0 -+ E0. It is not difficult to prove that the determinant, detTIE,, is well defined and does not depend on the choice of the finite dimensional space E0 containing Range K. Thus, it makes sense to denote by det T this common value.
We recall that a linear operator between vector spaces, L : E -+ F, is called (algebraic) Fredholm if Ker L and coKer L have finite dimension. Its index is the integer ind L = dim Ker L - dim coKer L.
Notice that, when L : R' --+ R", one easily gets ind L = m - n.
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If L : E -+ F is Fredholm and A : E - F is any linear operator with finite dimensional range, we say that A is a corrector of L provided that L + A is an isomorphism. This can be verified only if ind L = 0, since, as well known, L + A is Redholm of the same index as L. Assume therefore ind L = 0 and notice that, in this case, the set of correctors of L, indicated by C(L), is nonempty. In fact, any (possibly trivial) linear operator A :
E -' F such that Ker A ® Ker L = E and Range A ® Range L = F is a corrector of L. We introduce in C(L) the following equivalence relation. Given A, B E C(L), consider the automorphism T = (L + B) -'(L + A) of E. We have
T=(L+B)-1(L+B+A-B)=I-(L+B)-1(A-B). Clearly (L + B) -1(A - B) has finite dimensional range. This implies that det T is well defined and, in this case, not zero since T is invertible. We say that A is equivalent to B or, more precisely, A is L-equivalent to B, if det (L+B)-1(L+A) > 0. This is actually an equivalence relation on C(L), with just two equivalence classes (see [2]). We can therefore introduce the following definition.
Definition 2.1. An orientation of a Fredholm operator of index zero L is one of the two equivalence classes of C(L). We say that L is oriented when an orientation is chosen. The idea of dividing in two classes the set of correctors of a Fredholm operator is already present in Mawhin (see [10] and [11]) and Pejsachowicz-
Vignoli ([13]). In [3] the reader can find a comparison between these approaches and our own. According to Definition 2.1, an oriented operator L is a pair (L, c), where L : E -, F is a Fredholm operator of index zero and w is one of the two equivalence classes of C(L). However, to simplify the notation, we shall not use different symbols to distinguish between oriented and nonoriented operators (unless it is necessary).
Given an oriented operator L : E - F, the elements of its orientation will be called the positive correctors of L. Any isomorphism L admits a special orientation, namely the equivalence class containing the trivial operator 0. We shall refer to this orientation as the natural orientation v(L) of L. Moreover, if an isomorphism L is actually oriented, we define its sign as follows: sgn L = 1 if the trivial operator 0 is a positive corrector of L (i.e., if the orientation of L coincides with v(L)), and sgn L = -1 otherwise. Let us now see a property which may be regarded as a sort of reduction
of the orientation of an operator to the orientation of its restriction to a convenient pair of subspaces of the domain and codomain. This will be useful in the next section where we will show that our degree is exactly the Brouwer degree of a suitable restriction to finite dimensional manifolds.
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Let E and F be two real vector spaces and let L : E - F be an algebraic Fredholm operator of index zero. Let F1 be a subspace of F which is transverse to L (that is, F1 + Range L = F). Observe that in this case the restriction L1 of L to the pair of spaces El = L-1(F1) (as domain) and F1 (as codomain) is again a Fredholm operator of index zero. We can prove that an orientation of L gives an orientation of L1, and vice versa. To see
this, let E0 be a complement of El in E and split E and F as follows: E = Eo ® E1, F = L(E0) ® Fl. Thus L can be represented by a matrix Lo 0
0 L1
where Lo is an isomorphism. Then our claim follows immediately from the fact that any linear operator A : E F, represented by 0 C0
0 Al
is a corrector of L if and only if Al is a corrector of L1. According to the above argument, it is convenient to introduce the following definition.
Definition 2.2. Let L : E -- F be a Fredholm operator of index zero between real vector spaces, let F1 be a subspace of F which is transverse to L, and denote by L1 the restriction of L to the pair of spaces L-1(F1) and Fl. Two orientations, one of L and one of L1, are said to be correlated (or one induced by the other) if there exist a projector P : E -, E onto El
and a positive corrector Al of L1 such that the operator A = JA1P is a positive corrector of L, where J : F1 y F is the inclusion. The above concept of orientation of an algebraic Fredholm operator of
index zero L : E -+ F does not require any topological structure on E and F, which are supposed to be just real vector spaces. However, in the context of Banach spaces, an orientation of L can induce an orientation on the operators sufficiently close to L. Precisely, assume that L : E -- F is a bounded Fredholm operator of index zero between real Banach spaces. Given an orientation of L and a positive bounded corrector A of L (whose existence is ensured by the Hahn-Banach theorem), A is still a corrector of any operator L' in a convenient neighborhood U of L in the Banach space of bounded linear operators from E into F. Therefore, if L' belongs to U, then it can be oriented by choosing A as a positive corrector. From now on, unless otherwise specified, E and F will denote real Banach spaces, L(E, F) the Banach space of bounded linear operators from E into F and Iso(E, F) the open subset of isomorphisms. For the sake of simplicity, in the context of Banach spaces, the set of
continuous correctors of a Fredholm operator of index zero L will still be denoted by C(L), as in the algebraic case, instead of C(L) fl L(E, F). Therefore, from now on, by a corrector of L we shall actually mean a
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continuous corrector. It is clear that an orientation of L can be regarded as an equivalence class of continuous correctors of L. We recall that the set 4i(E, F) of the Fredholm operators from E into
F is open in L(E, F), and the integer valued map ind
:
di(E, F) - Z
is continuous. Consequently, given n E Z, the set
4o(E, F). Definition 2.3. Let A be a topological space and h : A --; 'o(E, F) a continuous map. An orientation of h is a continuous choice of an orientation a(A) of h(a) for each \ E A; where "continuous" means that for any A E A there exists Aa E a(A) which is a positive corrector of h(A') for any A' in
a neighborhood of A. A map is orientable if it admits an orientation and oriented when an orientation has been chosen. Remark 2.4. We can give a notion of orientation for subsets of '0(E, F). Precisely, a subset A of -to(E, F) is said to be orientable if so is the inclusion
Any orientable map h : A -p 1o(E, F) admits at least two orientations. In fact, if h is oriented by a, reverting this orientation at all A E A, one gets what we call the opposite orientation a_ of h. Observe also that two orientations of h coincide in an open subset of A, and for the same reason the set in which two orientations of h are opposite one to the other is open. Therefore, if A is connected, two orientations of h are either equal or one
is opposite to the other. Thus, in this case, an orientable map h admits exactly two orientations.
Remark 2.5. An orientation of a continuous map h : A -+ -to(E, F) can be given by assigning a family {(U1, Ai) : i E Z}, called an oriented atlas of h, satisfying the following properties: {Ui : i E Z} is an open covering of A; given i E Z, Ai is a corrector of any h(a), V.1 E Ui; if A E Ui fl Uj, then Ai is h(,\)-equivalent to Aj.
Let us now define a notion of orientation for nonlinear Fredholm maps of index zero between open subsets of Banach spaces. Definition 2.6. Let S2 be an open subset of E and f : St F be Fredholm of index zero. An orientation off is an orientation of D f : x '-- D f (x) E
co(E, F), and f is orientable (reap. oriented) if so is D f according to Definition 2.3.
Remark 2.7. A local diffeomorphism f : S2 - F between two open subsets of E and F can be oriented with the orientation v defined by 0 E v(Df(x)),Vx E 12. We will refer to this orientation as the natural orientation of the local diffeomorphism f.
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We have previously seen that the orientation of an algebraic FYedholm operator of index zero can be regarded as the orientation of the operator to a suitable pair of subspaces. Analogously, let f : SZ - F be oriented and let F1 be a subspace of F which is transverse to f ; that is F1 + Range D f (x) = F, Vx E f -1(F1). It is known that, in this case, M1 = f -' (F1) is a submanifold
of ) and the restriction f, of f to M1 (as domain) and F1 (as codomain) is again a F redholm map of index zero. Moreover, for any x E M1, the tangent space to M1 at x, T. M1 coincides with D f (x) -1(FI) (see for example [11
and [8] for general results about transversality). Therefore, according to Definition 2.2, given any x E M1, the orientation of D f (x) E 4io(E, F) induces an orientation on its restriction D f 1(x) E 4?o (Ty MI, F1), which is the derivative of f, at x. Such a collection of orientations of D f 1(x), x E M1,
is actually an orientation of fl : M1 - F1 that, from now on, we shall call
the orientation on f, induced by f. Observe also that, given x E M1, D f (x) : E - F is an isomorphism if and only if so is D f 1(x) : Tx M1 -> F1 and, with the induced orientation, one has sgn D f (x) = sgn D f 1(x); and this will imply one of the fundamental properties of the degree (Reduction property).
Definition 2.6 can be slightly modified in order to obtain a notion of orientation for continuous homotopies of Fredholm maps. Let SZ be open in E. We say that a continuous map H : 0 x [0, 1] -- F is a continuous family of Fredholm maps of index zero if it is differentiable with respect to the first
variable, the partial derivative D1H(x, t) : E -+ F is a Fredholm operator of index zero for any (x, t) E SZ x [0, 1], and it depends continuously on (x, t). We will say that a continuous family of Fredholm maps of index zero H
is orientable if so is the map D1H : (x, t) '- D1H(x, t) and its orientation is an orientation of D1 H. Clearly, given an oriented continuous family of Fredholm maps of index zero, H : S1 x [0, 1] -- F, any partial map Ht := H(., t) is an oriented map from SZ into F, according to Definition 2.6. One could actually show that, given a continuous family of Fredholm maps of index zero H : 11 x [0, 1] ->
F, if Ho is orientable, then all the partial maps Ht are orientable as well, and an orientation of Ho induces a unique orientation on any Ht which makes H oriented (see (3)). The properties of this concept of orientation are treated in depth in [2] and [3]. Here we limit ourselves to some remarks. If E and F are two finite dimensional (of the same dimension) Banach spaces, 4?o(E, F) coincides with L(E, F). In this case, one can prove that any continuous map with values in L(E, F) is orientable (moreover, recalling Remark (2.4), L(E, F) is orientable). If E and F are infinite dimensional, it is proved that continuous maps defined on simply connected and locally path connected topological spaces into 4?o(E, F) are orientable. However, because of the topological structure of 4io(E, F), we can find nonorientable maps with values in F). This
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is based on the fact that there exist Banach spaces E whose linear group GL(E) is connected. For example, an interesting result of Kuiper (see [9]) asserts that the linear group GL(E) of an infinite dimensional separable Hilbert space E is contractible. It is also known that GL(IP), 1 < p < oo, and GL(co) are contractible as well. There are, however, examples of infinite dimensional Banach spaces whose linear group is disconnected (see [6], [12]
and references therein). When GL(E) is connected, then it is possible to define nonorientable maps with values into (Do(E).
Theorem 2.8. Assume Iso(E, F) is nonempty and connected. Then there exists a nonorientable map y : S' -'o(E, F) defined on the unit circle of R2.
Proof. We give here a sketch of the proof (see [3] for more details). Let S+ and S1 denote, respectively, the two arcs of S' with nonnegative and nonpositive second coordinate. One can prove that there exists an oriented open connected subset U of 1o(E, F) containing two points in Iso(E, F), say L_ and L+, such that sign L_ = -1 and sign L+ = 1. Let y+ : S+ -a U be a path such that y+(-1, 0) = L_ and 'y+(1, 0) = L+. Since Iso(E, F) is an open connected subset of L(E, F), it is also path connected. Therefore there exists a path y_ S. - Iso(E, F) such that y_ (-1, 0) = L_ and y_ (1, 0) = L+. Define y : S1 4o (E, F) by y)
{
-f-+(x,
y)
if y < 0,
and assume, by contradiction, it is orientable. This implies that also the image -y(S') of -y is orientable, with just two possible orientations. Orient, for example, y(S') with the unique orientation compatible with the oriented
subset U of'o(E, F). Thus, being y(S+) C U, we get sign L_ = -1 and sign L+ = 1. On the other hand, since the image of y_ is contained in Iso(E, F), it follows that sign L_ = sign L+, which is a contradiction.
0
Remark 2.9. The above result can be used to prove that -to(E, F) is not simply connected, since it can be verified that simply connected and locally path connected subsets of -to(E, F) are orientable (see [3]).
By means of this example of a nonorientable map it is possible to define an example of a nonorientable Fredholm map of index zero from an open subset of a Banach space into another Banach space. To make this paper not too long, we omit this construction (which can be found in [3]).
3
Degree for oriented maps
As previously, E and F stand for two real Banach spaces. Let 11 be open
in E and f : S2 - F be oriented. Given an element y E F, we call the
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triple (f, fl, y) admissible if f -1(y) is compact. A triple (f, fl, y) is called strongly admissible provided that f admits a proper continuous extension to the closure fl of 11 (again denoted by f ), and y V f (8f1). Clearly any strongly admissible triple is also admissible. Moreover, if (f, fl, y) is strongly admissible and U is an open subset of Sl such that U n f -1(y) is compact, then (f, fl, y) is strongly admissible as well. Our aim here is to define a map, called degree, which to every admissible triple (f, Sl, y) assigns an integer, deg(f, fl, y), in such a way that the following five properties hold: i) (Normalization) If f : 11 y E f (S1), then
F is a naturally oriented diffeomorphism and deg(f, 91, y) = 1.
ii) (Additivity) Given an admissible triple (f, f1, y) and two open subsets U1, U2 of 11, if U1 n U2 = 0 and f (y) c U1 u U2, then (f, U1, y) and (f, U2, y) are admissible and deg(f, fl, y) = deg(f, U1, y) + deg(f, U2, y).
iii) (Topological Invariance) If (f, S2, y) is admissible, W : U - 11 is a naturally oriented diffeomorphism from an open subset of a Banach space : f (S1) -+ V is a naturally oriented diffeomorphism from f (11) onto an open subset of a Banach space, then
onto M and
1i
deg(f,M,y) = deg(t,bfcp,U,tl.'(y)), where 7pf co is oriented with orientation induced by the orientations of 1i, f and W.
iv) (Reduction) Let f : S] -+ F be oriented and let F1 be a subspace of F which is transverse to f. Denote by f, the restriction of f to the manifold M1 = f -1(F1) with the orientation induced by f. Then
deg(f,Sl,y) = deg(fl, MI, y), provided that f -1(y) is compact. v) (Homotopy Invariance) Let H : 11 x [0, 1] -+ F be an oriented continuous family of Fredholm maps of index zero, and let y : [0, 1] -+ F be a continuous path. If the set {(x, t) E S] x [0,1] : H(x, t) = y(t)}
is compact, then deg(Ht, 0, y(t)) is well defined and does not depend on t E [0, 1].
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In the sequel we shall refer to i) -v) as the fundamental properties of degree.
We define first our notion of degree in the special case when (f, St, y) is a regular triple; that is when (f, 11, y) is admissible and y is a regular value for f in Q. This implies that f -1(y) is a compact discrete set and, consequently, finite. In this case our definition is similar to the classical one in the finite dimensional case. Namely
deg(f,i,y) = E sgnDf(x),
3.1
xEf-' (y)
where, we recall, sgn D f (x) = 1 if the trivial operator is a positive corrector of the oriented isomorphism D f (x), and sgn D f (x) = -1 otherwise. It is easy to prove that the first four fundamental properties of the degree hold for the class of regular triples, and we will prove below that they are still valid in the general case. A straightforward consequence of the Additivity is the following property that we shall need (for the special case of regular triples) in the proof of Lemma 3,2 below.
vi) (Excision) If (f, !Q, y) is admissible and U is an open neighborhood of
f-1(y), then deg(f, SZ, y) = deg(f, U, V).
In order to define the degree in the general case we will prove that, given any admissible triple (f, St, y), if U1 and U are sufficiently small open neighborhoods of f -1(y), and yLVj E F are two regular values for f, sufficiently close to y, then
deg(f, U1, y1) = deg(f, U2, y Let us show first that the degree of a regular triple (f, 0, y) can be viewed as the Brouwer degree of the restriction of f to a convenient pair of finite dimensional oriented manifolds. Consider an admissible triple (f, 0, y) (for the moment we do not assume y to be a regular value of f). Since f-1(y) is compact, there exists a finite dimensional subspace FO of F and an open subset U of f-1(y) in which f is transverse to FO. Consequently, MO = f -1 (FO 0 U is a differentiable manifold of the same dimension as FO, f is transverse to FO in U, and the restriction fo : MO --f Fo of f is an oriented map (with orientation induced by f). Since Fo is a finite dimensional vector space and fo is orientable, MO is orientable as well. Therefore, the orientation of fo induces a pair of orientations of MO and FO, up to an inversion of both of them (which does not effect the Brouwer degree of fo at y). When a pair of these orientations is chosen, we say that MO and FO are oriented according to f.
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Before stating the following lemma, we point out that if a Fredholm map f : St -+ F is transverse to a subspace F0 of F, then an element y E F0 is a regular value for f if and only if it is a regular value for the restriction
fo: f-'(FO) --Foof f. Lemma 3.1. Let (f, 0, y) be a regular triple and let F0 be a finite dimensional subspace F, containing y and transverse to f . Then Mo = f -I (FO) is an orientable manifold of the same dimension as F0. Moreover, orienting Mo and F0 according to f , the Brouwer degree degB (fo, Mo, y) of fo at y coincides with deg(f, St, y).
The proof is not difficult and can be found in [2]. As a consequence of this lemma we get the following result which is crucial in our definition of degree.
Lemma 3.2. Let (f, SZ, y) be a strongly admissible triple. Given two neighborhoods U1 and U2 of f -1(y), there exists a neighborhood V of y such that for any pair of regular values yl, y2 E V one has deg(f, U1, y1) = deg(f, U2, y2)
Proof. Since (f, 11, y) is strongly admissible, f is actually well defined and proper on the closure SZ of Q. Let U1 and U2 be two open neighborhoods of f -1(y) and put U = U1 fl U2. Since proper maps are closed, there exists a neighborhood V of y with V fl f (Ii \ U) = 0. Without loss of generality we may assume that V is convex. With an argument similar to that used just before Lemma 3.1 one can show the existence of a finite dimensional subspace F0 of F containing yl and y2 and transverse to f in a convenient neighborhood W C U of the compact set f -1(S), where S is the segment joining yl and y2. Thus, Mo = f -1(Fo) n W is a finite dimensional manifold of the same dimension as F0, and they turn out to be oriented according to f (up to an inversion of both orientations). Denote by fo the restriction off to Mo (as domain) and Fo (as codomain). From Lemma 3.1 we obtain
degB (fo, Mo, yl) = deg(f, W, y1), degB (fo, Mo, y2) = deg(f, W, y2)
On the other hand, since f-1(yl) and f-1(y2) are contained in W, by the Excision property for regular triples, we get deg(f, U1, yi) = deg(f, W, y1) and.deg(f, U2, y2) = deg(f, W, y2). Therefore, it remains to show that degB (fo, Mo, y1) = degB (fo, Mo, Y2)
Consider now the path Clearly, the set
[0,1] - Fo given by t ' - ty1 + (1 - t)y2.
{x E Mo : fo(x) = y(t) for some t E [0, 11}
Orientation and Degree for Fredholm Maps of Index Zero
211
coincides with f -1(S), which is compact. Therefore, from the homotopy invariance of the Brouwer degree we get deg. (fo, Mo, yi) = deg.(fo,Mo,y2), and the result is proved.
0
Lemma 3.2 justifies the following definition of degree for general admissible triples.
Definition 3.3. Let (f, St, y) be admissible and let U be an open neighborhood of f -1(y) such that U C St and f is proper on U. Put deg(f, M, y) := deg(f, U, z),
where z is any regular value for f in U, sufficiently close to y. To justify the above definition we point out that the existence of regular values for flu which are sufficiently close to y can be directly deduced from Sard's Lemma. In fact, as previously observed, one can reduce the problem of finding regular values of a Fredholm map to its restriction to a convenient pair of finite dimensional manifolds.
Theorem 3.4. The degree satisfies the above five fundamental properties.
Proof. The first four properties are an easy consequence of the analogous ones for regular triples. Let us prove the Homotopy Invariance. Consider an oriented continuous family of Fredholm maps of index zero H : St x [0, 1] -' F and let y : [0,1] -+ F be a continuous path in F. Assume that the set
C = {xEfl:H(x,t)=y(t) for some tE [0, 1]} is compact. Since H is locally proper, there exists an open neighborhood U
of C in fl such that H is proper on U x [0,1]. Consequently, Ht = H(., t) is proper on U for all t E [0, 1], and, by the definition of degree, deg(Ht, fl, y(t)) = deg(Ht, U, y(t)), Vt E [0,1].
We need to prove that the function a(t) = deg(Ht, U, y(t)) is locally constant. Let r be any point in [0, 11. Since H is proper on U x [0, 1] and y(r) V HT(OU), one can find an open connected neighborhood V of y(r)
and a compact neighborhood J of r (in [0, 1]) such that y(t) E V for t E J and H(8U x J) n V = 0. Thus, if z is any element of V, one has a(t) = deg(Ht, U, z) for all t E J. To compute this degree we may therefore assume that z is a regular value for HT in U, so that Hr 1(z) is a finite set {XI, x2, ... , xn} and the partial derivatives Dl H(x;, r), i = 1, 2, ... , n, are all nonsingular. Consequently, given any x; in HT 1(z), the Implicit Function Theorem ensures that H-1(z), in a neighborhood W; x J; of (x2, r), is the graph of a continuous curve -y; : J; - Q. Since H is proper in U x J
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(recall that J is compact) and z 0 H(OU x J), the set H-1(z) n (U x J) is compact. This implies the existence of a neighborhood J0 of r such that
fort EJoonehas Ht 1(z) = {71(t),72(t),...,7n(t)}.
Moreover, by the continuity of D1H, we may assume that z is a regular value for any Ht, t E Jo. Finally, since H is oriented, the continuity assumption in the definition of orientation implies that, for any i, sgn D1 H(7t(t), t) does not depend on t E J0, and from the definition of degree of a regular
triple we get that a(t) is constant in Jo.
0
The notions of orientation and topological degree we have defined for nonlinear Fredholm maps of index zero between Banach spaces can be extended to Fredholm maps of index zero between Banach manifolds. This is treated in [2] and [3]. The degree theory we have introduced can be applied to obtain some bifurcation results. In a forthcoming paper, written in collaboration with M. Furi, this topic will be treated in depth. Here we are going to state the following global bifurcation result.
Let 11 be an open subset of E and H : St x R - F be a continuous family of Fredholm maps of index zero. Assume that H(0, A) = 0 for all A E R. Let us recall that a real number A0 is called a bifurcation point if any neighborhood of (0, Ao) contains pairs (x, A) such that H(x, A) = 0 and
x # 0. Those solutions of H(x, A) = 0 belonging to {0} x R are usually called trivial whereas the other ones are said to be nontrivial.
Theorem 3.5. Let H : Sl x R -' F be a continuous family of Fredholm maps of index zero with H(0, A) = 0 for all A E R. Let also A1, A2 E R be such that sign D1H(0, a1) sign DiH(0,A2) = -1. Then there exists a connected set of nontrivial solutions of H(x, A) = 0 whose closure in SZ x R has nonempty intersection {0} x [A1, A2] and either is not compact or contains a point (0, A) with a ¢ (A1, A2].
REFERENCES
[1] R. Abraham and J. Robbin, Transversal Mappings and Flows, Benjamin, N. Y., 1967. [2] P. Benevieri and M. Furi, A simple notion of orientability for Fredholm maps of index zero between Banach manifolds and degree, to appear in Annales des Sciences Mathematiques du Quebec 22:2 (1998), 131-148.
[3] P. Benevieri and M. Furi, On the concept of orientability for Fredholm maps between real Banach manifolds, preprint.
Orientation and Degree for Fredholm Maps of Index Zero
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[4] K.D. Elworthy and A.J.Tromba, Differential structures and Fredholm maps on Banach manifolds. In Global Analysis (S. S. Chern and S. Smale, eds.), Proc. Symp. Pure Math., Vol. 15, 1970, 45-94. [5] K.D. Elworthy and A.J.Tromba, Degree theory on Banach manifolds. In Nonlinear Functional Analysis (F. E. Browder, ed.), Proc. Symp. Pure Math., Vol. 18 (Part 1), 1970, 86-94.
[6] P.M. Fitzpatrick, The parity as an invariant for detecting bifurcation of the zeroes of one parameter families of nonlinear Fredholm maps. In Topological Methods for Ordinary Differential Equations (M. Furi and P. Zecca, eds.), Lectures Notes in Math., vol. 1537, 1993, 1-31.
[7] P.M. Fitzpatrick, J. Pejsachowicz and P.J. Rabier, Orientability of Fredholm Families and Topological Degree for Orientable Nonlinear Fredholm Mappings, J. Functional Analysis 124 (1994), 1-39. [8] V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, Englewood-Cliffs, 1974.
[9] N.H. Kuiper, The homotopy type of the unitary group of a Hilbert space, Topology 3 (1965), 19-30.
[10] J. Mawhin, Equivalence Theorems for Nonlinear Operator Equations and Coincidence Degree Theory for Some Mappings in Locally Convex Topological Vector Spaces, J. Differential Equations 12 (1972), 610636.
[11] J. Mawhin, Topological Degree and Boundary Value Problems for Nonlinear Differential Equations. In Topological Methods for Ordinary Differential Equations (CIME 1991) (M. Furi and P. Zecca, eds.), Lecture Notes in Math., No. 1537, 1993, 74-142.
[12] B.S. Mitjagin, The homotopy structure of the linear group of a Banach space, Uspehki Mat. Nauk. 72 (1970), 63-106. [13] J. Pejsachowicz and A. Vignoli, On the topological coincidence degree for perturbations of Fredholm operators, Boll. Unione Mat. Ital. 17-B (1980), 1457-1466.
Pierluigi Benevieri Dipartimento di Matematica Applicata UniversitA di Firenze Via S. Marta, 3 - 50139 Firenze Italy benevieriComa.unifi.it
On the Method of Upper and Lower Solutions for First Order BVPs Alberto Cabada Eduardo Liz Rodrigo L. Pouso 1
Introduction
The theory of differential equations with discontinuous nonlinearities has become more and more rich due to a still growing number of papers and books on the subject such as [2, 3, 4, 8, 9, 10]. The method of upper and lower solutions has been successfully applied in this framework using absolutely continuous upper and lower solutions [6, 9, 10, 12]. However, the regularity of these functions can be relaxed to obtain finer results in practical situations: see [7, 13], where piecewise absolutely continuous upper and lower solutions are considered. Moreover, we can still go a little further by defining the concepts of upper and lower solutions in the space of functions of bounded variation. The results we are going to present here are some of the conclusions reached by the authors in several works [5, 13, 14, 15, 16).
2
Ordered upper and lower solutions
Given the interval I = [0, T], with T > 0, we shall say a function f : I x R 1 R is an admissible function if it verifies the following assumptions:
(i) For every x E R, f
x) is measurable on I.
(ii) For almost every t E I, f (t, ) verifies the condition lim sup f (t, z) < f (t, x) < lim inf f (t, z) for all x E R. z -.r
(iii) For every R > 0 there exists 1P E L1(I) such that IxI < R implies If (t, x) I < i'(t) for a.e. t E I.
216
A. Cabada, E. Liz, R.L. Pouso
If an admissible function f is such that f (t, ) is continuous on R for a.e. t E I, we shall say that f is a Caratheodory function. Consider the BVP
x'(t) = q(x(t)) f (t, x(t)) for a.e. t E I, B(x(0), x(t`)) = 0, where t" E (0, T] and the following assumptions are verified:
(a) q : R - (0, +oo) is such that q, 11q E L' (R). (b) f is an admissible function.
(c) B : R2 -+ R is such that B(u, ) is nonincreasing in R for all u E R and lim inf
Z1-'X
>
lim Z1-Z+
,Z2
sup
Z2_.y+
y-
B(zl, z2) > B(x, y)
B(zl, z2) for all (x, y) E R2.
We search for Caratheodory solutions for (2.1), i.e., we say x : I -+ R is a solution of (2.1) if x E AC(I) and satisfies (2.1). We denote by AC(I) the space of absolutely continuous functions on I and by BV(I) the space of functions of bounded variation on I. By Lebesgue's decompostion theorem we know that if g E BV(I), then there exists a unique decomposition of the form
9ga+g3, where ga E AC(I) and ga(0) = 0 and g, E BV(I) is a singular function, i.e., such that g' (t) = 0 for a.e. t E I (see [17]). Function ga is called the absolutely continuous part of g and g, is the singular part of g. We denote by BV+(I) the set of bounded variation functions which have nondecreasing singular part, and by BV- (I) the set of bounded variation functions which have nonincreasing singular part (see [1]).
A function a : I
R is a lower solution of (2.1) if a E BV-(I),
f
E
I,
B(a(0), a(t')) < 0. The concept of upper solution is defined similarly in BV+(I) and reversing inequalities. A first result is the following whose proof can be found in [15].
Theorem 2.1. Assume conditions (a), (b) and (c) are fulfilled. Also assume problem (2.1) has a lower solution a and an upper solution Q such that a(t) _< p(t) for all t E I. Then problem (2.1) has a minimal and a maximal solution between a and Q.
Upper and Lower Solutions for First Order BVPs
217
We say a function f : I x R --+ R is an inversely admissible function if it satisfies conditions (i) and (iii) of the definition of admissible function and, instead of (ii): (ii') For almost every t E I, f (t, ) verifies the condition
-
-
lim inf f (t, z) > f (t, x) > lim sup f (t, z) for all x E R. Z4x_
Next, consider the problem
x'(t) = q(x(t)) f (t, x(t)) for a.e. t E I, B(x(t*),x(T)) = 0, where t' E [0, T) and the following assumptions are verified:
(a) q : R
(0, +oo) is such that q, 11q E L110 (R).
(b') f is an inversely admissible function.
(c') B : R2 - R is such that B(., v) is nondecreasing in R for all v E R and
lim sup B(zl, Z2) < B(x, y) Zl-.Z-,Z2..b-
<
lim inf ZI
Z+,Z2_,y+
B(zl, z2) for all(x, y) E R2.
By defining the concepts of lower and upper solutions as in the spirit of the previous case and reversing time, we deduce from Theorem 2.1 the following result.
Theorem 2.2. Assume conditions (a), (b') and (c') are fulfilled. Also assume problem (2.2) has a lower solution a and an upper solution 8 such
that a(t) > p(t) for all t E I. Then problem (2.1) has a minimal and a maximal solution between a and 13.
It is important to realize that if f : I x R --+ R is both an admissible and an inversely admissible function, then f is a Caratheodory function. On the other hand, a function B verifying (c) and (c') must be continuous on R2.
3
Upper and lower solutions without ordering
Consider the problem
x'(t) = q(x(t)) f (t, x(t)) for a.e. t E [0, 21, B(x(0), x(T)) = 0.
(3.1)
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A. Cabada, E. Liz, R.L. Pouso
Theorem 3.1. Assume f is a Carathedory function and B : R2 -, R is a continuous function, nondecreasing in its first variable and nonincreasing in the second one. Also, assume problem (3.1) has a lower solution a and an upper solution f3. Then problem (2.1) has a minimal and a maximal solution between m and M, where
m(t) = min {a(t), p(t) } and M(t) = max {a(t), 0(t) 1,
for ailtE I. The proof of this result can be found in [16]. It is a consequence of two respective analogous results for initial and terminal value problems, which are proved following the ideas of Marcelli and Rubbioni in [11]. There exist counterexamples for the case of admissible (or inversely admissible) nonlinearities, so we must limit ourselves to Caratheodory conditions (note that discontinuities in x on the right-hand side of the equation are due to q(x), so, in fact, our nonlinearity g(t, x) = q(x) f (t, x) might not be a Caratheodory function). REFERENCES [1] A. Adje, Sur et sons-solutions generalisees et problemes aux limites du second ordre, Bull. Soc. Math. Belgique 42 (1990), 347-368. [2] D.C. Biles, Continuous dependence of nonmonotonic discontinuous differential equations, Trans. Amer. Math. Soc. 339 (1993), 507-524.
[3] D.C. Biles, Existence of solutions for discontinuous differential equations, Differential and Integral Equations 8 (1995), 1525-1532.
[4] P. Binding, The differential equation i = f o x, J. Differential Equations 31 (1979), 183-199. [5] A. Cabada and R.L. Pouso, On first order discontinuous scalar differential equations, Nonlinear Studies 6:2 (1999), 161-170. [6] S. Carl, S. Heikkila and M. Kumpulainen, On solvability of first order discontinuous scalar differential equations, Nonlinear Times Digest 2 (1995), 11-24.
[7] C. De Coster, La methode des sur et sous solutions dans 1'etude de problemes aux limites, Doctoral Thesis, Univ. Catholique de Louvain, 1994.
[8] E.R. Hassan and W. Rzymowski, Extremal solutions of a discontinuous differential equation, Nonlinear Anal. 37 (1999), 997-1017.
[9] S. Heikkila and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Marcel Dekker, New York, 1994.
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[10] S. Heikkila and V. Lakshmikantham, A unified theory for first - order discontinuous scalar differential equations, Nonlinear Analysis T.M.A. 26 (1995), 785-797.
[11] C. Marcelli and P. Rubbioni, A new extension of classical Muller's theorem, Nonlinear Analysis T.M.A. 28 (1997), 1759-1767. [12] M.N. Nkashama, A generalized upper and lower solutions method and multiplicity results for nonlinear first-order ordinary differential equations, J. Math. Anal. Appl. 140 (1989), 381-395.
[13] E. Liz and R.L. Pouso, Upper and lower solutions with "jumps", J. Math. Anal. Appl. 222 (1998), 484-493. [14] E. Liz and R.L. Pouso, Approximation of solutions for nonlinear periodic boundary value problems with discontinuous upper and lower solutions, J. Comp. Appl. Math. 95 (1998), 127-138.
[15] R.L. Pouso, Upper and lower solutions for first order discontinuous ordinary differential equations, J. Math. Anal. Appl. 244 (2000), 466482.
[16] R.L. Pouso, Nonordered discontinuous upper and lower solutions for first order ordinary differential equations, Nonlinear Anal., to appear. [17] K.R. Stromberg, An Introduction to Classical Real Analysis, Wadsworth Inc., California, 1981.
Alberto Cabada and Rodrigo L. Pouso Departamento de Analise Matematica Facultade de Matemdticas Universidade de Santiago de Compostela Spain rodrigolpccorreo.usc.es
Eduardo Liz Departamento de Matematica Aplicada E.T.S.I. Telecomunicacidn Universidade de Vigo Spain
Nonlinear Optimal Control Problems for Diffusive Elliptic Equations of Logistic Type A. Canada J.L. Gamez J.A. Montero 1
Motivation of the problem -- Introduction
It is very well known that the evolution on the time of a species u, living in a bounded habitat and with a logistic growth, has been modelled by nonlinear reaction-diffusion equations whose equilibrium states originate in the study of problems of the form
-Du(x) = u(x)[r(x) - b(x)u(x)J, u(x) = 0, x E 852,
x E 9,
where .0 is a bounded and regular domain in R', u is the species concentration, r is its growth rate, and b represents the crowding effect. On the other hand, the presence of the Laplacian operator z points out the diffusive character of the species u, while the boundary condition may be interpreted so that the species may not stay on M. Let us suppose that we have the possibility of influencing the growth rate of the species in a linear way, through modification of the environment conditions. In this case, the function r(x) is of the form r(x) = a(x) - f (x), where a represents the intrinsic growth rate of the species u (in the absence of perturbation) and f plays the role of control. If we assume that, for each given f in a certain subset of functions A, we have a unique solution of (1.1), u f, and we sell the obtained product, we will obtain a benefit expressed
by a formula which usually depends on f and u f. Taking into account some previous works ([6], [7]), such a reasonable formula may be given by
J(f) = f (Ku f f - M f 2), where K > 0 and M > 0 mean, respectively, the The authors have been supported in part by Direcci6n General de Ensedanza Superior, Ministry of Education and Science (Spain), under grant number PB95-1190 and by EEC contract (Human Capital and Mobility program) nQ ERBCHRXCT 940494.
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A. Canada, J.L. G&nez, J.A. Montero
price of the species and the cost of the control. Our interest is to maximize the functional J.
By considering the functional J(f)/M, instead of J(f), we arrive at the control problem
-tu(x) = u(x) [a(x) - f (x) - b(x)u(x)], u(x) = 0, J(f) = mEaAx J(g),
x E Sl,
f E A, (1.2)
x E 03n,
J(g) = J (Augg - 9Z), a
where A is the quotient between the price of the species and the cost of the control. Such a function f will be called an optimal control for our problem in the set of functions A. Here, we will study the case where A = L+ (Sl) = {g E L°°(Sl) : g(x) > 0, a.e. in Sl}. In this case, we are allowed to modify, only in a negative way, the growth rate of the species u. Other related results and the detailed proofs of those presented here may be seen in [2], [3], [4] and [5].
2
Existence of the optimal control
If e E L' (Q), we denote e = ess inf e(x), e = ess sup e(x). From now on, xEf2
XC-0
we assume the hypotheses [H]
a, b E L°°(Sl), b > 0, f E L+ (Sl).
Also, if q E LOO (Q) we define vl (q) as the principal eigenvalue of the eigenvalue problem
-Au(x) + q(x)u(x) = cu(x),
u(x)=0,
x E Sl,
xEOn
where u E Ho '(9) (the usual Sobolev space). By using the lower and upper solution method, we can prove that the b.v.p. in (1.2) has a (weak) nontrivial and nonnegative solution u1, if and only if o 1(-a+ f) < 0. Moreover, in this case, the solution u f is the unique nontrivial and nonnegative solution of the b.v.p. in (1.2) and uf(x) < °¢f , dx E Sl, V f E LO (Sl). Also, we may extend the definition of uf. To that purpose, for each f E L+ (1l) (and in the same way, for each f E L°°(Sl)), we will denote by u f the maximal nonnegative solution of the b.v.p. in (1.2). Then u f - 0 iff of (-a + f) > 0
and u f is strictly positive in Sl if al (-a+ f) < 0. The next theorem shows that, under hypotheses [H], our control problem has a solution.
Nonlinear Optimal Control Problems
223
Theorem 2.1. Under hypotheses [H], the optimal control problem (1.2), with A = L+ (Sl), has a solution.
Sketch of proof. We may assume that a > 0. Otherwise, a < 0, and therefore, V f E L+ (1?), of (-a + f) > a, (-a) > o j (0) > 0. Then, u f 0, V f E L+ (1) and consequently, J(f) 0, V f E L+ (St). So, the optimal control must be the function f identically zero. Now, by assuming a > 0, for proving the existence of optimal control, the basic idea is to prove that the possible optimal controls must be bounded. Is this statement intuitive? First, recall that we must be interested only in those f E L' (Q) such that u f > 0 in fl, since if u f =- 0, then J(f) < 0 and J(0) = 0. Moreover, if u f > 0 in fl then the quantity \U f (x) f (x) - f 2 (x) may be written as
auf(x)f(x)
- f2(x) = (Auf(x) - f(x))f(x) < N - f(x)) f(x)
which is negative if f (x) > b . This reasoning suggests that the possible optimal control must be bounded by ba . This suggestion may be proved rigorously. In fact, if f E L+ (ft) and g = min { f,
b
} ,
then J(g) > J(f ).
Moreover, J(g) > J(f) if f > g on a subset of positive measure. This proves that J : L+ (St) - R is bounded from above. Let us take ba a maximizing sequence f,,, for J. Then we may assume 0 < f,,, < and by elliptic estimates, for a fixed ry > n, IIuf.,. II W2., < c, where c is independent of f,,, (W2.ti denotes the usual Sobolev space). Consequently, we may assume f,,, - f E L+ (St), weakly in L2(11) and u f. - u f strongly in HH (SZ) . Now, since 111112 < lim inf 11fm 112 (11.112 is the usual norm in
f
L2(SZ)) and lim fn Aufmfm = )tuff, n n
J(f) = f Au f f - I f 2 >_ lim sup J(fm) = sup n
n
J.
L+ (S?)
The previous result provides the existence of optimal control, but after it, we may state a logical question, related to the benefit associated to the optimal control problem: will it be always positive? The answer is contained in the next result.
Theorem 2.2. Let us consider the optimal control problem (1.2), under hypotheses [H] and A = L+ (SZ). Then,
sup 9E L+ (St)
J(g) > 0 a al(-a) < 0.
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A. Canada, J.L. G&mez, J.A. Montero
Proof. If the benefit is positive, then there exists g E L+ (S2) such that
J(g) > 0. So, u9 > 0 and al(-a) < a,(-a + g) < 0. Reciprocally, if of (-a) < 0 is assumed, then uo > 0 in Sl. Now, choosing a sequence of positive real numbers e,1 -, 0, and taking into account that ui,, -+ uo, in C1(SZ) (see [1]) , we obtain J(e,+) = e,,
Au, -
which is
positive if n is sufficiently large.
3
Uniqueness of the optimal control
With the purpose of studying the (possible) uniqueness of the optimal control we may try to investigate the regularity of the functional J and the monotonicity properties of its Frechet derivative X. As we are going to see, this may be carried out under an additional restriction on the size of
the parameter A. By looking at the functional J in (1.2), it is clear that, previously, we needed to examine the "differentiability" of u f with respect
to f. Lemma 3.1. Assume hypotheses [H] and a,(-a) < 0. Consider the open set B = If E LOO(11) : al(-a + f) < 0}. Then the operator U : B CI (U), U(f) = u f, is of class C'. Moreover, U'(f)(g) = vf,9,
V f E B,
V g E LO°(SZ),
(3.1)
where v f,9 is the unique solution of the linear problem
-Av = [a - f - 2bu f]v - gu f in St,
v = 0 on M.
(3.2)
The proof is rather technical and uses some properties about elliptic operators of the Schrodinger type -Au + q(x)u, q E L°°(c), some other properties of the eigenvalue of (q), monotonicity properties of u f with respect to f and LP theory of elliptic problems. First of all, if f E B is given, of satisfy the g E L°°(fI), and A is small, the functions vp = equation
-Avp +[-a+ f + b(uf+p9 + u f)]vp = -guf+p9 in 0,
vp = 0 on M. If f E B and g E L°° (S2) are fixed, it may be proved that the functions vp are bounded, independently of p, in W2,1(S2) for fixed ry > n. This, the previous expression, and standard reasonings, prove the Gateaux differentiability of U and that its Gateaux differential is given by (3.1), (3.2). From this fact, and from the uniqueness of solutions of (3.2), we obtain the Frechet differentiability of U.
Nonlinear Optimal Control Problems
225
So, the previous lemma implies that J : B - 1, f i--' J(f) = J (,\U f f n
f2) is Frechet differentiable and
J'(f)(g) = J (Avt,9f + \uf g - 2fg), V f E B, V g E L°°(fl). n
At this point, we need to study the quantity (J'(f) - J'(g)) (f - g). At first, it is convenient to find a more manageable expression for Y. In order to do so, let us define Pf as the unique solution of the linear problem
-APf + (-a + f + 2bu f)Pf = f in fl,
Pf =0 on all. Then an elementary computation proves quently,
J'(f)(g)
J
v f,9 f + f gu fPf = 0. Conse-
= j (-AgufPf + Au fg - 2fg) = j(Aufg(1 - Pf) - 2fg).
Now, if a,(-a) < 0 and A is sufficiently small, l0, as b
C B, and L-(0)
V(f) - J'(g))(f - g) < Jn [AL(f - g)2 - 2(f - g)2] where L is the Lipschitz constant of the Lipschitzian continuous mapping f0, aal L
b
L°°(11), f
uf(1 - Pf). It is clear that this reason-
L°D(n)
ing provides the uniqueness of the optimal control if the parameter A is sufficiently small. So, we have the following theorem.
Theorem 3.2. Let us consider the optimal control problem (1.2), where we assume that the domain Sl and the functions a and b are fixed and satisfy hypotheses [H] and al (-a) < 0. Then, if A is sufficiently small, the problem has a unique optimal control. By using similar ideas, it is possible to prove the uniqueness of the optimal control in other situations different from the previous one. For example, this is the case if we fix fl, the function a and the parameter A and consider b as a sufficiently large constant. Moreover, it is possible to treat other cases where b is not necessarily a constant function ([5], [8]).
226
A. Canada, J.L. Gdrnez, J.A. Montero
4 The optimality system. Further qualitative properties Until now, we know that, under hypotheses [H], the optimal control problem
(1.2) has a solution in the space L+ (ft), and that if the parameter A is sufficiently small and al(-a) < 0, then the optimal control is unique. But several questions remain unanswered. For instance: Has the optimal control some kind of regularity? Is it possible to approximate the optimal control? etc. Next, we treat this type of questions, by first deducing the optimality system.
Theorem 4.1. Assume hypotheses [H] and a, (-a) < 0. Then, if \ is sufficiently small, any optimal control f E L+ (f1) may be written in the form A
f = 2uf(1 - Pf), where the pair (u f, Pf)
(u, p) satisfies
00 a.e.inf2 and the optimality system
-Au = u (a - [b + 2 (1 - P)Au)
Za
inc,
-Op+p(-a+2bu) = 2u(1 -p)2 in f2,
u=p=Oonc. Sketch of proof. If f E L+ (fl) is an optimal control and g E LOO(fl) verifies f + fag E L+ (f l) as Q - 0+, then J(f + Q9 - J(f) < 0. That is,
I r of +R9Q - of (f + Q9) + auf9- 9f-Q9 2 21 J nL
< 0.
Letting Q -+ 0+, we have
+ Au fg - 29f) < 0,
in
where v is defined by (3.2). So, f g[Auf(1 - Pf) - 2f] < 0, d g E L+ (f2). n
Therefore,
f>
2
u f(1 - Pf) a.e. in f2.
(4.4)
Nonlinear Optimal Control Problems
227
Now, taking g = -f we obtain, by similar reasoning, that
f = 2uf(1 - Pf) a.e. in Sl fl {x E 1 : f (x) > 0}.
(4.5)
From (4.4) and (4.5) we obtain
f = 2 uf(1 - Pf)+ a.e. inn.
(4.6)
Moreover, if the parameter A is sufficiently small, then it may be proved
that the function Pf satisfies the inequality 0 < Pf < 1 a.e. in 11. This argument proves the theorem. Some consequences of the previous theorem are the following:
1. If the parameter A is sufficiently small, then, taking into account the equations satisfied by the pair (u, p), we obtain that any optimal control f E L+ (St) must belong to C(S1) and therefore to W2,7(c2), V -y E (1, oo).
2. Uniqueness of the optimal control: it is possible to prove, without using the regularity of the functional J, that if A is sufficiently small, then there is a unique solution of the optimality system (4.3), satisfying the additional conditions (4.2) ([4]).
3. Approximation of the optimal control: by using the optimality system and the expression (4.1) it is possible to provide, for A sufficiently small, a constructive scheme, based on the upper and lower solution notion for systems of equations, which gives a sequence of functions converging to the unique optimal control. To see this, let us observe that the optimality system is of the form -Du(x) = B(x, u(x), p(x)), x E S2, -1p(x) = C(x, u(x), p(x)) + D(x, u(x), p(x)), x E 11, u(x)=p(x)=0, xE8S1,
(4.7)
where
B(x,u,p)=u(a-[b+2(1-p)]u) C(x, u, p) = p(a - 2bu), D(x, u, p) = 2u(1 - p)2. Let us point out that the functions B, C and D fulfill the following hypotheses of regularity and monotonicity, if the variable (u, p) belongs to a bounded subset S of R2:
B, C and D are continuous with respect to (u, p) E R2, [Hi]
for fixed x E tions L°°(D).
St.
u(.), p(.)),
Moreover, Vu, p E L°°(Q), the funcu(.), p(.)) and p(.)) belong to
228
A. Canada, J.L. Gnmez, J.A. Montero
3 M > 0 such that the function B(x, u, p) + Mu is increasing in u and the functions C(x, u, p) + p, D(x, u, p) + 2p are increasing in p, for (x, u, p) E Si X S. Moreover, B(x, u, p) is increasing in p, the function C(x, u, p) is decreasing in u and
2
[H2]
the function D(x, u, p) is increasing in u, for (x, u, p) E S2 x S.
Observe that, from an abstract and general point of view, for systems like (4.7) and functions B, C and D satisfying hypotheses [H1] and [H2], if C =_ 0, we have a system of cooperative type, whereas if D - 0, we have a system of predator-prey type. In our case, C and D are both nonidentically zero. However, it is possible in this case (and, in general, under hypotheses [H1] and [H2]) to prove an existence and approximation theorem about the solutions of (4.7), based on the notion of a system of upper-lower-solutions for systems of equations (see [4]). This allows us to prove the next theorem.
Theorem 4.2. Let us assume hypotheses [H], o1(-a) < 0 and A sufciently small. Consider system (4.3) and let us define u = w, -u = b , p = 0,
$ = AQ, where w is the maximal nonnegative solution of
-Ow=wla-2w-bw] in St, w=0 on d12, and Q is the unique solution of
-iQ + (-a + 2b(uo - e))Q =
6
in S2, Q = 0 on 852,
with a sufficiently small. Define by induction the sequences {un}, {un}, {Pn}, {pn}, as
u1=u, u1=u, P1=P, P1=P, tun + Mun = B(x, un-1 , Pn-1) + Mun-1 in St, un = 0 on 8Sl,
Dun + Mun = B(x, 0-1,pn-1) + Mun-1 in 0, un=Oon8S2,
-tPn + MPn = C(x,un-',Pn-1) +
M 2
Pn-1 + D(x, un_i,p _1) + 2Pn-1 in 0,
Pn=0on8St,
Nonlinear Optimal Control Problems
229
-Ap" +Mp" = D(x,u"-',p"-1) +
+Mp"-' +
M
p"-' in Q.
p"=0on&1, where M is a sufficiently large positive constant. Then the following order relation is fulfilled: U1
u,2 < ... < u < un C un-1 < ... < ul,
P1
P2 <...
for all x E fZ and
un / u, u" \ u, pn / p,
pn
\ p,
(pointwise), where (u, p) is the unique solution of (4.3) which satisfies (4.2).
5
Final remarks and questions
1. Let us observe that in the problem that we have studied here we are allowed to modify the intrinsic growth rate of the species u, i.e., the function a, only in a negative way (the control f must belong to L`0(0)). But, what happens if we allow that this modification may be positive and negative, i.e., f belongs to LO°(SZ)? Let us remark that we are expanding the control space. However, until now, we have not been able to prove even the existence of optimal control. The main difficulty for proving this is that, in this case, we do not know if the possible optimal controls are bounded.
2. The second remark is related to the case where we may change the domain Q. More precisely, let us consider the case where St is an open ball
in R' of radius R > 0. If we assume that the functions a and b (which must be defined on R') and the parameter A are fixed, what is the range of values of R for which the optimal control is unique? Is it possible to do similar things for other kinds of domains in R"?
REFERENCES
[1) J. Blat and K.J. Brown, Bifurcation of steady-state solutions in predator-prey and competition systems, Proc. Royal Soc. Edinburgh 97 (1984), 21-34.
[2) A. Canada, J.L. Gamez and J.A. Montero, An optimal control problem for a nonlinear elliptic equation arising from population dynamics. In Proceedings of the Second European Conference on Elliptic and Parabolic Problems, Pont-t -Mousson, France, 1994 (C. Bandle et al.
A. Canada, J.L. Gamez, J.A. Montero
230
eds.), Longman, Pitman Research Notes in Mathematics Series 326 (1995), 35-40.
[3] A. Canada, J.L. Gimez and J.A. Montero, Uniqueness of the optimal for a nonlinear elliptic control problem arising from Population Dynamics, Proceedings of XIV C.E.D. Y.A., Vic, Spain, 1995 (http://www-mal. upc.es/cedya/cedya.html). [4] A. Canada, J.L. Gamez and J.A. Montero, Study of an optimal control problem for diffusive nonlinear elliptic equations of logistic type, SIAM J. Control Optim. 36 (1998),1171-1189 (http://www.siam.org/ journals/sicon/36-4/29332.html).
[5] J.L. Gamez and J.A. Montero, Uniqueness of the optimal control for a Lotka-Volterra control problem with a large crowding effect, ESAIM: Control Optimization and Calculus of Variations 2 (1997), 1-12 (http://www.emath.fr/COCV). [6] A. Leung and S. Stojanovic, Direct methods for some distributed games, Differential Integral Equations 3 (1990), 1113-1125. [7] A. Leung and S. Stojanovic, Optimal control for elliptic Volterra -Lotka equations, J. Math. Anal. Appl. 173 (1993), 603-619.
[8] J.A. Montero, A uniqueness result for an optimal control problem on a diffusive elliptic Volterra-Lotka type equation, J. Math. Anal. and Appl. No. 243 (2000), 13-31.
A. Canada, J.L. Gamez and J.A. Montero Departamento de Analisis Matem£tico Universidad de Granada 18071-Granada
Spain jmonteroougr.as
On The Use of Time-Maps in Nonlinear Boundary Value Problems Anna Capietto ABSTRACT Existence and multiplicity of solutions (with prescribed nodal prop-
erties) for the two-point boundary value problem associated to a second order ODE is proved. First, we consider an autonomous sublinear problem; secondly, we deal with a nonautonomous superlinear asymmetric one.
1
Introduction
In this paper we deal with the two-point boundary value problem r u11 + j(
f (t, u) = 0, u(0) = 0 = u(ir),
(1.1)
where f : [0, ?r] x R - R is continuous. Recently, in collaboration with W. Dambrosio, existence and multiplicity of solutions to (1.1) have been studied by a topological degree method associated with a time-map technique. More precisely, in [6] we studied the "asymmetric superlinear" case, while in [7] a "sublinear" condition near the origin is assumed. In both situations, an autonomous problem related to (1.1) is first studied by a time-map technique; then, a suitable continuation theorem (on the lines of [27,28]) is shown to be applicable by the performance of suitable a priori bounds on the number of zeros of the (possible) solutions to a parameter dependent problem associated to (1.1). Problems of this kind have been widely treated in the literature; see e.g., the works [2, 3 ,4, 9, 10, 11, 14, 16, 17, 23, 24, 26, 32]. For more references, we refer to the bibliography in [6, 7, 8, 13).
The aim of this paper is to illustrate and develop some of the ideas in [6, 7].
Work performed under the auspices of GNAFA-C.N.R and in the frame of the EEC project "Nonlinear boundary value problems: existence, multiplicity and stability of solutions", grant CHRX-CT94 - 0555. Supported by I.N.D.A.M., Italy. Keywords: two-point boundary value problem, continuation theorem, time-maps.
232
A. Capietto
In Section 2 we deal with an autonomous equation of the form u"+g(u) = 0, where g(x)/x = +oo. In this situation (arguing as in [7]), we prove the existence of infinitely many solutions with prescribed nodal properties. Moreover, by adding an assumption at infinity a lower bound on the number of zeros of the solutions is obtained. Then, we give a result which guarantees the existence of exactly four solutions with n zeros, for every n sufficiently large. For related results see [5, 25, 29]. In Section 3 we study a nonautonomous equation of the form u" + Au +
g(t, u) = 0, where A > 0 and lima-.o g(t, x)/x = go > 0, limx.+00 g(t, x)/x
= +00, limx,og(t, x)/x = 0, uniformly in t. It is easily shown that Theorem 4.2 in [6] is applicable, and for A sufficiently large the existence of solutions with an arbitrarily large number of zeros is proved. For related results see [1, 15,18,33,34].
2 An autonomous problem We are concerned with the second order differential equation
u" + g(u) = 0 together with the two-point boundary condition u(0) = 0 = u(7r).
We assume that g : R -a R is continuous. Let us set G(u) = fu g(s)ds and suppose g(u)u > 0 Vu # 0
(2.3)
and
lim
JuI+00
G(u) = +00.
(2.4)
In what follows, we describe a "time-map" technique for (2.1)-(2.2); to do this, we follow Section 2 in [12] (see also [13]). At this stage, it is important
to remark that in the sequel assumptions stronger than (2.3)-(2.4) on g shall be made; however, for the moment we describe the (almost) minimal set of assumptions when the "time-map" is defined. First of all, we observe that (2.3) guarantees that every Cauchy problem associated to (2.1) has exactly one global solution; we also recall that the solutions of equation (2.1) satisfy the energy relation 21
H (u(t), u'(t)) = u1(t)2 + G(u(t)) = const.
Vt E [0, 7r].
On The Use of Time-Maps in Nonlinear Boundary Value Problems
233
In the phase-plane (x, y) _ (u, u') this means that every orbit of (2.1) belongs to the closed curve defined by the equation 1
H(x,y) = y2 + G(x) = c 2 for some c > 0. In particular for every a > 0 the (unique) solution of the Cauchy problem
I un + 9(u)
(2 5)
u(0) = 0, =u'(0) 0, = a,
which will be denoted by
.
0, a), satisfies the relation
2 u'(t)2 + G(u(t)) = 2 a2
Vt E [O, ir1.
(2.6)
We observe that (2.6) holds true for u(.; 0, -a) as well. Let us define the functions T; : (0, +oo) - (0, +oo) (i = 1, 2) by C2(a)
Ti
(a) = Jo
1
a2 - 2G(s)
ds
(2.7)
ds,
(2.8)
and 0
T2 (a)
1
J-Ci(a) VG-272G(s)
where G(-Ci(a)) = G(C2(a)) = Za2 and C;(a) > 0 (i = 1,2). It is straightforward to check, integrating (2.6), that they represent the time needed for a rotation along the orbit of energy 2a2 in the upper (lower) half plane from the point (0, a) to the point (C2(a), 0) (from (C2(a), 0) to (0, -a) ) and from the point (-Cl (a), 0) to the point (0, a) (from (0, -a) to (-Cl(a),0)), respectively. A simple analysis in the phase-plane (whose details can be found in [6, Prop. 2.21) leads to the following (see Figure 1):
Proposition 2.1. Problem (2.1) - (2.2) has a solution of energy 2a2 if and only if there exists an integer n E N such that 2nTi(a) + 2nT2(a) = 7r or
2nTi(a) + 2(n + 1)T2(a) = a or
2(n + 1)T1 (a) + 2nT2(a) = ir.
A. Capietto
234
Moreover, if u is a solution of (2.1) - (2.2) with energy za2, then (i) u has exactly 2n + 1 zeros in [0, ir) and u'(0) > 0 q (Tl (a),T2(a)) E an :_ {(x, y) E Q : 2(n + 1)x + 2ny = a} (n > 0); (ii) u has exactly 2n + 1 zeros in [0, ir) and u'(0) < 0 q (Tj(a), T2 (a)) E c,, := {(x, y) E Q : 2nx + 2(n + 1)y = 7r} (n > 0); (Tl(a),T2(a)) E b := {(x,y) E Q : (iii) u has exactly 2n zeros in [0,7r) 2nx + 2ny = 7r} (n > 1). (We have set Q := {(x, y) E R2 : x > 0,
01
W
2n
y> 0}).
Z 4
FIGURE 1. The set S and the regions A;,n.
By Proposition 2.1, in order to study (2.1)-(2.2) we are led to discuss the existence of intersections between the support of the curve T : (0, +oo) -'
S
2
On The Use of Time-Maps in Nonlinear Boundary Value Problems
235
R2 defined by T(a) = (Tl(a),T2(a)) and the set S = {(x, y) E Q : 2(n + 1)x + 2ny = it or 2nx + 2ny = Tr or 2nx + 2(n + 1)y = 7r for some n E N) = (Un>oan) U (Un>lbn) U (Un>ocn) (see Figure 1). The existence (and number) of these intersections can be investigated by means of the mutual position of the points Po(xo, yo) and P.,, (x00, yam), where
xo = limx.oTi(a), yo = limx-.oT2(a) and
xOO = limy.+o0Ti(a),
(2.10)
Y. = limy....+0T2(a).
The set S is called "generalized FuLik spectrum" (cf.[19]). As a matter of fact, the limits in (2.9)-(2.10) can be computed as soon as information on the behaviour of the ratio g(x)/x near the origin and at infinity is available. Indeed, Z. Opial [30] proved the following
Proposition 2.2. ([30, Cor.12]) Let g satisfy (2.3) - (2.4). If lim inf g(x) = k and lim sup g(x) = K (0 < k < K < +oo), x x-+oo x g(x) ( Run inf = h and lim sup g(x) = H (0 < h < H < +oo),
x-.-oo
-
-
x
X
respectively) , then lim sup Tl (a) = a--0+00
(lim sup T2(a) _ a--.+00
2h
2 7r
and lim inf Tl (a) =
0-,+
and lim inf T2 (a) =
a-+00
2y n
,
r 27, respectively).
Remark 2.3. According to [30], a result analogous to Proposition (2.2) is valid when the asymptotic behaviour of the ratio g(x)/x as x tends to zero is known.
Now, we are in position to state the following multiplicity result for (2.1) - (2.2), which can be found (up to minor modifications) in [7].
Theorem 2.4. Assume that g satisfies gx lim X-0 x = +oo;
(2.11)
then, there is no E N such that for every n > no problem (2.1) - (2.2) has two solutions un and vn with u;,(0) > 0 and vn(0) < 0, both having exactly n zeros in [0, ir). Moreover, limn-+00 IIujI = limn.+o0 IIvnlI = 0.
236
A. Capietto
Proof. We follow the ideas already developed in [7]. According to Proposition 2.2 (cf. Remark 2.3), assumption (2.11) guarantees that Po = (0,0), i.e., the support of the time-map "emanates" from the origin. Then, it is
sufficient to recall that the lines constituting the set S accumulate near the origin in order to ensure that, for n sufficiently large and for some a, the point (Tl (a), T2 (a)) is over the lines an, bn, cn, and the first part of the statement is proved. The nodal properties of the solutions easily follow from Proposition 2.1.
Theorem 2.4 guarantees the existence of solutions with an arbitrarily large number of zeros; one can then raise the question whether it is possible to establish a lower bound on the number of zeros of the solutions to (2.1)-(2.2). Indeed, this can be obtained by adding an assumption on the behaviour of g(x)/x at infinity, which (using again Proposition 2.2) provides the coordinates of the point P. More precisely, we have:
Theorem 2.5. Assume that g satisfies (2.11) and lim sup g(x) = p < +00; x--.+oo
x
(2.12)
let I E N be the smallest integer such that 6 < 12. Then, for any n > 21 problem (2.1) - (2.2) has two solutions un and vn with u' (0) > 0 and v;,(0) < 0, both having exactly n zeros in [0, ir). Moreover, limn+,, I Iun1l _ llmn-+oo [1vn11 = 0.
Proof. According to Proposition 2.2, by (2.12) we have lim info.+oo Ti (a)
= 277; by the choice of 1, this means (cf. Figure 1) that the point P,,. belongs to the half-plane {(x, y) E R2 : x > ' }; hence, the support of the curve T : (0, +oo) - R2 meets all the lines a,n, b,,,, c,n with m > 1; in particular, the existence of a pair of solutions with 21 zeros, together with solutions with a higher number of zeros is ensured.
The superlinear case when lima-+oo g(x)/x = +oo, which corresponds to the situation when the time-map is infinitesimal (for a -- +oo), will be treated separately in Theorem 2.8 below.
Remark 2.6. Theorem 2.5 guarantees that if the number 6 in (2.12) is "small", then the existence of solutions with a "small" number of zeros is also ensured, while in case ,0 is "large", then only solutions with a "sufficiently large" number of zeros are obtained. Indeed (cf. Figure 1), the former case corresponds to the situation when the abscissa of the point P , is "large", the latter to the situation when the abscissa of the point P... is "small" (recall that, according to Proposition 2.1, the closer to the origin the lines constituting the set S, the greater the number of zeros of the corresponding solutions). As a simple consequence of Theorem 2.5, we can ensure the existence of positive solutions.
On The Use of Time-Maps in Nonlinear Boundary Value Problems
237
Corollary 2.7. Assume that g satisfies (2.11) - (2.12); if Q < 1, then problem (2.1) - (2.2) has one positive solution.
Proof. It is sufficient to observe that condition p < 1 guarantees that the support of the curve T : (0, +oo) - R2 intersects the line T1 = x/2.
It is clear that a result analogous to Corollary 2.7 on the existence of negative solutions is valid when assumption (2.12) is replaced by a condition
on the behaviour of g at -oo. We end this section by considering the superlinear case. In this situation, by adding a technical assumption, one can ensure the existence of exactly
four solutions with n zeros, for all n > N, for some N which could be explicitly computed when g is given. More precisely, we have
Theorem 2.8. Suppose that g satisfies (2.11); assume lim g(x) = +oo, x_.±oo X
(2.13)
and that there exist x2 < 0 < x1 such that g( ) > 0
Vx
> xl,
j
9(x )
<0
b'0 < x < xl,
( 2 . 14 )
g(am )
Vx
< x2i
d g(x )
<0
V X2
<0 .
( 2 . 15 )
x
>0
<x
Then, there is N such that for all n > N problem (2.1) - (2.2) has exactly four solutions un, vn, wn and zn with u;,(0) > 0, v;,(0) < 0, w;,(0) > 0 and z'n(0) < 0, all having exactly n zeros in [0, rr). Moreover, limn-+o. IIunII = Hurl[ = 0, limn-»+oo IIwnII = limn-»+oo IIznII = +00.
Proof. According to assumptions (2.11) and (2.13), we have Po = P"', = (0, 0). Hence, arguing as in [7], we can prove that the support of the curve T intersects infinitely many lines of the set S twice (for two different values of a). This is sufficient to guarantee the existence of at least four solutions with a fixed number of zeros. Moreover, conditions (2.14)-(2.15) imply (cf. [31, Prop. 1.3.4]) that the functions T1 have a unique maximum T1(6) (see also [30, Th. 8]) ; then, there is N such that the point (T1(a),T2(a)) belongs to each region Ai,n for all n > N and does NOT belong to the regions Ai,n for n < N (cf. Figure 2). This is sufficient to conclude the proof.
Example 2.9. Consider the function g(x) = f + x3 for x > 0, g(x) = -g(-x) for x < 0. It satisfies all the assumptions of Theorem 2.8; the existence of the points x1, x2 implies the existence of unique maxima T; (a)
for the time maps Ti; as a consequence, the time map T "evolves" in the
set Q as shown in Figure 2. i.e., the regions Ai,n, n < N = N(a) are not "touched" by the time-map. Note that Figure 2 illustrates only the qualitative behaviour of the functions examined.
238
A. Capietto
FIGURE 2.
3 A nonautonomous problem In this section, we deal with the following nonautonomous BVP U" + Au + g(t, u) = 0,
(3.1)
j u(0) = 0 = u(zr),
where g : [0, 7r] x R - R is continuous and A > 0. We assume the following conditions: (K1 )
= + oo
limx-+OO ALx X
(K2)
li mz-_OO
xx = 0 unifo rml y in
limx-o 21zx
(K3)
= go
t;
unifo r mly in
uniformly in
t;
t.
Existence and multiplicity of solutions (with prescribed nodal properties) shall be established in what follows for (3.1) by the application of Theorem 1 in [6]. For the reader's convenience, we state this result below, in a simplified version which is sufficient for the application we have in mind.
Theorem 3.1. ([6, Th. 4.2]). Let f : [0, ir] x R -+ R be a continuous function satisfying the following conditions:
(H1)
limX-+00 L y _ +oo uniformly in t;
(H2)
there exists a positive continuous function a such that lim
x-+-oo
f (t, x) = a(t) x
uniformly in t;
On The Use of Time-Maps in Nonlinear Boundary Value Problems
(H3)
239
there exists a constant h > 0 such that lim f (t' x) = h
x-+o*
X
uniformly in t.
Let no = [Ohi], and let 1 be the largest non-negative integer such that 12 < a(t), for all t E [0, ir]. Then, if no + l < 21,
(H4)
for every integer k E [no + 2,21], problem
I u" + f (t, u) = 0, 1 u(0) = 0 = u(n),
(3.2)
has at least one solution uk with uk(0) > 0 and one solution vi, with vk(0) < 0, both having exactly k zeros in [0, 7r).
We are now ready to state the following
Theorem 3.2. Thereis Ao > 0 such that for all A > A0 there areK(A) E N and two solutions UKlal, VK(A) with uK(A) (0) > 0, v'Kial (0) < 0 of (3.1) having exactly K(A) zeros in (0, 7r). Moreover, lima K(A) = +oo.
Proof. We apply Theorem 3.1, with f (t, x) = Ax + g(t, x). Indeed, it is immediate to check that (K1) - (K2) - (K3) guarantee the validity of (H1)-(H2)-(H3) in Theorem 3.1, with a(t) - A, h = go+A. In particular, no = [ g + Aj. As far as condition (H4) in Theorem 3.1 is concerned, we observe that it is implied by go < (21 - 1)2 - A. It is immediate to check that for I > 2 the right-hand side of this inequality is positive; moreover, recalling that I is a function of A, one can see that lima.,,.(2l - 1)2 - A = +oo. This fact guarantees the validity of (H4) for A sufficiently large, and
the first part of the theorem is proved. The same argument used above enables us to say that the amplitude of the interval [ [ go + + 2,211 tends to infinity.
4
0
Final remarks
1. We point out that the function which associates to each solution its number of zeros is well defined (and continuous) because of the sign con-
dition on g (as for Section 2) and of assumption (K3) on the behaviour of g near the origin (as for Section 3); in other words, these conditions guarantee that solutions have a finite number of zeros. For details on this subject, we refer to [21, 6, 7].
240
A. Capietto
2. With the addition of a technical assumption, results analogous to Theorems 2.4, 2.8, 3.2 in this paper can be obtained when existence and multiplicity of radially symmetric solutions to a BVP on the ball of the form
f Du + f (IxI, u) = 0
ju=0
in B, on 813
(4.1)
is investigated (cf. [8]). 3. All the above results can be generalized to the situation when the differential operator u .-- u" is replaced by a strongly nonlinear operator of the form u '- cp(u), where cp : R - R is an odd increasing homeomorphism satisfying suitable assumptions; for example, we can deal with the p-laplacian operator u '-- IuIp-2u, p > 2. We refer for details to (8, 13, 20, 21, 22] and references therein.
Acknowledgments. It is my pleasure to thank the organizers of the Autumn School on Nonlinear Analysis and Differential Equations for the invitation and the hospitality. REFERENCES [1] D. Arcoya and S. Villegas, Nontrivial solutions for a Neumann problem
with a nonlinear term asymptotically linear at -oo and superlinear at +oo, Math. Z. 219 (1995), 499-513.
[2] M. Arias and J. Campos, Exact number of solutions of a onedimensional Dirichlet problem with jumping nonlinearities, Differential Equations and Dynamical Systems 5 (1997), 139-161.
[3] A. Bahri and H. Berestycki, Forced vibrations of superquadratic Hamiltonian systems, Acta Math. 152 (1984), 143-197.
[4] A.K. Ben-Naoum and C. De Coster, On the existence and multiplicity of positive solutions of the p-Laplacian separated boundary value problem, Differential Integral Equations 10 (1997), 1093-1112. [5] G.J. Butler, Periodic solutions of sublinear second order differential equations, J. Math. Anal. Appl. 62 (1978), 676-690. [6] A. Capietto and W. Dambrosio, Multiplicity results for some two-point superlinear asymmetric boundary value problem, Nonlinear Analysis TMA 38 (1999), 869-896.
[7] A. Capietto and W. Dambrosio, Boundary value problems with sublinear conditions near zero, NoDEA 6 (1999), 149-172.
[8] A. Capietto, W. Dambrosio and F. Zanolin, Infinitely many radial solutions to a boundary value problem in a ball, Quad. Dip. Mat. Univ. Torino, Annal. Mat. Pura Appl., to appear.
On The Use of Time-Maps in Nonlinear Boundary Value Problems [9]
241
A. Castro and R. Shivaji, Multiple solutions for a Dirichlet problem with jumping nonlinearities, II, J. Math. Anal. Appl. 133 (1988), 509528.
[10] Y. Cheng, On the existence of radial solutions of a nonlinear elliptic bvp in an annulus, Math. Nachr. 165 (1994), 61-77. [11] D.G. Costa, D.G. De Figueiredo and P.N. Srikanth, The exact number of solutions for a class of ordinary differential equations through Morse index theory, J. Differential Equations 96 (1992), 185-199. [12] W. Dambrosio, Time-map techniques for some boundary value problem, Rocky Mountain J. Math. 28 (1998), 885-926.
[13] W. Dambrosio, Boundary value problems for second order strongly nonlinear differential equations, Ph.D. thesis, 1998.
[14] H. Dang, K. Schmitt and R. Shivaji, On the number of solutions of boundary value problems involving the p-Laplacian and similar nonlinear operators, Electr. J. Differential Equations 1 (1996), 1-9.
[15] D.G. De Figueiredo and B. Ruf, On a superlinear Sturm-Liouville equation and a related bouncing problem, J. Reine Angew. Math. 421 (1991), 1-22.
[16] G. Dinca and L. Sanchez, Multiple solutions of boundary value prob-
lems: an elementary approach via the shooting method, NoDEA 1 (1994), 163-178.
[17] M.J. Esteban, Multiple solutions of semilinear elliptic problems in a ball, J. Differential Equations 57 (1985), 112-137. [18] C. Fabry and P. Habets, Periodic solutions of second order differential equations with superlinear asymmetric nonlinearities, Arch. Mat. 60 (1993), 266-276. [19] S. Fucik, Solvability of Nonlinear Equations and Boundary Value Problems, Reidel, Boston, 1980.
[20] M. Garcia..Huidobro, R. Manasevich and F. Zanolin, Strongly nonlin-
ear second order ODE's with rapidly growing terms, J. Math. Anal. Appl. 202 (1996), 1-26. [21] M. Garcia-Huidobro, R. Manasevich and F. Zanolin, Infinitely many solutions for a Dirichlet problem with a non-homogeneous p-Laplacian like operator in a ball, Advances in Differential Equations 2 (1997), 203-230.
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[22] M. Garcia-Huidobro and P. Ubilla, Multiplicity of solutions for a class of nonlinear second-order equations, Nonlinear Analysis TMA 28 (1997), 1509-1520. [23] G. Harris and B. Zinner, Some remarks concerning exact solution numbers for a class of nonlinear boundary value problems, J. Math. Anal. Appl. 182 (1994), 571-588.
[24] J. Hempel, Multiple solutions for a class of nonlinear boundary value problems, Indiana Univ. Math. J. 20 (1971), 983-996.
[25] M.A. Krasnosel'skii, A.I. Perov, A.I. Povolotskii and P. P. Zabreiko, Plane Vector Fields, Academic Press, New York, 1966. [26] A.C. Lazer and P.J. McKenna, On a conjecture related to the number of solutions of a nonlinear Dirichlet problem, Proc. Roy. Soc. Edinburgh Sect. A 95 (1983), 275-283. [27] J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS Series, vol. 40, Amer. Math. Soc., Providence, RI, 1979.
[28] J. Mawhin, C. Rebelo and F. Zanolin, Continuation theorems for Ambrosetti-Prodi type periodic problems, preprint. [29] V. Moroz, Solutions of superlinear at zero elliptic equations via Morse theory, Topological Methods in Nonlinear Analysis 10 (1997), 387-397. [30] Z. Opial, Sur les periodes des solutions de 1'equation differentielle x" +
g(x) = 0, Ann. Polon. Math. 10 (1961), 49-72. [31] R. Schaaf, Global Solution Branches of Two Point Boundary Value Problems, LNM 1458, Springer-Verlag, Berlin, 1990.
[32] B.L. Shekhter, On existence and zeros of solutions of a nonlinear twopoint boundary value problem, J. Math. Anal. Appl. 97 (1983), 1-20.
[33] M. Struwe, Multiple solutions of anticoercive boundary value problems for a class of ordinary differential equations of second order, J. Differential Equations 37 (1980), 285-295. [34] B. Zinner, Multiplicity of solutions for a class of Superlinear SturmLiouville problems, J. Math. Anal. Appl. 176 (1993), 282-291. Anna Capietto Dipartimento di Matematica University di Torino Via Carlo Alberto, 10 - 10123 Torino Italy capiettoOdm.unito.it
Some Aspects of Nonlinear Spectral Theory Pavel Drabek
Let us give the following simple motivation which arises in such a fundamental subject as the Sobolev imbedding theorems. It is well known that for fl C RN, a domain, the continuous imbedding Wo'p(fl)
`-
L9(0)
(1.1)
holds provided p > 1, q > 1 and N > 1 satisfy certain relations and that under some additional restrictions this imbedding is compact,
Wo'p(IZ) -- LQ(fl)
(1.2)
(see e.g. Adams [1) or Kufner, John and Futile [16]). Denoting by II ' Il9 and Il Il1,p the norm in L9(fl) and in WW'p(Sl), respectively, the imbedding (1.1) expressed in terms of norms reads as follows: there exists C > 0 independent of u E Wo'p(Sl) such that (1.3)
IIuII9 <_ CIIuII1,p
holds for any u E W01"(0). Due to the Friedrichs inequality (see (1), [16)) the last assertion can be restated also as 11U119 :5 Clloullp
(1.4)
for any u E Wo'p(Sl), where C > 0 does not depend on u. To make the notation clear we note that 1/p
IIu!I1,p := Iloullp = I I IVu(x)Ip dx)
The following natural question arises when studying more carefully (1.4). The research of the author was partially supported by the Grant Agency of Czech Republic 201/97/0395, Ministery of Education VS 97156 and FONDAP de Maternaticas Aplicadas, Chile.
244
P. Drabek
Question 1. Does the best constant C > 0 exist in (1.4) ? Let us remark that "the best constant" in (1.4) means the least number C > 0 for which the inequality (1.4) holds. In fact, such a constant can be characterized as
C =sup
IIUII4
IIouIIP'
or equivalently as 1
C
=inf IIouIIP IIulIq
where sup and inf are taken over all u E W0"(1) \ {0}. Due to the homogeneity of the fractions in (1.5), (1.6), the best constant Cp,q > 0 in (1.4) can be expressed as CP,q = inf{IIVuIIP; u E W0,P(cz), Ilullq
=1}.
(1.7)
Now, the answer to Question 1 generates
Question 2. Does u E Wo'P(1l) exist in which the infinum in (1.7) is achieved ?
If the imbedding (1.2) is compact, then the standard minimizing argument provides a positive answer to Question 2. So, let us assume (1.2) and denote by up,q E Wa''(11) the minimizer for (1.7). Then straightforward
application of the Lagrange multiplier method yields that there exists a real number A > 0 such that
in
IDup,glp 20up,gVW -A
IuP,glp-2uP,gW = 0
(1.8)
holds for any cp E Wo'P(Q). Substituting W = up,q in (1.8) one easily sees
that (1.9)
A= C'P,9.
Moreover, using the standard notation Opu := div(IVulp-2Vu) for the p-Laplacian, the integral identity (1.8) means that up,q is a nontrivial weak solution (i.e., eigenfunction) of the nonhomogeneous eigenvalue problem -APU = AIulq-2u
lu=0
in SZ,
on Of ,
(1.10)
and A given by (1.9) is an associated eigenvalue. Then the following question
arises in a natural way.
Some Aspects of Nonlinear Spectral Theory
245
Question 3. What is the meaning of the spectrum of the nonhomogeneous eigenvalue problem (1.10) and what are its fundamental properties ? If we call nontrivial solutions of (1.10) the eigenfunctions and corresponding values of the spectral parameter A associated eigenvalues of (1.10), then u = uP,q and A = Cy.4 are the principal eigenfunction and associated principal eigenvalue. Now we have to distinguish between two cases p = q and
pi4 q
If p = q (i.e., the problem (1.10) is homogeneous but nonlinear if p 34 2) and p > 1, most properties of the principal eigenvalue and associated eigenfunction are the same regardless if p = 2 or not (Anane [2], Lindqvist [17]). The properties like positivity, isolatedness and simplicity of the principal eigenvalue as well as positivity of the principal eigenfunction are preserved. Also the second eigenvalue can be characterized variationally, see Anane of and Tsouli (3]. Moreover, a sequence of variational eigenvalues (1.10) satisfying a standard minimax characterization can be constructed, but if N > 1 it is not known if this represents a complete list of the eigenvalues. For N = 1, completeness follows from the uniqueness theorem for
the associated initial value problem and was proved e.g., by Drabek [8], Otani [18] and DelPino, Elgueta and Manasevich (6]. The case p 0 q (i.e., the problem (1.10) is nonhomogeneous) is more complicated. First of all it follows from a simple renormalization argument that if A0 > 0 is an eigenvalue of (1.10), then any \ > 0 is also an eigenvalue of (1.10) and the corresponding eigenfunctions are real multiples of those associated with A0. Hence speaking about the eigenvalue of (1.10) we have always to add the normalizing condition for the corresponding eigenfunction. So we can restrict our attention for instance to the eigenfunctions lying on the unit sphere IIufIq = 1. It was proved in Garcia and Peral [13]
that for 1 < p < N, 1 < q < p', where p*: _-, the problem (1.10) has a sequence of variational eigenvalues. However, completeness of the set of eigenvalues as well as its basic properties (eveness of the principal eigen-
value) are not clear at all. It was proved in Huang [15] that the principal eigenvalue of (1.10) is simple if p < q. The proof follows more or less the same lines as that for p = q and does not work for p > q. In fact, an example of ring-shaped domain 11 is given in Garcia and Peral [14], for which the
principal eigenvalue of (1.10) is not simple if q is close enough to p'. (see Drabek also [9]). For N = 1 we can benefit again from the global existence and uniqueness theorem for the associated inital value problem and to get a very transparent picture of the whole spectrum, including analytic expressions for the eigenvalues and associated eigenfunctions (see Drabek and Manhsevich [10]). This picture also suggests how some bifurcation diagrams
should look in the PDE case. Let us go back to the case p = q, p > 1. In this case CP : = CPq satisfies CP1=inf {IIVuIIP:UE WW"P(sl),IIu1IP=1}
(1.11)
246
P. Drabek
and there exists a unique positive in 1 function ul E Wa'p(Q), llul lip = 1, such that
Cpl=llVu,llp. We derive easily that A l = C; P and ul are the principal eigenvalue and associated eigenfunction of the homogeneous (for p = 2 linear) eigenvalue problem
-Apu = Alulpu
in Sl,
lu=0
on all.
(1.12)
It is well known from the linear Fredholm alternative that the boundary value problem
-Du-Alu= f inn, 1 u=0
on 00
(1.13)
f E W - 1,P' (Q) satisfies has a weak solution if and only ifffui=o. (1.14)
Moreover the solution set is an unbounded one dimensional linear set in Wo'p(Sl). Several questions arise if we consider a similar situation for general p > 1. Namely, consider the boundary value problem
Opu -
Allulp-2u
= f inn,
lu=0
on an.
(1.15)
Question 4. How does the condition (1.14) affect the solvability of (1.15) ?
Here the striking difference between p = 2 and p 54 2 appears. The condition (1.14) is not necessary for the solvability of (1.15). A counterexample in the ODE case (N = 1) is constructed in Binding, DrAbek and Huang [4] and DelPino, Drabek and Manbsevich [5]. In the latter paper even more than that is shown: if Sl = (0, 1) there is a function fo E C2[0,1] and
p > 0 such that for any f E L'(0, 1), Ill -folly < p we have fo f uldx # 0 and the boundary value problem
_/lu/lp-2ui)' _ \I
lulp-2u
=f
in (0, 1),
U(0) = u(1) = 0
(1.16)
has at least two solutions. In particular, this result and the homogeneity of the left-hand side of (1.16) imply that the range of the operator A: Wo,p(0,1) -' W-"p'(0,1),
A: U H -(lu'lp-2U,), - allulp-2u,
Some Aspects of Nonlinear Spectral Theory
247
contains a cone with nonempty interior in L°°(0,1) if p 2. A similar result, but with .11 in (1.16) replaced with a certain higher eigenvalue, is proved in Drbbek and Takac [11]. On the other hand it is well known that the range of A for p = 2 is a linear subspace of W-1,p'(0,1) of codimension 1 (and hence it has an empty interior). It should be pointed out here that the range of A is not the whole space W-l"p (0, 1) for p 0 2. For example, taking f - 1, one can show that (1.16) has no solution (see DelPino and Manasevich [71). Coming back to the meaning of the condition (1.14), another interesting phenomenon occurs. Namely, this condition appears to be sufficient in a certain sense. More precisely, it is proved in [5] that given f E C' [0, 1] satisfying fo fuldx = 0, the boundary value problem (1.16) has at least one solution. The following question then appears in a natural way.
Question 5. What is the solution set of (1.16) in that case (i.e., if f E C'[0,1] satisfies (1.14)) ? Also here the case p 34 2 is very different. It is proved in [5] that the set of all solutions of (1.16) is bounded in C1 norm. The picture of the nonlinear Fredholm alternative can be completed by considering solvability of
_(Iu'Ip-2u')' - AIulp-2u = u(0) = u(1) = 0
f in (0,1),
(1.17)
when \ is not an eigenvalue. It is well known that for p = 2 the boundary value problem (1.17) has a unique solution for any f E W-lip (0, 1). It follows from the Leray-Schauder degree theory that for p 54 2 the problem (1.17) has at least one solution for any f E W-1,p'(0,1). Uniqueness, however, holds only for \ < 0, due to the monotonicity of the operator u'-+ -(Iu'Ip-2u)' - \IuIP-2u. If A > 0 and p # 2, one can find f such that the problem (1.17) has at least two distinct solutions as shown in Fleckinger, Hernandez, Takac and deThelin [12], DelPino, Elgueta and Manasevich [6] and Drabek and Takac [11]. Let us consider now the energy functional Ef,a : W1"p(11) -' R associated with the boundary value problem
-Ayu - AIuIp-2u = f in St, on M. 1u=0 Obviously,
f f
(1.18)
fu, IuIp p and the critical points of E f,.\ are in one-to-one correspondence with the weak solutions of (1.18). The functional E1,,, has a global minimum (and in fact it is coercive) if A < \1 due to the variational characterization (1.11) Ef,A(u)
p fn IVuIp -
248
P. Dribek
while Ef,a has a saddle point geometry if \ > A1i A is not an eigenvalue. For p = 2, Ef,,, has always a unique critical point in the above mentioned cases, for p 2, E f,a has a unique critical point only if A < 0. A counterexample showing that there exists f for which Ef,,, has at least two distinct critical points is given in [6] (for A E (0, A1) and p > 2), (12] (for A E (0, A1) and p E (1, 2)) and [11] (for A > 0 and p > 1, p 54 2). Let us consider A = Al and study the energy functional Ef,,\.
Observation 1. If f E
W-l,P' (Il), fn f u1 # 0, then E f,.\, is unbounded
from below (in the direction of u1).
Observation 2. If p = 2 and f E W-1,2(SZ), fn ful = 0, then Ef,a, is bounded from below.
Again the following question arises in a natural way.
Question 6. Let p E (1, 2) U (2, oo), f E W-lip (St), fo f u1 = 0. Is Ef,A, bounded from below ?
The answer is known in the ODE case (N = 1, Sl = (0,1)) and it is quite interesting. The following assertions are proved in (5) : (i) Let p E (1, 2), f E C' [0, 1], fo f ul = 0; then Ef,a, is unbounded from below. (ii) Let p E (2, oo), f E C' [0, 11, fo fu1 = 0; then Ef,a, is bounded from below.
Observation 3. Let p = 2, f E W-"2(SZ), f ful # 0. Then Ef,p,, has no critical point.
Question 7. Let p E (1, 2) U (2, oo) and fo f u1 # 0. Does Ef,a, have any critical point? In the above mentioned papers [4] and [5] examples are given showing that the answer is positive for certain f E W-"P'(0,1). On the other hand there are f's for which Ef,a, has no critical point (cf. [71).
Observation 4. Let p = 2, f E W-1'2(S?), fn fu1 = 0. Then Ef,a, has an unbounded continuum (linear set of dimension one) of critical points.
Question 8. Let p E (1, 2) U (2, oo), f E W-1,v' (fl), fn f ul = 0. What is the structure of the set of all critical points of Ef,A, ?
The answer is known in the ODE case (N = 1, ) = (0,1)) and it is proved in [5] that for p E (1, 2) U (2, oo), f E Cl [0, 1], f, f u1 = 0 the set of all critical points of E f,a, is nonempty and bounded in C' norm. Let us conclude by mentioning the relation between the above mentioned results and the sensitivity of optimal Poincare inequality under linear perturbations. It follows from (1.7) and the simplicity of the first eigenvalue of AIuIP-2u m (0,1), -(lu'IP-2u')' =
u(0) = u(1) = 0
Some Aspects of Nonlinear Spectral Theory
249
that the expression
Cp
lu'Ip -
1
Jo
1
(1.19)
gulp
Jo
minimizes (and equals zero) just in the one dimensional linear subspace of Wo'p(0,1) spanned by ul. Let us add perturbation term - f0 fu to the left-hand side of (1.19) and consider f E W-l,p (0, 1), J' ful = 0. Case p = 2. We have
C2J1 lu'l2
-j
lul2
-J
f u > Cf > -00
(1.20)
0
and the left-hand side of (1.20) minimizes and equals CJ just on the linear set of all solutions of the boundary value problem
-C2U"-u= f in(0,1), lu(0) = u(1) = 0. Case p > 2. For f E C' [0,1] we have again i
i
Cp
f lu'lp - f lulp 0
0
Jfu>Cf>-00
(1.21)
0
but the left-hand side of (1.21) minimizes and equals Cf on the bounded set of all solutions of the boundary value problem -CP(lu'lp-2ui)i _ lulp-2u
=f
in (0, 1),
1u(0) = u(1) = 0. Case 1 < p < 2. For f E C' [0,1] we can always find a sequence {un } E W01'(0, 1) such that i
CPP
I
ri
lunlp -
J
fi
lunlp -
J
fun \1 -00
250
P. Drabek
REFERENCES [1]
R.A. Adams, Sobolev Spaces, Academic Press Inc., New York, 1975.
[2]
A. Anane, Simplicit6 et isolation de la premiere valeur propre du pLaplacien avec poids, C.R. Acad. Sci. Paris, Ser. I. Math. 305 (1987), 725-728.
[3] A. Anane and N. Tsouli, On the second eigenvalue of the p-Laplacian.
In: Nonlinear Partial Differential Equations (From a Conference in Fes, Maroc, 1994) (A. Benkirane and J.-P. Grossez ed.), Pitman Research Notes in Math. 343, Longman, 1996. [4] P. Binding, P. Drabek and Y.X. Huang, On the Fredholm alternative for the p-Laplacian, Proc. Amer. Math. Soc. 125 (1997), 3555-3559. [5] M. DelPino, P. Drabek and Manasevich, The Fredholm alternative at the first eigenvalue for the one dimensional p-Laplacian, J. Differential Equations, 159 (1999), 386-419.
[6] M.A. DelPino, M. Elgueta and R. Manasevich, A homotopic deformation along p of a Leray-Schauder degree result and existence for
(Iu'I"-2u')' + f(t,u) = 0, u(0) = u(T) = 0, p > 1, J. Differential Equations 80 (1989), 1-13. [7]
M. DelPino and R. Manasevich, Multiple solutions for the p-Laplacian under global nonresonance, Proc. Amer. Math. Soc. 112 (1991), 131138.
[8] P. Drabek, Ranges of a-homogeneous operators and their perturbations, Casopis pro Pcstovdni Matematiky 105 (1980), 167-183. [9] P. Drabek, A note on the nonuniqueness for some quasilinear eigenvalue problem, Appl. Math. Letters 13 (2000), 39-41.
[10] P. Drabek and R. Manasevich, On the closed solution to some nonhomogeneous eigenvalue problems with p-Laplacian, Differential and Integral Equations 12 (1999), 773-788. [11] P. Drabek and P. Takac, A counterexample to the Fredholm alternative for the p-Laplacian, Proc. Amer. Math. Society 127 (1999), 1079-1087.
[12] J. Fleckinger, J. Hernandez, P. Tak6Z and F. deThelin, Uniqueness and positivity for solutions of equations with the p-Laplacian. In: Proceedings of the Conference on Reaction - Diffusion Equations, Trieste, Italy, October 1985. Marcel Dekker, Inc., New York, Basel, 1997. [13] J. Garcfa and I. Peral, Existence and non-uniqueness for the p-Laplacian: Non-linear eigenvalues, Comm. Partial Differential Equations 12 (1987), 1389-1430.
Some Aspects of Nonlinear Spectral Theory
251
[14] J. Garcia and I. Peral, On limits of solutions of elliptic problems with nearly critical exponent, Comm. Partial Differential Equations 17 (1992), 2113-2126.
[15] Y.X. Huang, A note on the asymptotic behavior of positive solutions for some elliptic equation, Nonlinear Analysis T.M.A. 29 (1997), 533537.
[16] A. Kufner, O. John and S. Fucik, Function Spaces, Academia, Prague, 1977.
[17] P. Lindqvist, On the equation div (IVuIp-'Vu) + XIuIP-2u = 0, Proc. Amer. Math. Soc. 109 (1990), 157-164. [18] M. Otani, A remark on certain nonlinear elliptic equations, Proc. Fac. Tokai Univ. XIX (1984), 23-28.
Pavel Drabek Department of Mathematics University of West Bohemia P.O. BOX 314, 306 14 Plzet Czech Republic pdrabekCKMA.ZCU.CZ
Asymmetric Nonlinear Oscillators Christian Fabry Alessandro Fonda ABSTRACT We review some results on large-amplitude periodic or almost periodic solutions of second order differential equations with asymmetric nonlinearities, when the system is close to "nonlinear resonance".
1
Statements
One of the simplest nonlinear second order differential equations is the following model for an asymmetric oscillator, where the restoring force is assumed to be piecewise linear.
x" + cx' + g(x) = f (t)
,
with
g(x) _
µx vx
if x > 0
ifx<0
Here it, v are positive real numbers, c is either positive or equal to zero, and f (t) is assumed to be continuous and periodic with period T > 0. The restoring force is often written as
g(x) = µx+ - vx-, where x+ = max{x, 0} is the positive part of x, and x- = max{-x, 0} is its negative part. We are interested in the existence of periodic solutions of (1.1). Let us recall the following existence theorem dealing with a "nonresonant" situation (cf. [2, 5]).
Theorem 1.1. If c > 0, or if c = 0 and
+1 then equation (1.1) has at least one T-periodic solution.
254
C. Fabry, A. Fonda
Assume now that there exists a positive integer n for which
T
1
1
71, W n7r The homogeneous equation V" +
AW+
- vv- = 0
(1.3)
then has nontrivial T-periodic solutions. We call this situation "nonlinear resonance."
Theorem 1.2. ([2, 6J) Assume (1.2) holds for some positive integer n. Then, if c = 0, there are functions f (t) for which equation (1.1) has no periodic solutions.
If (1.2) holds, let cp be a nontrivial T-periodic solution of the homogeneous equation (1.3). That solution has minimal period T/n. Let us also introduce the T/n-periodic function (P (6) = I JT f (t)cp(t + 0) dt
.
0
Theorem 1.3. ([3)) Assume (1.2) holds for some positive integer n. Let It only have simple zeros, and let 2k be their number in the interval [0, T/n[ .
If c = 0 and k # 1, then equation (1.1) has at least one T-periodic solution.
There is an R > 0 such that, for any R > R there is a CR > 0 such that, if 0 < c < CR, then equation (1.1) has at least k periodic solutions with period T. Among these, exactly k of them are such that
min{lx(t)l + Ix'(t)l : t E R} > R, and they are asymptotically stable. The following result deals with the situation when 4i is of constant sign (a Landesman-Lazer type situation). Theorem 1.4. ([4[) Assume (1.2) holds for some positive integer n. Let 4) be positive and nonconstant. Then, for every R > 0 there is a 5 > 0 such
that, if T
nzr
-b<
1a 1 +
/7J
<
T n7r
and lit - al + Iv - #1 < b, then the equation
x" + ax+ - /3x- = f (t) has at least three T-periodic solutions. Among these, at least two solutions are such that min{lx(t)I + Ix'(t)I : t E R} > R.
Asymmetric Nonlinear Oscillators
255
Alonso and Ortega [1] proved that, if c = 0 and 4) has simple zeros, then every solution of (1.1) having large amplitude must be unbounded. On the other hand, Liu [7], assuming f sufficiently smooth, proved that, if c = 0, µ 54 v, and 4i is of constant sign, then all solutions of (1.1) are bounded on R.
Concerning the existence of large amplitude subharmonic solutions, i.e., solutions whose minimal period is an integer multiple of T, we have the following.
Theorem 1.5. ([4]) Let (1.2) hold for some positive integer n. Assume that 4i is positive and that f has minimal period T and nonzero Fourier coefficients of arbitrarily high order. Then, there is a neighborhood V of (p, v) with the following property. For every R > 0 and any positive integer p, there is a set D, which is dense in V, such that, for every (a, 13) E D, the equation x" + ax+ - ox- = f (t) has a subharmonic solution x whose minimal period is greater than p-T and which is such that mini jx(t)I + Ix'(t) I : t E R} > R .
(We emphasize the fact that the set V is independent of the amplitude of the solutions.)
When the forcing function f is not assumed to be periodic but is almost periodic, we let cp satisfy (1.3) (µ and v being here arbitrary positive constants) and define JT
-P(9) =
T
lim o 7,
f (t)cp(t + 0) dt
.
Then we can prove the existence of large amplitude almost periodic solutions to equation (1.1).
Theorem 1.6. ([4]) Let 0* be such that 4)(0') = 0
,
4i'(9) > 0.
There is a c' > 0 such that, if 0 < c < c", then equation (1.1) has an asymptotically stable almost periodic solution x(t; c) of the form (x(t; C),
(t; c)) =
p(t;
0(t; c)), V, (t + 9(t; C)))
,
the functions p(t; c), 0(t; c) being almost periodic in t and such that lim 0(t; c) = 0* C
+
,
lim p(t; c) = p' C+
256
C. Fabry, A. Fonda
REFERENCES
[1] J. M. Alonso and R. Ortega, Roots of unity and unbounded motions of an asymmetric oscillator, J. Dif. Eq. 143 (1998), 201-220. [2] E. N. Dancer, Boundary-value problems for weakly nonlinear ordinary differential equations, Bull. Austral. Math. Soc. 15 (1976), 321-328. [3] C. Fabry and A. Fonda, Nonlinear resonance in asymmetric oscillators,
J. Dif. Eq. 147 (1998), 58-78. [4] C. Fabry and A. Fonda, Bifurcations from infinity in asymmetric nonlinear oscillators, Nonlinear Dif. Eq. Appl. 7 (2000), 23-42. [5] S. Fu6fk, Solvability of Nonlinear Equations and Boundary Value Problems, Reidel, Boston, 1980.
[6] A. C. Lazer and P. J. McKenna, Existence, uniqueness, and stability of oscillations in differential equations with asymmetric nonlinearities, Trans. Amer. Math. Soc. 315 (1989), 721-739. [7] Bin Liu, Boundedness in asymmetric oscillators, J. Math. Anal. Appl. 231 (1999), 355-373.
Christian Fabry Institut de Mathematique University Catholique de Louvain Chemin du Cyclotron, 2 B-1348 Louvain-la-Neuve Belgium fabry@amm.ucl.ac.be
Alessandro Fonda Dipartimento di Scienze Matematiche
University di Tieste P. le Europa 1 1-34127 Trieste
Italy fondass@univ.trieste.it
Hopf Bifurcation for a Delayed Predator-Prey Model and the Effect of Diffusion Teresa Faria 1
Introduction
Consider the Lotka-Volterra predator-prey system
ii(t) = u(t)[ri - au(t) - alv(t - a)], v(t) = v(t)[-r2 + blu(t - T) - bv(t)], where T, r1i r2, a1, b1 are positive constants and a, a, b are non-negative constants. In biological terms, u(t) and v(t) can be interpreted as the densities of prey and predator populations, respectively, and a, b self-limitation constants. In the absence of predators, the prey species follows the logistic
equation it(t) = u(t)[rl - au(t)]. In the presence of predators, there is a hunting term, a,v(t - a), al > 0, with a certain delay a, called the hunting delay. In the absence of prey species, the predators decrease. The positive feedback biu(t -T) has a positive delay 'r which is the delay in the predator maturation. Models involving delays and also spatial diffusion are increasingly applied to the study of a variety of situations. For this reason, we consider a second model, the delayed reaction-diffusion system with Neumann conditions, resulting from considering one spatial variable and adding diffusion terms
d1tu, d20v, d1i d2 > 0, respectively to the first and second equations of (1.1): 8u((tt, x)
x)
= di 8 2u8(xt2 x) = d2
a2
+ u(t, x) [r1 - au(t, x) - a1 v(t - o,, x)],
a(t2x) + v(t, x)[-r2 + blu(t - T, x) - bv(t, x)], (1.2) t>0,xE(0,7r)
&U(t, x)
8x
- ev(t,8x x)
=0, x=0,7r.
Work partially supported under projects PRAXIS/PCEX/P/MAT/36/96 and PRAXIS/2/2.1/MAT/125 /94 of FCT (Portugal).
258
T. Faria
Systems of type (1.1) or similar, and also predator-prey models with distributed delays, have been widely studied (e.g., [1, 8, 9, 13] and references therein). However, we note that most of the literature considers a = b = 0 or some additional constraints on the constants. Here, we assume the existence of a positive equilibrium E. for system
(1.1), therefore also an equilibrium of (1.2). Taking the delay r > 0 as a parameter, for (1.1) it is proven that a Hopf bifurcation occurs at E. as T crosses some critical values r,,. In order to determine the direction of the bifurcation and the stability of the periodic orbits for r near the first bifurcating point r0, the normal form theory for functional differential equations (FDEs) in [6] is used. This technique allows us to obtain the ordinary differential equation (ODE) giving the flow on the center manifold
at the singularity, explicitly given in terms of the original FDE. Unfortunately, for (1.1) the application of the normal form algorithm involves hard computations; although formulas are presented here, their resolution is not done in the general framework. Nevertheless, in the last section we illustrate the use of the formulas with a particular example. Our purpose is also to relate the dynamics of the two systems (without and with diffusion) in the neighbourhood of E., and determine the effect of the diffusion terms, regarding the stability and the Hopf bifurcation, near r = To. For the second model, system (1.2), the adjoint theory for partial functional differential equations (PFDEs) is used (see e.g., [10, 12] ). However, the linearized equation about the equilibrium is given by an operator that mixes the modes of the Laplacian, and the theory must be adjusted to take this into account. The study of the Hopf bifurcation is then based on the existence of a center manifold [10, 12] and on the normal form procedure for FDEs with diffusion [3, 41. In this setting, the results for the reaction-diffusion system are deduced from the previous analysis of the Hopf singularity for (1.1). Finally, we refer the reader to [5], where the material presented here can be found with detail.
2
The model without diffusion
Consider system (1.1). Through the change of variables u --4 blu, v - alv,
we may assume that al = bl = 1. Also normalizing the delay T by the time-scaling t - t/T, (1.1) is transformed into
u(t) = Tu(t)[rl - au(t) - v(t - r)], '!)(t) = Tv(t)[-r2 + u(t - 1) - bv(t)],
(2.1)
where r = a/T. Without loss of generality, let max(1, r) = 1. With the assumptions rl > O,r2 > O,r > O,a > 0, b > 0, rl - are > 0,
(2.2)
Hopf Bifurcation for a Delayed Predator-Prey Model
259
there is a unique positive equilibrium E. for (2.1), E. = (u., v.), with r2 + brl
ab + 1 '
v.
rl - are (2.3)
ab + 1
By the translation z(t) _ (u(t), v(t)) - E. E R2, (2.1) is written as an FDE in C:= C([-1, 0]; R2) as
z(t) = N(r)(zt) + fo(zt, r),
(2.4)
where zt E C, zt(0) = z(t + 0), -1 < 0 < 0, and N(r) : C -' R2, fo:CxR+ -'R2 are given by N(r)(W) fo(W, r)
- r(
(0) +'P(-r))
v.('Pi(-1) - `^02(0))
'
- r (V2(0)(wai(1 1)) bp(0)))) '
for cp = (V 1, cp2) E C. Throughout this note, we refer to [7] for notation and
classic results on FDEs. The characteristic equation for the linear equation z(t) = N(r)(zt) is a(.1, r) := A2 + A. 7-,\ + B. r2 +
C.r2e-,\(t+r)
= 0,
(2.5)
where A. = au. + bv., B. = abu.v., C. = u.v.. Using the material in [9, pp. 74-82] and in [11], one can prove the following results (see [5] for more information and proofs):
Theorem 2.1. Assume (2.2). If ab > 1, all the roots of the characteristic equation A(.\, r) = 0, r > 0, have negative real parts. If 0 < ab < 1, let p. = p. (rl, r2i a, b) be the unique real positive solution of p4 + (a2u; + b2v,)p2 + (a2b2 - 1)U2 V.2 = 0. Then, for a > 0, r > 0, 4(ia, r) = 0 if and only if there is an n E No such that r = rn and a = an, where cos(an(1 + r)) = P! an
_ 2(n + 1)ir
1+r
- B. C.
rn = an ,
P
and
(2.6)
2nir (2n + 1)ir )ifa2+b2>0. ifa=b=0, anE (1+' 1+r
(2.7)
Futher'more, ±ian are simple roots, Re A'(rn) > 0 and a Hopf bifurcation occurs for (2.4) at z = 0, r = rn.
Theorem 2.2. Assume (2.2), ab < 1, and define ao, TO as above. (i) If a2 + b2 > 0 and 0 < r < To, all the roots of A(A, r) = 0 have negative real parts; for r = ro, ±iao are simple roots of A(.\, ro) = 0 and the remaining roots have negative real parts. (ii) If a = b = 0 and r > 0, there is at least a pair of complex conjugate roots of 4(A, r) = 0 with positive real parts; for r < ro, that pair is unique.
260
T. Faria
As we see from the results above, ab > 1 implies that the equilibrium E. is asymptotically stable for all r > 0. If (i) ab < 1 and a2 + b2 > 0, the equilibrium E. is asymptotically stable for 0 < T < ro and unstable for T > TO; for (ii) a = b = 0, E. is unstable for all r > 0. Assuming ab < 1, ±iao are the only eigenvalues on the imaginary axis for the linearized equation at T = ro, (u, v) = E.. Therefore, the center manifold theory for FDEs [7] guarantees the existence of a local center manifold of dimension 2, where a Hopf bifurcation takes place. This manifold is stable in case (i) and unstable in case (ii). Throughout the rest of this section we always assume (2.2) and ab < 1. For the equilibrium E. of (1.1), or equivalently, for the zero solution of (2.4),
we want now to determine the Hopf singularity at T = To. To accomplish this, we use the normal form theory (we refer to [6] for explanations of the algorithm involved). Consider (2.4) in the phase space C and let Ao = {-iao, iao}. Introducing the new parameter a = T - To, (2.4) is rewritten as z(t) = N(TO)ze + Fo(zt, a),
(2.8)
where Fo(cp, a) = N(a)(cp) + fo(cp, To + a). Using the formal adjoint theory
for FDEs in [7], we decompose C by Ao as C = P ®Q, where P is the center space for z(t) = N(TO)(zt). Considering complex coordinates, P = span { O1 ,4 6 21 , with 01(9) = ewooV1, j2(9) = q51(0), -1 < 0 < 0, where the
bar means complex conjugation, and v1 E C2 is such that N(To)(01) = ioovl.
(2.9)
For 4) = [01 452], note that 4 ='B, where B is the 2 x 2 diagonal matrix B = diag (iao, -ia0). Choose a basis IQ for the adjoint space P', such that I2, where (,) is the associated bilinear form on (IF,') _ (')i, e-sooa,.u1 e`c08), C* xC. Thus, (s) = COQ (t,bi(s), 02(S)) = COI (ui / s E [0, 1], for ul E C2 satisfying (01, 01) = 1,
(01, 02) = 0.
(2.10)
The normal form method gives for (2.8) an ODE describing the flow on the center manifold of the origin near a = 0, written in normal form as
i = Bx + C Alxla Blx2a
\
J
A2xlx2 + C B2xlx2 ) + O(I xI a2) + O(I xI4),
(2.11)
where x = (xl, x2) E C2. It was shown in [5] that B1 = Al, B2 = A2. The change to real coordinates w, where xl = w1 -iw2iX2 = w1 +iw2, followed by the use of polar coordinates (p,.), w1 = p cos C w2 = p sin C, transforms the normal form (2.11) into Klap+ K2p3 +O(a2p+ I(p,a)I4), a + o(I (p a)1}
(2 . 12)
Hopf Bifurcation for a Delayed Predator-Prey Model
261
with K1 = Re Al, K2 = Re A2. It is well known (e.g., (2]) that the sign of K1K2 determines the direction of the bifurcation and that the sign of K2 (if K2 # 0, which is the case of generic Hopf bifurcation) determines the stability of the nontrivial periodic orbits. Using the algorithm of normal forms, explicit formulas to compute the coefficients K1, K2 in terms of the original FDE (2.8) (or (1.1)) were obtained in [5], without having to compute the center manifold beforehand. Such normal forms are given in the next theorem. We point out that computations are particularly difficult here (mainly because the original equation is two dimensional, rather than scalar, and because there are two delays).
Theorem 2.3. The flow on the center manifold of the origin for (2.8) at a = 0 is given in polar coordinates by equation (2.12), with K1 = Re (ip.ul vi),
(2.13)
K2 = 2 Re c3i
(2.14)
where: p = ao/To; v1i ul are vectors in C2 such that (2.9) and (2.10) hold; c3 is given by 7
-((a+OL)Cl(0)+C2(-r)]'Ul,l
C3 = ul
(b+iPV-.')C2(0)]v1,2
where v1 = col (v1,1, v1,2); E = t
v2,) ; h(2 0) _ ((1, (2) is the solu-
\ v1,22i
tion of
f
h(2,o)
/v..
- 2iooh(2,o) = 2iao(ui E 01 + ui E 02),
h(2.0) (0) - N(To)(h(2,o)) = 2iao
3
(2.15)
E.
(2.16)
The reaction-diffusion model
Consider now (1.2), where d1, d2, r, r1, r2, a1, b1 are positive constants and
a, a, b are non-negative constants. Similarly to what was done for (1.1), we may assume al = b1 = 1. Under (2.2), E. = (u., v.) given by (2.3) is now the unique positive stationary solution for (1.2). After the time-scaling
t -- t/T, Eq. (1.2) is given in abstract form as
4ju(t) = dirAu(t) + ru(t) (r1 - au(t) - v(t - r)],
gv(t) = d2rAv(t) + rv(t)(-r2 + u(t - 1) - bv(t)], where r = a/r and, for simplification of notation, we use u(t) for u(t, ),
v(t) for v(t, ), and (u(t), v(t)) = (u(t, ), v(t, )) is in the Hilbert space
262
T. Faria
= }, with the X = {(u, v) u, v E W2'2(0, 7r), jx- = 0 at x = 0, inner product < , > induced by the inner product of the Sobolev space :
W2,2(0, 7r). Translating E. to the origin by setting U(t) = (u(t), v(t))-E. E X, (3.1) is transformed into the equation in C := C([-1, 0]; X) (see [12] for
notation)
dt U(t) = TdtU(t) + L(r)(Ut) + f (Ut, r),
(3.2)
Wt-
where dL = (d10, d20) and L(r) : C -+ R2, f : C x R+ -p R2 are given by
u.(aW,(0) +W2(-r))
T
v.(wl(-1) - b,2(0))
-T \.
1)
'
-b'P2(0))))'
for cP = (vi, w2) E C. The characteristic equation for the linearized equation
TU(t) = rdtU(t) + L(r)(Ut) is ([10, 12])
Ay - rdIy - L(T)(e"'y) = 0,
y E dom(0), y # 0.
(3.3)
The eigenvalues of rdA on X are yk = -dirk2, i = 1 , 2, k = 0,1, 2, ... , with corresponding normalized eigenfunctions,Qk, where Qk =
Ilk
0 J
, Qk =
( 0 ) , 7k (X)
II cos(kx) 12,2'
k E No.
We note that in general L(T)(wjiQk) ¢
L(7-)(W2f3) V span{/3k}, for (WI, W2) E C = C([-1, 0]; ]R2). However, we have
L(r)(cP1Qk+'2Qk) = -u.r(aW1(0)+co2(-r))Q,+v.7-(W1(-1)-b'P2(0))Qk, or, equivalently,
Qk
)
t Qk
=
)
,
(3.4)
implying that L(r) does not mix the modes of the generalized eigenspaces span {/3k, /3k }. For any y E X, consider now its Fourier series relative to the basis {/3k : i = 1, 2; k = 0,1, ... }, written in such a way that the Fourier coefficients relative to /3k, /3k are kept together: 00
Y
- k=o Yk =
01
(
k
)
Yk
(
< Y0/3k > < y,
)
Hopf Bifurcation for a Delayed Predator-Prey Model
263
Using this decomposition and (3.4), we note that for y E dom(0), y # 0, the characteristic equation (3.3) is equivalent to °O EYk
k=0
/ [AI0
k2d1
au,, -u.e-a'
0
-k2d2-T (v.e'a
-bv,
l
(O,,kI
)J
)
=0.
Hence, we conclude that (3.3) is equivalent to the sequence of characteristic equations
\2 + (dl k2 + d2 k2 + au. + bv. )T,\
Ak (A, T)
u,v,T2e-all+r) =
+(d1k2 + au.)(d2k2 + bv.)T2 +
0
(3.5k)
with k = 0, 1, 2, .... It is important to remark that for k = 0 the above equation (3.5o) is the characteristic equation (2.5) obtained for the system without diffusion. The analysis of the characteristic equations (3.5k), for k > 1, shows that all their roots have Re A < 0, provided some additional conditions are made. See [5] for proofs.
Theorem 3.1. Assume (2.2) and define To, vo as in Theorem 2.1. Suppose also that a2 + b2 > 0
and
ab(au. + bv.)2 < u.v.,
(3.6)
or
a = b = 0,
and di + d2 > 27r rlr2
or did2 > rir2.
(3.7)
Then, for 0 < T < To and k > 1, all the roots of the characteristic equations (3.5k) have negative real parts.
Remark 3.2. If a = b = 0 and the coefficients of the diffusion terms dl, d2 are small, the instability of the stationary solution E. of the reactiondiffusion equation (3.1) at To might increase. Actually, one can prove that
for a = b = 0, d1 =d2and'=To=27r/((l+r) rlr2),ifk>1issuch that dike <
3rlr2/2, then Ak(A,To) = 0 has at least a pair of complex
conjugated roots with positive real parts [5]. This shows the effect of small diffusion terms creating more instability of E. at the first bifurcation point To.
Remark 3.3. In [10] the authors proved a center manifold theorem for PFDEs in C = C([-r, 0]; X), r > 0, X a Hilbert space, of type
d U(t) = diU(t) + L(UG) + g(Ut)
(t > 0),
where d > 0, dom(A) C X, L : C -' X is a bounded linear operator and g : C -' X is a Ck function (k > 2) such that g(0) = 0, Dg(0) = 0. Some quite general hypotheses were assumed, plus the very restrictive assumption that L does not mix the modes of eigenspaces of the Laplacian,
264
T. Faria
i.e., L(< v(.), /3 > /3) E span {/3}, for all v E C and 0 an eigenfunction of dO. Obviously, from [10], the existence of a local center manifold follows for equations with parameters. We can also consider equations where dO(d E R) is replaced by dO = for d = (d1,.. . , d,,,) E R'. Clearly, for the PFDE model presented here, the above mentioned condition fails. However, it was proven in [4] that a center manifold theorem is still valid when, instead, the following weaker assumption is fulfilled: the set of eigenvalues of dA can be written as Uk=o {µk ik = 1, ... , Pk },
such that the subspaces Qk C C, l3k := span{<
vE
C, ik = 1, ... , pk }, where /3k" is the normalized eigenfunction corresponding to µk, satisfy L(t3k) C span{/3k, ... , f3k" }, for all k E No. We observe that this condition is satisfied by the reaction-diffusion system (3.2), with Bk = {co1Qk + p2/3 I ( 1, V2) E C([-1, 0]; R2)}, k E No.
Theorem 3.4. Assume (2.2) and either (3.6) or (3.7). Then, for (3.1) and r near ro, there is a 2-dimensional locally center manifold of the stationary point E.. In the rest of this section, we always assume hypotheses of Theorem 3.4. Now, we show how the Hopf bifurcation analysis for (3.1) at E. and ro (if it is generic) can be deduced from the case without diffusion. To be more precise, for the ODE giving the flow on the center manifold for (3.1), we shall determine the terms up to third order from the terms of the equation on the center manifold for (2.4). Again let r = ro + a and A0 = {-ioo, iao}. Theorem 3.1 implies that Ao is the set of eigenvalues on the imaginary axis of the infinitesimal generator associated with the flow of
dtU(t) = rodOU(t) + L(-ro)(UU).
(3.8)
Defining F(v, a) := adzv(0) + L(a)(v) + f (v, ro + a), equation (3.2) is written as
d U(t) = rodtU(t) + L(ro)(U=) + F(UU, a).
(3.9)
Following [4], we consider the phase space C decomposed by Ao, C = P ® Q,
where P is the center space for (3.8), given by P = span {4iT C Qo and 4? = [01 02] is the basis for the space P C C = C([-1, 0]; C2) defined
in Section 2. Note that u E P necessarily has the form u = with (cpli 02) E C. According to [4], for Eq. (3.9) at U = 0, a = 0, we define its associated FDE by A0 as the retarded FDE in C:
i(t) = R(xl) + G(xt, a),
(3.10)
Hopf Bifurcation for a Delayed Predator-Prey Model
265
where R, G are given by L(ro) (coT
(L)) = [R(w)]T G(cp, a) =
('001
)
'802
< F c0T
Q2
Q2
cpT
ao Qo
From (3.4), we observe that R = N(ro). Also, we have O/I = 0, i = 1, 2,
#01 =(1//F,0),Q0=(0,1/V7r),=1,=0,if i$j, i, j = 1, 2. Using again (3.4), we obtain G(cp, a) = N(a) (cp) + T fo(cp, ro + a) and (3.10) becomes
i(t) = N(ro + a)(xc) +7= fo(xc, ro + a).
(3.11)
Note that the scaling x = fz transforms (3.11) into (2.8). This proves that the dynamics for (2.8) and for (3.10) are the same near the origin and
a = 0. On the other hand, it was proven in [3, 4J that if a condition of nonresonance on the second order terms of (3.9) is satisfied, then there is a change of variables such that the normal forms giving the flow on the center manifolds for both Eq. (3.9) and its associated FDE by A0 are the same, up to cubic terms. The mentioned condition on the second order terms of F is translated here as < DjF2(u,a)(5f3 ),Ao >= 0, for i, j = 1, 2, u E F, b E C([-1, 0J; R) and k > 1, 2F2(v, a) are the quadratic terms of F in (v, a)). The above definition of F yields where
1
2
F2(v, a) = adOv(0) + L(a)(v) + ro
vl (0)(av1(0) + v2(-r))
(
v2(0)(vl(-1) - bv2(0))
(3.13)
for v=(vl,v2)EC.Using =0,k>1,i,j=1,2,and (3.13),one can prove that condition (3.12) is fulfilled. Since the terms up to third order are sufficient to determine the dynamics of a generic Hopf bifurcation, the above arguments lead to the following conclusions:
Theorem 3.5. Assume (2.2), and (3.6) or (3.7); let ro be as in Theorem 2.1. Then, for a suitable change of variables, the equations on the center manifold of E. at r = ro for both Eq. (1.1) and Eq.(1.2) are the same, up to third order terms. In particular, if the Hopf bifurcation on the 2dimensional local center manifold for (1.1) is generic, the same is true for
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T. Faria
(1.2), and the bifurcation direction and the stability of the periodic orbits on the center manifold for (1.1) and (1.2) are the same.
Remark 3.6. Assuming (2.2) and either condition (3.6) or (3.7), the results above show that the local stability of E. for 0 < r < ro, as well as the Hopf singularity at ro, if generic, are reduced to the case without diffusion. In this sense, the diffusion terms are irrelevant in our model.
4
Example
Consider system (2.1) with a = b = 0 and r1 = r2:
f ti(t) = u(t) [rl - v(t -a)], j v(t) = v(t)[-r1 + u(t - r)].
(4.1)
The positive equilibrium E. of (4.1) is E. = (u., v.) = (r1ir1). According to Theorem 2.1, oo = 1+r, 7-0 = 1+; rl Let v1, u1 be defined as in Section 2. From (2.9) and (2.10), one can choose
1-i1r
1
1
-ie-ioo
vl
hence 45(6) = [eiooevl e-'aoe'Vi], q, (0) = col(ui , 7.
7rr1
K1 = r1 Re (iu1 vl) = 1 +
a2
1
iei
ul = 2(1 + ir2)
(
°0
Formula (2.13) gives > 0.
(4.2)
To compute the coefficient K2 of the cubic term in (2.12), we first note
that E _
(
-e-
2so°
).
From (2.15), we get c3 = 2(1+2) A(
iG (0) -
e2taO(i (-1) + ie2i°0C2(0) - (2(-r)), where h(2,o) = (S1, (2) is the solution of
(1/(6) - 2ioo(1(6) = 1+r
21(1+R
(Cleiaoo + C2e iaoe),
2,ri 1+r r, 1+a (-C1etO0(9 -1) +
C2(6) -
C2e-ioo(e-1)),
(4.3a)
with C1 = (1 -iir)(e-i°0 +i), C2 = (1
+iir)(-a-3io° +i), that satisfies the
conditions
4i,, 1(0) + rlro(2(-r) = l+r rl (4 . 3 b ) 4i7r (2(0) - r1roc i(-1) = - 1+r r, e - 2ioo Here, we shall pursue the computations only for r = 0 or r = 1. For r = 0, it is ao = 21r, ro = 27r/rl. From (4.3Q,b), we obtain (1(-1) = (1(0) and (1(0),(2(0) given by
I
2i
-1
1
2i)
(1(0)
(2(0)) =
2iir ri(1 + 1r2)
(
1 + it 1 - 1r
)
(4.4)
Hopf Bifurcation for a Delayed Predator-Prey Model
267
Therefore (i (0)
3r1(1 + r2)
C2(0)
3rI(1 + jr2)
[-2(l + 7r) + i(-1 + 7r)], [-2(1 - 7r) + i(1 + 7r)].
Finally, we get c3
= 2(1 +
K2 =
2) (-(1 + i)(i(0) + (-1 + i)(2(0)1 _
3r1(1 + 7r2)
,
(4.5)
2
-Rec3 =
2(l + jr2) < 0.
(4.6)
Consider now the case r = 1. Then, oo = 7r, ro = 7r/rl. Applying (4.3Q,b), we can prove that once more c3 has the expression in (4.5), and then K2 = r1 Re c3 < 0. In the situations above, K2 < 0, KI K2 < 0. Theorems 2.3 and 3.5 are used now to describe completely the Hopf bifurcation at r = ro.
Proposition 4.1. Let a = 0 (respectively a = r). Then, the Hopf bifurcation occurring for (4.1) for the positive equilibrium E. = (rl,r1) at ro = 2ir/r1 (respectively ro = 7r/rl) is supercritical, with the bifurcating non-trivial periodic solutions being stable. If d1 + d2 > 2irri or d1d2 > r2l, the same conclusions hold for x)
=d av
&
X)
1
= d2
x) + u(t a2a a 2
a
2
x)
+ v(t, x) [-ri + u(t - r, x)]
,
t > 0, x E (0, ir),
azt2
Ou(t, x) ax
av(t, x) ax =
0
, x= 0 ,ir.
REFERENCES [1] E. Beretta and Y. Kuang, Convergence results in a well-known delayed predator-prey system, J. Math. Anal. Appl. 204 (1996), 840-853.
[2] S.-N. Chow and J.K. Hale, Methods of Bifurcation Theory, SpringerVerlag, New York, 1982.
[3] T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, Transactions of the A.M.S. 352 (2000), 22172238.
[4] T. Faria, Bifurcations aspects for some delayed population models with diffusion, Fields Institute Communications 21 (1999), 143-158.
268
T. Faria
[5] T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, to appear in J. Math. Anal. Appl.
[6] T. Faria and L.T. Magalhaes, Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation, J. Differential Equations 122 (1995), 181-200. [7] J. K. Hale and S. M. Verduyn-Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New-York, 1993.
[8] X.-Z. He, Stability and delays in a predator-prey system, J. Math. Anal. Appl. 198 (1996), 335-370.
[9] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993. [10] X. Lin, J. W.-H. So and J. Wu, Centre manifolds for partial differential equations with delays, Proc. Roy. Soc. Edinburgh 122A (1992), 237254.
[11] P. Tbboas, Periodic solutions of a planar delay equation, Proc. Roy. Soc. Edinburgh 116A (1990), 85-101.
[12] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996.
[13] T. Zhao, Y. Kuang and H. L. Smith, Global existence of periodic solutions in a class of delayed Gauss-type predator-prey systems, Nonlinear Anal. 28 (1997), 1373-1394.
Teresa Faria Departamento de Matematica, Faculdade de Ciencias, and Centro de Matematica e Aplicacoes Fundamentais Universidade de Lisboa 1749-016 Lisboa, Portugal tf ariaClmc . f c . ul . pt
Galerkin-Averaging Method in Infinite-Dimensional Spaces for Weakly Nonlinear Problems Michal Feckan ABSTRACT We present a survey of our recent achievements based on an asymp-
totic approximation method carried out by projection and averaging for initial value problems of infinite-dimensional difference equations with small parameters.
Applications are given to delay integro-differential equations and to semilinear Schrodinger equations as well.
1
Introduction
The purpose of this note is to present our recent results (4) on combination of the Galerkin approximation method [1] with the asymptotic-averaging procedure (2], (13]. We formulate our abstract results for semilinear difference equations with small parameters satisfying certain properties. By using the Galerkin method, we reduce infinite-dimensional semilinear difference equations with initial value conditions to finite-dimensional ones.
We also derive asymptotic approximation error bounds. Then we study finite-dimensional semilinear difference equations with initial value condi-
tions on the discrete time scale 1/e, where we embed our problem in an ordinary differential equation with the small parameter C. In this way, we are able to apply the classical averaging method [2], [13]. We also study, on the discrete time scales 1/c and 1/e2, infinite-dimensional semilinear difference equations with stable and center linear parts. Abstract results are at first demonstrated on an example of an integrodifferential equation with a memory and delay coupled by the small parameter e. Its dynamics is studied on the time scale 1/c2. Then we study a semilinear Schrodinger equation. This part is related to former studies on weakly nonlinear wave equations with fixed ends [5], [8], [9], [11], [12], [14]. The influence of almost periodic perturbations on the dynamics is in1991 Mathematics Subject Classification: 34C29, 35A40, 65L60, 65M15. Key words and phrases: Galerkin-averaging method, differential-diference equations, Schrodinger equations. This work was supported by Grant GA-SAV 2/5133/98.
M. Feekan
270
vestigated on the time scale 1/E. We show that, in the first order Galerkinaveraging method, waves with a higher number of modes can be excited by starting initially with a lower number of modes; for most large initial value conditions, this excitation is only a small turbulence in higher modes. We then give applications to certain neutral differential difference equations with the small parameter E. Generally in this case, averaged equations are very difficult because they are nonlinear wave equations. Only quantitative properties based on the Galerkin-Picard iteration procedure can be established. On the other hand, we also study an asymptotic stability of a linear problem. We end the note with a simple "blow-up" result for solutions on a finite-time, and we refer the reader to [41 for more details.
2
Galerkin-averaging method
Let X, Y be normed linear spaces with norms I - Ix, I - IY, respectively, and Y continuously embedded into X. Let hk : X X X x (0, oo) -' X and fk : X x X --# Y, k E Z+ = N U {0}. Let A : X - X be a continuous linear mapping satisfying IIAIIL(x) 5 1. Finally, let Pn : Y - X, n E N be linear mappings such that (i) dim Im Pn < oo, Im Pn C Y, I IPn I I L(Y,x) 5 K4 for a constant K4 > 0,
(ii) III-PnIIL(Y,X) =an -0 as n- oo. Theorem 2.1. ([41) Consider the sequences {xk}kEZ+ C X and {xn,k}kEZ+ C Im Pn given by Xk+1
xn,k+l
= AXk + Efk(Xk, Xk+1) + E2hk(xk, Xk+1, E),
(2.1)
= PnAxn,k + EPnfk(xn,k, xn,k+1) ,
(2.2)
where c > 0 is a constant and n E N is fixed. Moreover, we suppose (iii) Ifk(xk, xk+1) - fk(xn,k, xn,k+1)IY 5 K3,1Ixk - Xn,klX + K3,2Ixk+1 xn,k+l Ix b k E Z+, 0 < k < L/E"' for positive constants K3,1, K3,2, L,
1>w>0,
(iv) Ifk(xk,xk+1)IY < K1, Ihk(xk,xk+1,E)IX 5 K2Vk E Z+, 0 < k < L/EW for positive constants K1, K2,
(v) PA/Ira Pn = A/Im Pn `d n E N. Then
Ixk - Xn,klX 5 exp {3K4LE1-'(K3,1 + K3,2)/2}Ixo - xn,OI X +El-"L4exp {3K4LE1-'(K3,1 +K3,2)/2} (anKl
2 + 3E(K3,1 + K3,2)K4
+ K2e)
Calerkin-Averaging Method for Weakly Nonlinear Problems
271
whenever 0 < k < L/E& and EK3,2K4 < 1/3. Here exp is the Euler exponential function.
We note that usually we take xn,o = Pnxo, xo E Y to obtain Ixo xn,olx <- anlxolY Similarly to Theorem 2.1, when the fk are not compact, we have the following result.
Theorem 2.2. ([4]) Theorem 2.1 is valid when {xk}kEZ+ C Y in addition for (2.1) and the conditions
fk:XXX -'Y, Pn:Y-'X, IIPnIIL(Y,X) SK4 are replaced by
hk:YxYx(0,00)-IX, fk:YxY-'Y, Pn:X --+ X, IIPnIIL(X)
(iii)' Ifk(xk,Xk+1) - fk(Xn,k,Xn,k+l)IX :5 K3,1lxk -Xn,kIX +K3,2lXk+1 xn,k+1 IX,
d k E Z+, 0< k< LIE' for positive constants K3,1,
K3,2, L, 1 > w > 0. Now we focus our attention on (2.2). So we consider a difference equation of the form (2.3) Zk+1 = BZk + EFk(zk), k E Z+,
where B E L(Rn), IIBIIL(R-) < 1, Fk E C(Rn,Rn). Let us suppose that there are positive constants 3, ry such that IFk(x)I !5A for every
IFk(x) - Fk(y)I <- 'rlx - yI (k, x, y) E Z+ x Rn x Rn .
If IIBIIL(Rn) < 1 and Fk are independent of k, then (2.3) has a unique global stable fixed point. So we consider that IIBIIL(Rn) = 1 and take an invariant decomposition Rn = Rn ® Ker B with respect to B. Then (2.3) has the form ak+1 bk+1
= Dak + EFk,1(ak, bk), ak E R1 , = EFk,2(ak, bk), bk E Ker B ,
where zk = ak + bk. Let us consider the averaged equation z = EW (x) ,
where
k
W(x) = lim 1 ED-'-1F,,1(D`x,0) k-.oo k
whenever the above limit exists.
:.o
272
M. Fe&kan
Theorem 2.3. ([4]) Assume that (2.4) holds and IIDIIL(R;) = IID-'IIL(Ri ) = 1 in (2.5). Moreover suppose the existence of an mo E N such that D_8_mo-1F;+mo,l(Dt+mox,0) = D-`-1F,,1(D1x,0)
d(x,i) E Ri x Z+.
Then there is a constant K > 0 such that Izk - DkrkI < Ke whenever 1 < k < L/e and e > 0 is sufficiently small, where {zk}kEZ+ is given by (2.3), Tk = TE (ao, k) and ; (x, t) is the solution of (2.6) satisfying ; (x, 0) = x. We note that we use the norm I - Ix on Im P,, for (2.2) with the connection to (2.3). The above arguments provide also an asymptotic approximation method for infinite-dimensional difference equations with a small parameter. It is enough to assume X = Y and P,a = I in the above considerations. Then (2.1) is reduced to (2.3) with X replacing Rn and (2.4) holds with X replacing R" as well. Finally, when W = 0 in (2.7) then (2.3) is studied on the time scale 1/e2 in [4]. For more details, we refer the reader to that paper.
3
Retarded differential difference equations
Let us consider the delay equation
y(t) = e f (y(t), y(t - 1),t) + e2h (y(t), y(t - 1), t, e)
,
(3.1)
where f E C1 (R3, R) and h E C(R4, R) are uniformly bounded. By putting
xk}1(t) = y(k + t), 0 < t < 1, (3.1) is equivalent to xk+1(t) = Ef (xk+1(t), xk(t), k + t) + e2h(xk+1(t), xk(t), k + t, E) . k+1
Now (2.7) has the form W (x) = k im
00
exists.
f f (x, x, s) ds [4], when this limit 0
Theorem 3.1. ([4]) Assume that there is an mo E N such that 1
1
IfxIk+38_Ix+mo+8)d8 `d(k,x) E Z+ x R. For any e > 0 sufficiently small, the solution y of (3.1) with the initial value condition yo(t) E C[-1,01 has the asymptotic approximation y(t) = r, (t) + O(e) for 0 < t < L/e, where TE is the solution of the averaged equation t = eW(r), r(0) = yo(0). Related results are derived in [6]. As another example different from (3.1) and [6], we consider an integro-differential equation with a memory
Galerkin-Averaging Method for Weakly Nonlinear Problems
273
and delay given by It)
±(t) = -2
x(s) ds + E f (x(t), x(t - 1)) ,
J
(3.2)
(t) -1
where [t] is the integer part of t and f E C3 (R2, R) is even, i.e., f (-z, -y) = f (z, y). By a solution of (3.2), we mean a continuous function x possessing the derivative apart from t E N. By putting xn+1(t) = x(n + t), 0 < t < 1, (3.2) is equivalent to r1
xn+1(t) = -2
J
xn (s) ds + C f (xn+1(t), xn (t))
0
It can be shown that now W = 0 in (2.7). For constants A1,2, let us define a 4-periodic function v : R -> R given by
v(t) =
-1 < t < 0,
-A2t + A1(1 + t), -A2t + A1(1 - t),
0 < t < 1,
A2(t - 2) + A1(1 - t),
1 < t < 2,
A2(t-2)+A1(t-3),
2
Theorem 3.2. ([4]) The solution x of (3.2) with the initial value condition xo E C[-1, 0] satisfies x(t) = v(t) + O(E) uniformly on the time scale 1/f, where v is given by (3.3) with Al = xo(0), A2 = 2 f °1 xo(s) ds - xo(0).
Since W = 0 in Theorem 3.2, we study (3.2) on the time scale 1/E2 of the form (t)
x(t) = -2
J
x(s) ds + e f (x(t), x(t - 1)) + E2g(x(t), x(t - 1))
,
(3.4)
(t) -1
where f (x, y) = Qlx2+Q2xy+Q3y2, g(x, y) = alx3+a2x2y+a3xy2+a4y3. The corresponding averaged equation of (3.4) has the form [4] Ti = aT2 + bTl + crl T2 + dr1T2 , T2 = -ail + bT2 - CTZ T1 + dr2T1
.
A simple but tedious computation gives that by fixing 01, /32, A3, Q4 and varying al, a2, a3, 014i we can get any a, b, c, d E R in (3.5). The dynamics of (3.5) can be roughly, but completely, classified [4] for different a, b, c, d as follows:
1. The origin is surrounded by periodic solutions.
274
M. Feekan
2. The origin is an attractor. 3. The origin is a repeller. 4. There are two D-equivariant couples of perpendicular, flow-invariant lines crossing the origin and the flow is hyperbolic-like inside each of eight sectors limited by these lines. The only equilibrium is the origin.
5. There is a D-equivariant couple of perpendicular lines consisting of equilibria and crossing the origin such that: - either any solution outside these lines is attracted to some of these equilibria, - or any solution outside these lines is attracted to the origin, - or any solution outside these lines is repelled off the origin, - or the flow outside these lines is hyperbolic-like.
We remark that for generic a, b, c, d, only one of the cases 2, 3, 4 can happen.
4
Semilinear Schrodinger differential equations
and the Let X be a complex Hilbert space with an inner product corresponding norm I Ix. Let B : D(B) C X - X be a self-adjoint operator with a complete orthonormal system {un}nEN of eigenvectors with positive eigenvalues {An}nEN such that AT, <_ an+I V n E N and An - oo as n - oo.
We put Y = XI = D(B) [71, [151. For any 0 < e < 1, we consider the following abstract semilinear Schrodinger differential equation
u(t) + 2Bu(t) = EF(u) + eh(t),
(4.1)
where F : Y -> Y, F : X -> X is continuous and h E C(R, Y). Moreover, we suppose the existence of a constant L1 > 0 and a nondecreasing function : [0, oo) - [0, oo) such that (4.2)
IF(xl) - F(X2)IX <- L1Ix1 - x21X, IF(yl) - F(y2)I Y <- -O(max{IyI IY, Iy2IY}) IyI - y2IY
Let us put Pn = span {u1, u2i -
- -
,
.
(4.3)
un} for any n E N.
Theorem 4.1. ([4]) Let them be n E N and T > 0 satisfying
a;T 21r
Ell V1
and h be T-periodic. Let xo E Y, and let L > 0, r > 0 be constants such that (iF(0)IY + mR Ih(t)IY + IxoiY + 1) exp {2LO(r)} < r.
Galerkin-Averaging Method for Weakly Nonlinear Problems
275
Then for any e > 0 sufficiently small, the solution u of (4.1) with the initial value condition u(0) = xo satisfies l
- e-ttBW(t)l
O(1/A"+1) +O(eT)
x=
uniformly on the interval [0, L/e], where w is the solution of the ordinary differential equation T
T
r
to = ET
e'SBPnF(e-"aBw)
0
ds + ET r e',B Pnh(s) ds,
w(0) = Pnxo .
0
(4.4)
Remark 4.2. As above for (4.1), the same arguments also hold for the equation
u(t) - zBu(t) = eF(u) + eh(t), where B, F, h satisfy the same assumptions as for (4.1). A more careful analysis of the proof of Theorem 4.1 shows that (4.2) can be weakened to IF(yi) - F(y2)Ix 5 9(max{IyIIY,Iy2IY})IYI -y2IX,
(4.2')
where 9 : [0, oo) --+ [0, oo) is a nondecreasing function. Of course, now yl, y2 E Y in (4.2').
For a given sequence {ak}k 1 C C such that F,k I k°IakI2 < oo and rlk, wk E N, k E N, we note that the almost periodically excited nonlinear Schrodinger equation given by
Ut = -s(u:= + elul2u + e k
1
ake'nkt/c
sin kx),
u(0, t) = u(rr, t) = 0 satisfies the assumptions of Theorem 4.1 (see Remark 4.2) with any n E N and T = 27r(wl , W2, , wn], where [... ] is the least common multiple. It can be shown that (4.4) has now the form
to=-etIwI2w, wECn, w(O)=xoECn with the solution w(t) = e-'=1Xo12txo. Theorem 4.1 and Remark 4.2 imply the following result.
Theorem 4.3. ((4]) For an initial value condition 00
ak sin kx, ak E C,
Ho" (0, rr) n H2,c(0, rr) D u(x, 0) = rE2 k=1
276
M. FeEkan
the solution of (4.5) satisfies max
tE(0,L/e)
lu(x,t)
_.L E n 2 re`t(k' k=1
-
la;l')ak sin kxl
k=1
= 0(1/n2) + Q(E[W1,W2, ...
Lz(o,x)
W111),
when c > 0 is sufficiently small and L > 0 is a constant independent of E. Instead of (4.5), we can consider ak(tjkt/Wk) sin kx),
Ut = -z(uyz + EIul2u + E k=1
u(0, t) = u(9r, t) = 0,
where ak E C(R, C) are 27r-periodic satisfying E' l 0 maxR Jak 12 < oo, and 17k, Wk E N. We suppose that there is a ko such that 2www,,o
ako(nkos/wko)e-i°k! ds # 0. J 0
The averaged equation of (4.6) in the variable r = Et and for n sufficiently large has the form [41
W' = - _ Iwl2w - ZBn,T,,,
w(0) = xo E C"
W E C",
,
(4.7)
where
B
b
)
6T,k =
f
ak(ilks/Wk)e-'.k2
ds,
0
T = Tn = 2ir[wi, W2, ... , Wn1 ,
By analyzing (4.7), we can show [41 that the waves with a higher number of modes for (4.6) can be excited by starting initially with a lower number of modes in the first order Galerkin-averaging method. This excitation is wholly determined by the almost periodic term of (4.6). A similar result is derived in [141 for the autonomous nonlinear Klein-Gordon equation for the second order Galerkin-averaging method. Moreover, for most large initial value conditions, this excitation is only a small turbulence in higher modes. Finally, we remark that we can similarly study the rapidly oscillating equation
u(t) + zBu(t) = F(u) + h(t/E),
(4.8)
where F satisfies (4.2'), (4.3) and h E C(R, Y) is T-periodic. The projected averaged equation is given by Tr
u(t) + zBu(t) = P"F(u) + T
J0
Pnh(s) ds .
(4.9)
Galerkin-Averaging Method for Weakly Nonlinear Problems
277
on The error in X between the solution of (4.8) and (4.9) is [0, L] for any initial value condition u(0) = xo E Y and a certain constant L > 0.
5
Neutral differential difference equations
We start with a special neutral differential difference equation of the form
i(t) = x(t - 1)(1 + f f (x(t), x(t - 1))) + eg(t) , (5.1) where f E C1(R2, R) is uniformly bounded and g E C'(R, R) is 1-periodic satisfying g(s) = -g(1-s). Such equations can occur in studying a collision problem in electrodynamics [3]. We put X = {x E C1[0,1] (x(t) = x(1 t), x(0) = 0}. Theorem 5.1. ([4]) The averaged equation for (5.1) on the space X is given by tb(t, z) = cG(w(t, z), z),
w(0, z) = xo(z) E X
,
t
where G(x, t) = f f (s, s) ds+f g(s) ds. The asymptotic shape of the solution 0
0
of (5.1) with t - 1/c, fore > 0 small, is a piecewise defined function whose value at z is either a zero of G(., z) or ±oo. Now we apply Theorems 2.2 and 2.3 to (5.1). For Pn we take n
('2)x(m/n)tm(1 - t)n-"`
(Pnx)(t) _ M=O
Hence Pnx is the Bernstein polynomial B(n, x) of the order n for x [10]. (2.6) has now the form of a system of n + 1 equations n k _ )n-k dm(t) = eG( ()ak(t) m (n nm ), am(0) = xo(m/n).
n
k=O
(5.2)
From [[10], Theorem 1.6.1 and p. 25] we know that Ix - Pnx(c <_ 9(x(c2/
(4n'/2) and (Pnx(c2 < (x(c2 for any x E C2[0,1]. Hence now an = 9/(4n1/2) and K4 = 1 in (i), (ii).
Theorem 5.2. ([4]) Let x(t) be the solution of (5.1) with the initial value condition xo E C2[-1,0]. Let am(t), m = 0, 1, ,n be the solutions of (5.2) with the initial value conditions an(0) = xo(111 1). Then it holds that
-
n
Ix(t + k -
1) - > m=0
(n)am(k)tm(1
- t)n-mI = 0(1/n 1/2) + O(e)
0
uniformly on the time scale 1/e.
278
M. FeLkan
Now we consider the linear problem
th(t) = x(t - 1) - ex(t - 1) sin 21rt,
e>0
(5.3)
which is considered on X = {x E L2(0,1) 1 x(s) = x(1 - s)} in the form t
xn+1(t) = xn(t) - e f x (s) sin 27rs ds, t E [0, 1]. It can be shown [4] that 0
x = 0 in (5.3) is approximately attracting on [0, L/e] for e > 0 sufficiently small and for any initial value condition x0 E Y = {x E W1"2(0,1) 1 x(s) _
x(1 - s), x(0) = 0). Finally, we note that generally the averaged equation is very difficult. For instance, let us consider
i(t) = ±(t - 1) + eh(x(t - 1)) sin 27rt,
(5.4)
where h E C1(R, R). The averaged equation of (5.4) on X is the Goursat problem for a wave equation [4]. On the other hand, to get some quantitative information for (5.4) on [0, L/e], we can apply the Picard iteration procedure to the projected averaged equation. Of course, L may be bounded. For instance, by taking h(u) = u2 in (5.4), there is the "blow-up" of its solution after a finite time for e > 0 sufficiently small and for any initial value condition x0 E W1"2(0,1) which is even around t = 1/2 and satisfies f o x0 (t) cos 21rt dt < 0.
REFERENCES [1] N.A. Bobylev, YU. M. Burman and S.A. Soloviev, Error bounds for Galerkin procedures, Nonlinear Anal. TMA 24 (1995), 1515-1532. [2] N.N. Bogoliubov and YU.A. Mitropolskii, Asymptotic Methods in the
Theory of Nonlinear Oscillations, Gordon and Breach, New York, 1961.
[3] R.D. Driver, A functional differential system of neutral type arising in a two-body problem of classical electrodynamics. In: Nonlinear Differential Equations and Nonlinear Mechanics, Academic Press, 1963, 474-484. [4] M. Fe?-1kan, A Galerkin-averaging method for weakly nonlinear equations, Nonlinear Anal. 41 (2000), 345-369.
[5] J.P. Fink, W.S. Hall and A.R. Hausrath, A convergent two-time method for periodic differential equations, J. Differential Equations 15 (1974), 459-498. [6] J.K. Hale, Averaging methods for differential equations with retarded arguments and a small parameter, J. Differential Equations 2 (1966), 57-73.
Galerkin-Averaging Method for Weakly Nonlinear Problems [7]
279
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lee. Not. Math. 840, Springer-Verlag, Berlin, 1981.
[8] M.S. Krol, On a Galerkin-averaging method for weakly nonlinear wave equations, Math. Meth. Appl. Sci. 11 (1989), 649-664.
[9] J. Kurzweil, The averaging principle in certain partial differential boundary problems, Gas. Pest. Mat. 88 (1963), 444-456 (in Russian). [10] G.G. Lorentz, Bernstein Polynomials, Math. Expos. 8, Toronto, 1953.
[11] H. Pals, The Galerkin-averaging method for the Klein-Gordon equation in two space dimensions, Nonlinear Anal. TMA 27 (1996), 841856.
[12] G.G. Rafel, Application of a combined Galerkin-averaging method. In Asymptotic Analysis II, Lec. Not. Math. 985, Springer-Verlag, Berlin, 1983, 349-369.
[13] J.A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems, Springer-Verlag, New York, 1985.
[14] A.C.J. Stroucken and F. Verhulst, The Galerkin-averaging method for nonlinear, undamped continuous systems, Math. Meth. Appl. Sci. 9 (1987), 520-549. [15] E. Zeidler, Applied Functional Analysis; Applications to Mathematical Physics, Appl. Math. Sciences 108, Springer-Verlag, New York, 1995.
Michal Fei`kan
Departament of Mathematical Analysis Comenius University Mlynska dolina, 842 48, 15 Bratislava Slovakia Michal.FeckanCfmph.uniba.sk
PBVPs for Ordinary Impulsive Differential Equations Daniel Franco Juan J. Nieto* Introduction
1
Impulsive differential equations occur in many biological, physical and en-
gineering applications (see [2, 3, 5]). In consequence, the study of such systems has gained prominence. In this work we present a new expression for the solution of a linear first
order PBVP with impulses at fixed points. As an application, we prove certain new existence results for a nonlinear PBVP with impulses.
Green's functions
2
Let J = [0, 27r], 0 < tl < t2 . . . < tp < 27r and define
J' = J \ {t1it2,...,tp}. We introduce the following spaces of functions:
PC(J) = {u: J -+ R : u E C(J'), there exist u(0+) = u(0), u(27r-) = u(21r), u(tk ), u(tk ), and u(tk) = u(tk), k = 1, ...,p}
PC'(J) = {u E PC(J) : u' E C(J'), there exist u'(0+), u'(27r ), u'(tk ), and u'(tk ), k = 1, ..., p}. It is easy to show that PC(J) and PC' (J) are Banach spaces with the norms IIuIIPC(J) = sup lu(t)I fEJ
and
IIuIIPC'(J) = IIuIIPC(J) + IIu IIPC(J),
respectively. The author's research was partially supported by Ministerio de Educacion y Cultura, DGESIC, project PB97-0552 and Xunta de Galicia, project XUGA207011398.
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We consider the following PBVP with impulses at fixed points:
u'(t) + Mu(t) = a(t), t E J', u(tk) = Cku(tk), k = 1, 2, ... , p,
(2.1)
u(O) = u(27r),
where or E PC(J) and M, ck, k = 1, 2, ... , p are constants. We say that a function u is a solution for (2.1) if u E PC1(J) and it satisfies (2.1). It is known, see for example [1], that a necessary and sufficient condition for (2.1) to have a unique solution is that P
ft Ck # e2TM k=1
In what follows we assume that (2.2) holds. Moreover, when M 96 0, it is possible to give the following expression of the solution (see [4]): P
2w
u(t) = f
g(t,s)a(s)ds + E g(t,tk)(ck - 1)u(tk). k=1
o
(2.3)
Here 1
f e-M(t-8)
if 0< s < t< 21r,
g(t , s) = 1 - e-2rM t e-M(2r+t-a) if 0 < t < s < 27r,
(2 . 4)
is the Green's function associated to the nonimpulsive problem
u'(t) + Mu(t) = a(t), t E J,
u(0) = u(21r).
(2.5)
Recall that (2.5) has a unique solution if and only if M # 0 and this solution is precisely u(t) = for g(t, s)o(s)ds.
Expression (2.3) is a basic tool in the qualitative theory of impulsive differential equations and it is used in many papers. However, it presents two fundamental difficulties. Firstly, we do not obtain an explicit formula for the solution. We obtain a new equation, since the right-hand side of (2.3) depends on the solution u at the instants of impulse tk, k = 1, 2,. .., p. Obviously, we do not know those values. Secondly, as we said before, there exists a unique solution for the problem
if and only if (2.2) holds. However, expression (2.3) may be not valid in this case. More precesily, (2.3) is not valid when M = 0 independently of the values Ck, k = 1, 2, ... , p. Therefore it is possible that (2.2) holds and that (2.3) has no sense. In consequence, we do not have an expression of the solution for all cases.
PBVPs for Ordinary Impulsive Differential Equations
283
To solve those problems we calculate a new Green's function, intrinsic to the impulsive problem, which allows us to express the unique solution of (2.1) in the form 27r
u(t) = fo g(t, s)v(s)ds.
(2.6)
Thus, we eliminate the impulsive term Flk=i g(t, tk)(ck - 1)u(tk) by including this information in the function g that depends on the constants M and ck, k = 1, 2,... , p. We define the auxiliary function
{(x,y)EJxJ:x
-'
(x, y)
(x, y)
[J
ck
{k:z
Using the variation of parameters formula (see [3]), we have that the solution of (2.1) satisfies u(t)
= u(O)
cke-Mt + fi 1 {k:tk
H cke-M(t-s)a(s)ds
0t
t
t)cke-Mt +
= u(0)e(0,
{k:a
(2.7)
f Vs, t)cke-Mt-')a(s)ds. 0
Evaluating (2.7) at t = 2ir and using the periodic condition we have 2R
u(0)
=1
- (0,
(s,
27r)e-2aM
27r)e-M(21r-a)U(s)ds.
Now, substituting into (2.7) the value of u(O) in (2.8), we obtain that
u(t)
=
1
21r)e-M(t-S)a(s)ds ie(s,
1 - (O, 21r)e- 2RM
+ 1 - (0, 27r)e-2xM 1
2w
itt
21r)e-M(2a+t-s)a(s)ds.
C(0, t)C(s,
Thus we have the desired expression, that is (2.6), where 1
9(t s) = 1 - t(0, 27r)e-2AM
{
t(s, (0, t)t(s,
0 < s < t < 27r,
t)e-M(t-s),
27r)e-M(2A+t-s),
0< t < s < 2ir.
We note that (2.6) is a valid formula since the right-hand side of the equality does not depend on the solution u.
284
D. Franco, J.J. Nieto
Before we give some applications of this representation we point out some of the properties of g.
(i) The sign of g depends on the sign of the constants ck, k = 1, 2.... , p. If all of them are nonnegative, then the sign of g is constant and it depends on the sign of IIk=1 ck - e27rM, but if there exists a constant ci negative, then the function g changes sign. (ii) The function g is discontinuous. This is similar to the discontinuity of g, which is not continuous in the set
D={(t,t):tEJ}. Moreover, g may be continuous in points of this set and discontinuous in other points. In general, § may be discontinuous in the set
D=DU{(ti,t):tEJ,i=1,2,...,p}U{(t,ti):tEJ,i=1,2,...,p}. (iii) g is bounded and, in particular, there exists a constant CO, which depends on the values of M and Ck, k = 1, 2, ... , p, such that 2r
J0
I9(t, s)jds < Co,
t E J.
The following figures are examples of the graphics of the functions g and
g. Figure 1 shows the graphic of the Green's function associated to the impulsive problem
u'(t) - u(t) = o(t), t E [0, 41 \ {1, 2, 3}, u(1+) = Zu(1), u(2+) = -u(2), u(3+) = 3u(3), u(0) = u(4). Note that in this case g is discontinuous in the set {(t,t) : t E [0,4]} U {(j,t) : t E [0, 4], j = 2, -1, 2}
U{(t,j):tE [0,4],j =2,-1,2}.
PBVPs for Ordinary Impulsive Differential Equations
285
Figure 1. And Figure 2 (see next page) shows the graphic of the nonimpulsive Green's function associated to the problem without impulses
u'(t) - u(t) = Q(t), t E [0, 4], u(0) = u(4).
3
Application to the nonlinear problem with impulses
In the recent work [41, Nieto presents several existence results for the nonlinear nonresonance PBVP with impulses
u'(t) + Mu(t) = f (t, u(t)), t E J', u(tk) = u(tk) + Ik(u(tk)),
k = 1,2,...,p,
(3.1)
u(O) = u(27r),
with f:JxR--#R,M#0. Here, we study the particular, but important case, where the impulse functions are linear, i.e., I k (x) = (ck - 1)x f o r k = 1, 2, ... , p,
u'(t) + Mu(t) = f (t, u(t)), t E J', u(tk) = cku(tk), k = 1, 2, ... , p, u(O) = u(27r).
(3.2)
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D. Franco, J.J. Nieto
Figure 2. In this situation it is possible to improve the results of [4] by using the new expresion (2.6). Note that in [4] relation (2.3) was used. We define the operator A: PC(J) -+ PC(J) as 2a
g(t, s)f(s, u(s))ds.
[Au] (t) = fo
Note that this operator is a compact operator and that to find a solution for (3.2) is equivalent to finding a fixed point for this operator. The following unsophisticated result is new and very useful.
Theorem 3.1. If f is bounded, i.e., there exists Cl > 0 such that If (t, x)I < Ci,
t E J,
x E R,
then there exists at least one solution for (3.2). Proof. For p E 10, 1], we consider the equation
u = iAu. Let u be a solution of (3.3) for a given p. Then, for any t E J we have that :5.U
Iu(t)I
f
2,,
I9(t, s) I If (s, u(s) Ids < .CoC1 <_ CoCI.
0
Thus, the solutions of (3.3) are bounded independently of µ E [0, 1]. Therefore, taking into account the properties of A and Schaeffer's theorem [6], we have that A has at least a fixed point.
If the nonlinearity f is not bounded but it has a sublinear growth, then we have the same conclusion.
PBVPs for Ordinary Impulsive Differential Equations
287
Theorem 3.2. Suppose that there exist a E PC(J), b E R, and a E [0,1) such that I f (t, x)I < a(t) + bIxl°, t E J, x E R, then there exists at least one solution for (3.2).
Proof. If u is a solution of (3.3), then for each t E J we have that Iu(t)I < µ
f
s
I9(t, s)a(s) Ids + µCob.
0
Therefore
IIuII <- a + b IIuIIa,
where
r2
a' = µ sup{ / tEJ
(3.4)
I9(t, s)a(s)Ids} and b' = µCob.
o
The function g(x) = x - b'x*, x E R+ satisfies lim g(x) = 00,
x-.+oo
since 0 < a < 1. By (3.4) we have that all solutions of (3.3) are bounded independently of µ E [0,1]. Using again Schaeffer's theorem, the proof is
0
complete.
We remark that it is not possible to use the proof in [4) and to consider
the special case Ik(x) = ckx to show the validity of the results that we have proved. That is, the case of linear impulses does not simplify either the proof or the hypotheses in [4]. Finally, we note that it is easy to generalize both theorems to deal with the problem
u'(t) + Mu(t) = f (t, u(t)), t E J', u(tk) = Ckt1(tk) + dk,
k
(3.5)
u(O) = u(21r),
with dk, k = 1, 2, ... , p constants.
REFERENCES
[1] D. Franco and J.J. Nieto, A new maximum principle for impulsive first-order problems, Int. J. Theor. Phys. 37 (1998), 1607-1616. [2] D. Franco, E. Liz, J.J. Nieto and Y. Rogovchenko, A contribution to the study of functional differential equations with impulses, preprint.
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D. Franco, J.J. Nieto
[3] V. Lakshmikantham, D.D. Bainov and P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. [4] J.J. Nieto, Basic theory for nonresonance impulsive periodic problems of first order, J. Math. Anal. Appl. 205 (1997), 423-433.
[5] A.M. Samoilenko and N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.
[6] D.R. Smart, Fixed Points Theorems, Cambridge University Press, Cambridge, U.K., 1980.
Daniel Franco and Juan J. Nieto Departamento de Analisis Matematico Universidad de Santiago de Compostela Spain
First author's present address: Departamento de Matematica Aplicada I Universidad Nacional de Educaci6n a Distancia Madrid Spain danie1@zmat.usc.es
Homoclinic and Periodic Solutions for Some Classes of Second Order Differential Equations M.R. Grossinho F. Minhos S. Tersian 1
Introduction
In this paper we state the existence of positive homoclinic solutions of the second order equation
u" -a(x)u+ 3(x)u2 +^/(X)U'=O, x E R,
(I)
where the coefficient functions a (x) , f3 (x) and y (x) are continuous, positive and 27r-periodic. We obtain, in some sense, generalizations of results contained in [10] and [6], where 6 (x) is assumed identically zero. The homoclinic solution u of equation (I) is obtained as the limit of 2nir-periodic solutions of (I). We establish the fact that the quadratic form associated to the linear operator is positive definite and the particular type of the nonlinearity considered introduces simplicity and clearness in the proof, namely when we use the mountain pass lemma to study some periodic approximating problems. We present only the main ideas and sketch the proofs briefly. In Section 2, we study equation (I). The approximating procedure used in the proofs appears in several papers concerning the existence of homoclinics, namely in the case of Hamiltonian systems. We refer to Rabinowitz [10], Ambrosetti and Bertotti [1], Korman and Lazer [6], Arioli and Szulkin [2]. However those results do not apply to equation (I). Essentially, not only does the nonlinearity we consider not satisfy the hypotheses assumed there, but also [1], [10] and [2] do not concern positive solutions. For further details concerning Section 2, see[5]The first author has been supported by FCT, PRAXIS XXI, FEDER, under projects PRAXIS/PCEX/P/MAT/36/96 and PRAXIS/2/2.1/MAT/125/94.
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We also study the equation
u"+a(x)u+g(x,u)=0. Here, the quadratic form is not anymore positive definite. The nonlinearity g is allowed to interfere with two consecutive eigenvalues of the underlying linear operator, m2 and (m+ 1)2, and non-uniform non-resonance may occur. Existence of periodic solutions of (II) is established using a saddle point theorem. In the autonomous case, it can be easily seen by the phase plane that there are no homoclinic solutions. Therefore, homoclinic solutions of (II) are not considered. In Section 3, we study equation (II). The result we present extends one contained in [4], where the function a (x) was supposed not to touch the eigenvalues m2 and (m + 1)2and the nonlinearity g(x, u) was the particular one g(x, u) = -,3 (x) u2 + y (x) u3. We also refer to [3] where the case a (x) = m2 is studied. The proof uses a saddle point theorem due to E.A. Silva [9], [12] (in [4] was used an approximating procedure by working in finite dimensional spaces and applying a linking result due to Rabinowitz [11]).
2
Positive homoclinic solutions of equation (I)
In this section we consider the existence of positive solutions of the problem
u" - a (x) u +'3 (x) u2 + 'Y (x) u3 = 0,
x E R, (P)
U (±oo) = u' (±oo) = 0, where a,,Q and ry are continuous, 27r-periodic functions such that
0
(2.1)
with a, b, B, C, C E R.
Theorem 2.1. Let a, 0 and 'Y satisfy the above assumptions. If
4ac > B2 -b2 ,
(2.2)
then the problem (P) has a positive homoclinic solution. Sketch of proof. Step 1 - Approximate problem. For every n E N, consider the periodic problem
u"- a (x)u+f3(x)U2+7(x)u3=0, u (-nir) = u (nir) ,
xE (-nir,nir), (Pn)
u' (-nir) = u' (n7r) ,
and the modified problem
U" -a(x)u+i3(x)u2+ry(x)u+ = 0, u (-n7r) = u (nir) ,
u' (-n7r) = u' (nor)
(Pn) ,
Homodinic and Periodic Solutions for Second Order Differential Equations
291
where u+ = max (u, 0). It is easy to see that the solutions of (Fn) are non-negative solutions of (Pn). Indeed, if u (x) is a solution of (P,) with negative minimum at some xo E (-nrr, nrr), from the equation we derive the contradiction
0 = u"(xo) -a(xo)u(xo) +$(xo)u2 (xo) > 0. Then u (x) > 0 and so u is a solution of (Pn). Consider the space
Hn={uEH'(In):u(-nrr)=u(nrr)}, with the norm I IuH In =
(_I (u'2 (x) + u2 (x)) dx
.
To prove the existence of a solution of (Pn) we consider the functional
fn:Hn -'Rdefinedby n7r
fn (u) =
J
(2 (u12 + a (x) u2) -
a (x) u3 - 4 ry (x) u+) dx.
-nI
3 It can be shown that fn satisfies the Palais-Smale (PS) condition and the geometric assumptions of the mountain-pass theorem which guarantees the existence of a nontrivial critical point of fn. So, there exists a weak solution of (P,) (and, by a standard way, a classical one), and, so, a non-negative solution of (Fn), un E Hn, such that c
inf = fn (un) =
tma i) fn
(a (t)),
fn (un) _ 0,
where Ian = {v E C ((0,1] , Hn) : a (0) = 0, a (1) = uo (x)}
with vo E Hl a function of norm large enough such that fn (uo) < 0. Step 2 - Uniform estimates. There is a constant K > 0, independently of n, such that IIuIIn < K.
(2.4)
Let m > n > 1 and, by continuation with a constant, consider the natural imbeddings Hn C H,n and rn c rm. Using the variational characterization (2.3) we can infer c,,, < cn < cl and then n7r
J -nx
1
2
(un + a (x) u,2,) - 3Q (x) U.3 - 4 (x) un 1 dx < cl.
(2.5)
M.R. Grossinho, F. Minhos, S. Tersian
292
Multiplying the equation of (Pn) by un and integrating by parts, nn
J
nir (U12
+ a (x) un) dx = J (7 (x) un + Q (x) un) dx.
-na
-n7r
Then, by (2.5), we have na
nir Cl
>_
1
J
(u'n + a (x) un)
dx -
1
- n'A nA
6
I
J
(Q (x) un + y (x) un) dx
-nir
(un + c (x) un) dx
- 6IIunI In,
which shows that (2.4) holds. Step 3 - Passing to the limit. By the embedding of Hn in C [-na, n?r] , it follows from the equation of (Pn), that there is K1, a positive constant independent of n, such that IIunIIC2[-nir,nirj <- K1. Denote by the same symbol the periodic extension of un to R. Then un is a 2n7r-periodic solution of equation (I). Moreover, there exists a subsequence of (un) which converges in C12,, (R) to a solution u of (P) that satisfies 00
J
(ui2 + u2) dx < oo.
(2.6)
-00 Since
T+1
max
xE[T-1,T+1J
j u (x) I < lim
T-.±oo
J T-1
(ui2 (x) + u2 (x)) dx,
(see [5] and [10]), it follows that T/+' 1
Tl
xE(maaT+1J Iu (x) 15 Tlim J (u"2 (x) + u2 (x)) dx = 0, T-1
so u (±oo) = 0. It remains to show that u is nonzero and u' (±oo) = 0. Let xn E [-n7r, nzr] be a point where un attains its maximum value. Since un (xn) > 0 and un (xn) < 0, it follows by the equation (I) that
U. (xn) (-a (xn) +0 (xn) un (xn) + 7 (xn) un (xn)) _ -u'n', (xn) > 0. Then, by assumptions (2.1) and (2.2), for all n,
un(x,)?
-B+ b2+4ac 2C
> 0.
(2.7)
Homoclinic and Periodic Solutions for Second Order Differential Equations
293
Let us denote by H, the space that consists of the periodic extensions of the functions of Hn and by fn the functional defined on Hn by f (u) = fn (uI [-nx,nwj) Then, replacing un (x) by un (x + 2 jnx) , if necessary, where in is an adequate integer, we still obtain 2nzr-solutions of (I), satisfying the above bounds derived by using variational arguments, and such that those solutions have maximum points xn appearing in the interval [-7r, 7r]. Therefore, we can assume xn - xo in [-7r, 7r] , and, by the uniform convergence of {un } on [-7r, 7r] and by (2.7), it follows that u (xo) > 0. Therefore, the
solution u (x) of (I) is nontrivial and nonnegative. By assumption (2.1), there exists M > 0 such that lu" (x) I _< M in R. Suppose, by contradiction, that u' (+oo) 0 (the case u' (-oo) # 0) is analogous). Then, there exists e > 0 and a sequence yn --+ +oo such that Iu' (yn) I > e, for all e n. By the Lagrange theorem, Iu' (x) I > for x E (yn - b, yn + 5) , where b E (0, ZM ) . Therefore Y.+6
ber e
u2(x)dx
2,
Y.-6
0
which contradicts (2.6).
Further, using the above method, we can prove the existence of a symmetric positive homoclinic solution of (I) under adequate assumptions on the coefficient functions. Our result is based on a Lemma proved by Korman and Ouyang [7] and is an extension of a result due to Korman and Lazer [6], where equation (I) is considered with 0 (x) = 0. Let a (x) , b (x) and c (x) be differentiable functions that satisfy (2.1) and such that
a (x) = a (-x), b (x) = b (-x), c (x) = c (-x), xa'(x) > 0,
xb'(x) < 0,
xc'(x) < 0.
(2.8) (2.9)
Using a lemma contained in [7], we can state the following result (for the proof, see [5]).
Lemma 2.2. Let assumptions (2.1), (2.8) and (2.9) hold. Then, for any T > 1, the problem
u"-a(x)u+/3(x)u2+y(x)u3=0, xE (-T,T), u (-T) = u (T) = 0,
(PT)
has a unique positive solution uT (x). Moreover, uz. (x) < 0, for x E [0, T] and there exists a constant K > 0, independent of T, such that
J (u4(x)+4(x))dx
,
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M.R. Grossinho, F. Minh6s, S. Tersian
Theorem 2.3. Let assumptions (2.1), (2.8) and (2.9) hold. Then problem (P) has exactly one positive solution u. This solution is an even function
andu'(x) <0 ifx>0. Sketch of proof. Take Tn - oo and let un be the solution of problem By Lemma 2.2 T.
J -T
(un 12 (x) + un (x)) dx < K,
independently of n. Consider the extension of un to R that takes the value
0 in R\ [-Tn, Tn] and denote it by the same symbol. Arguing as in the proof of Theorem 2.1 we can derive that un - u in C,2,,, (R). By Lemma 2.2, un attains its maximum at 0. Then, since un (0) (a (0) - b (0) un (0) - c (0) un (0)) = u' (0) < 0,
it follows that un (0)
-
b (0) + (b2 (0) + 4a (0) c (0)) 12 2c (0)
_ Pi > 0,
and, therefore u (0) pi > 0. Moreover, u is an even function that attains its only maximum at 0, since the same holds for the functions un. Arguing as in [7], we easily obtain u' (x) < 0 if x > 0.
To prove uniqueness, observe that if u, v are two solutions, it follows
that
Juv((x)(u_v)+(x)(u2 - v2)) dx = 0. -00
The existence-uniqueness theorem of the Cauchy problem and the last iden-
tity imply that u (x) and v (x) cannot be ordered and so they must intersect. Two cases are possible: either u (x) and v (x) have at least two positive
points of intersections or only one positive point of intersection. In both cases the proof continues as in the proof of Theorem 2.1 of Lazer and Korman [6]. To prove u (+oo) = u' (±oo) = 0, we proceed as in the proof of Theorem 2.1.
3
Periodic solutions of equation (II)
In this section we consider the equation (II). Define G(t, s) = f g(t, ) dd. 0
Homoclinic and Periodic Solutions for Second Order Differential Equations
295
Theorem 3.1. Let a (x) be a measurable function, 27r-periodic, such that m2
a (x)
N
N
(m + 1)2 ,
but strictly "<" on a subset of positive means where m E N and measure. Assume that g : [0, 27r] x R -+ R satisfies the following conditions:
(gl) g(t, s) is an L' Caratheodory function such that g(t, 0) = 0 for a. e. t E (0, 21r);
(92) there are constants B > 2 and M > 0 such that 0 < BG(t, s) < s g(t, s), for Isl > M and a.e. t E (0, 21r); (g3) there exist E1iE2 > 0 and so > 0 such that
G(t, s) > -E 1s2, for s E R and a.e. t E (0, 27r) G(t, s) < E2s2, for IsI < so and a.e. t E (0, 27r).
Then, if E1,,-2 are small enough, the equation (II) has a nontrivial 27rperiodic solution.
We recall the saddle point theorem due to E.A. Silva [9], [12] used next.
Theorem 3.2. Let E = X, ® X2 be a real Banach space, with X1 a finite dimensional subspace. Suppose f E C1 (E, R) satisfies (PS) condition and: (i) f (u) 5 0, for every u E X1 (ii) There exists p > 0 such that f (u) > 0, for every u E X2 with IIull = p. (iii) There exists l; E X2, IKII = 1, and p E R such that f (u) < Q, for -
every u E X 1
®R+t.
Then f possesses a critical point in E, u $ 0. Sketch of the proof of Theorem 3.1. Let H2,, be the (Sobolev) space of absolutely continuous real functions u defined on [0, 27r] such that u(0) _ u(27r) and u' E L2(0,2ir), with the usual inner product 2w
(u, v) =
J
(u'(x)v'(x) + u(x)v(x)) dx.
The corresponding norm is denoted by 11.11. Let I.Ip denote the usual norm in the LP space. Consider the C'-functional 4 ): H2,r -' R defined by r2,r
4(u) = ,1
Jo
1
(2 u'2 - a(x)u2 - G(x, u))dx.
It is clear that 4 E C' (E, R) and that the critical points of provide weak solutions of (II). Due to the Euler identities, the expansion of u E H2,r as a series of eigenfunctions can be written in the form
u(t) = r Cke'kx kEZ
296
M.R. Grossinho, F. Minhbs, S. Tersian +oo
with c_k = ck and
Ick l2 (1 + k2) < oo. Consider k=-oo
X1 =
Cketka,
u E H2W : u =
C-k = Ck
k2<m2
X2 = JuEH2ir:u=
ckeikx,C_k=Ck k2>(m+1)2
It is obvious that H21W = X, E) X2 and each u E H2,r can be written as
u = u1 + u2i with ul E X1 and U2 E X2. Taking into account that the functions g and G verify (92) and (g3), it can be proved in a standard way
that the functional satisfies (PS) condition. Let us show that also satisfies the geometric type hypotheses of the above theorem. We recall that, by a result of Mawhin [8) on coercivity of some quadratic forms on H2,r, there is am > 0 such that for every u E H2,r, 2a
J a (x) (U2 (X) - u2 (x)) - u'12(x) + u2 (x)dx > amllull2 0
Let u E X1. Then, for El small enough, 2a
(u)
<
-6,r+IIuII2
- f G(x, u)dx 0
-amllUll2 + el Iu12 < 0
Let K1 be the Sobolev imbedding constant of H2,r in LO (0, 21r). Take u E X2 such that llull 5 s0 . Since Iuko < Klllull, we obtain, by (g3), G(x, u) < e2u2 and so 2,r
4 (u) > bmllUIl2
- f G(x, u)dx > bmIIuII2 - e2lUI2, 0
which shows that, if e2 is small enough, 4 (u) > 0 for u E X2 and I lull 5 p, for some p > 0. Let f (x) = VW(1+(m+1)2) 1 cos (m + 1) x E X2. By integration, condition
(92) implies that G (x, u) > gllul" - q2,
Homoclinic and Periodic Solutions for Second Order Differential Equations
297
with ql, q2 E R+. So, for u = E cke'k2 + At;, with A > 0, we obtain k2 <m2 2A
4i (u)
< (2m + 1)
> ck
+
A2
k2 <m2
<
G (x, u) dx o
(2m+1)IuI2-giline+2irg2.
And, since 2 < 9, it follows that
(u) < /3 for some 0 E R. The assertion
follows by Theorem 3.2.
Note. An existence result for the case a (x) ,< 1 can be derived in a similar way using in the appropriate step the result of Mawhin [8] on coercivity of some quadratic forms on HZA that states that if, moreover, a1 = f2A a(t)dt > 0, then there exists 61 > 0 such that for every u E H2A, 2A
a14 + fux ) - a (x) u2 (x)) dx > j1 IIuI12 0
REFERENCES [1] A. Ambrosetti and M. Bertotti, Homoclinics for second order conservative systems, preprint, Scuola Normale Superiore di Pisa n. 107, 1991.
[2] G. Arioli and A. Szulkin, Homoclinic solutions for a class of systems of second order equations, preprint, Dept. Math., Univ. Stockholm, n. 5, 1995.
[3] M. R. Grossinho, Periodic solutions of some second order differential equations at resonance. In: Advances in Difference Equations, I (S. Elaydi, G. Ladas, and I. Gyori eds.), Gordon and Breach, 1995, pp. 271-280. [4] M. R. Grossinho and L. Sanchez, A note on periodic solutions of some
nonautonomous differential equations, Bull. Austral. Math. Soc. 34 (1986), 253-265.
[5] M. R. Grossinho, F. Minhos and S. Tersian, Positive homoclinic solutions for a class of second order differential equations, J. Math. Anal. Appl., December 1, 240/1 (1999), 163-173.
[6] P. Korman and A. Lazer, Homoclinic orbits for a class of symmetric hamiltonian systems, Electr. J. Differential Equations 1 (1994), 1-10. [7] P. Korman and T. Ouyang, Exact multiplicity results for two classes of boundary value problems, Differential and Integral Equations 6 (1993), 1507-1517.
298
M.R. Grossinho, F. Minhos, S. Tersian
181
J. Mawhin, Compacite, monotonie e convexite dans etude de problemes aux limites semi-lindaires, Seminaire d'Analyse Moderne, 1981.
191
E. A. Silva, Linking theorems and applications to semilinear elliptic problems at resonance, Nonlinear Anal. T.M.A. 16 (1991), 455-477.
110] P. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh 114A (1990), 33-38. [11] P. Rabinowitz, Free vibrations for a semilinear wave equation, Comm. Pure Appi. Math. 31 (1978), 31-68.
1121 M. Ramos, Teoremas de enlace na teoria dos pontos crlticos, Universidade de Lisboa, 1993.
Maria do Rosario Grossinho Departamento de Matematica, ISEG, Universidade T6cnica de Lisboa, Rua do Quelhas, 6, 1200-781 Lisboa, Portugal and CMAF, Universidade de Lisboa Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal
mrgalmc.fc.ul.pt Feliz Minhas C.I.M.A.-U.E. Universidade de Evora Evora, Portugal
Stepan Tersian Center of Applied Mathematics and Informatics University of Rousse 8, Studentska, 7017 Rousse, Bulgaria
stepanQuami.ru.acad.bg
Global Bifurcation for Monge-Ampere Operators Jon Jacobsen 1
Introduction
In this paper I wish to describe some recent results concerning bifurcation for nonlinear equations of Monge-Ampere type. Many of the results below
may be found, with more detailed proofs, in [8]. I would like to thank Professor Klaus Schmitt under whose guidance this work was carried out. Let S C R" be a bounded, open set. For a real valued function u E C2(Sl), the Monge-Ampere operator M : C2(Q) - C(Sl), is defined by M[u] = det D2u.
Partial differential equations containing the nonlinearity M[u] are often called Monge-Ampere equations. Equations of Monge-Ampere type have a beautiful history, stemming from early applications in differential geometry (e.g., the Minkowski and Weyl problems [11], the Louwner-Nirenberg problem [10]). More recently equations of Monge-Ampere type have appeared in problems related to Gaussian curvature flow [18], meteorology [1], and geometric optics [7]. For nonlinear equations of the form
F(x, u, Du, D2u) = 0,
(1.1)
such as Monge-Ampere equations, the type of the equation may depend on the function u. Equation (1.1) is called elliptic (at the function u) if
8u
(x,
u(x), Du(x), D2u(x))
is a positive definite matrix. For Monge-Ampere equations, the matrix 37.7 is the cofactor matrix of D2u, hence Monge-Ampere equations will be el-
liptic at u if u is strictly convex (or concave). Thus, it is natural to restrict oneself to this class of functions. There is a well established existence and uniqueness theory for Monge-Ampere equations of elliptic type (e.g., [2, 15, 6]). For reference we recall the following theorem due to Trudinger:
300
J. Jacobsen
Theorem 1.1. (Trudinger, [15]) Let f) be a strictly convex bounded domain in Rn, tai E L'(0) with tfi > 0. Then there exists a unique admissible weak solution u E C°"1(fl) of the equation
f det D2u = tp(x),
u = 0,
x E S1,
x E Ofl,
(1.2)
where a function u E C(fl) is an admissible weak solution of (1.2) if it solves (1.2) in the viscosity sense N.
2
Principal eigenvalue
In this section we consider the Dirichlet problem det D2u = I,\uln, x E S2,
xE&I
lu=0, and seek nontrivial solutions.
The Monge-Ampere operator shares a property with another popular operator at this conference (Op), namely, it has a certain degree of homogeneity. The Monge-Ampere operator is positively homogeneous of degree n. Therefore, if we find a nontrivial solution u to equation (2.1), then, by the homogeneity, the function v = bu will also solve (2.1) for 6 > 0. Let us now assume SZ is strictly convex and let E denote the Banach space
of continuous functions defined on St, with sup norm IIuII = sup.,n Iu(x)I Let K C E denote the cone of nonpositive functions. The cone K induces
a partial order on E via u <_ v if v - u E K. The restriction of f l to be strictly convex is natural here as 8S2 is the level set u = 0 of the strictly convex function u. In connection with the Dirichlet problem (2.1), we introduce the solution
operator A : E -- E defined by A(u) = z where
detD2z =
Iu(x)I',
1z=0,
x E S2,
xE81.
It follows from Theorem 1.1 that A is well defined. A satisfies many desirable properties, which we now enumerate:
1. A : K -+ K is completely continuous. 2. A is positively homogeneous of degree 1.
3. A is monotone with respect to the partial ordering induced by the cone K, i.e., x < y implies A(x) 5 A(y).
Global Bifurcation for Monge-Ampere Operators
4. If u E Im(A), then there exists 0 < M < oo such that 0 < for x E Q, where v denotes the outward unit normal to M.
301
<M
An operator A : K - K is said to be strong provided for all u, v E Im(A)\{0}, there exists 8, ry > 0 such that
u-BvEK
and
v-yuEK.
The following theorem is a generalization of the Krein-Rutman theorem which is suitable for our purposes:
Theorem 2.1. (Jacobsen, [8]) Let E contain a cone K. Let A : E -' K be a completely continuous operator with ASK : K -- K homogeneous, monotone, and strong. Then there exists a unique a0 > 0 with the following properties:
1. There exists u E K\{0}, with u = \0A(u). 2. If V E K\{0} and A > 0 such that v = \A(v) then A = A0.
To apply this theorem to the operator A above, we need to know that A is strong, however, this follows from the gradient bound (Property 4 above). Therefore Theorem 2.1 may be applied directly to establish: Theorem 2.2. The Monge-Ampere operator A has a unique positive eigenvalue 1/a0. Therefore, there exists a unique positive constant A0 such that the problem (2.1) admits a nontrivial admissible solution u E K. As the degrees of homogeneity are the same on each side of (2.1), the functions 6u0 also form
admissible solutions to (2.1), for all J > 0. Note that u = 0 is a solution to (2.1) for all A > 0, thus A0 corresponds to a bifurcation value of (2.1), where a line of solutions bifurcates from the trivial branch at (A0i0). Further properties of \0 are: Proposition 2.3. (Simplicity of A0) If v E K\{0} also satisfies (2.1), then v = 9u for some 0 > 0.
Proposition 2.4. If ST
fl, both strictly convex, then the eigenvalue associated with 11' is strictly greater than the eigenvalue associated with 91.
The properties of A0 above agree with the usual properties for the principal eigenvalue for linear second-order elliptic operators. For this reason, A0 is referred to as the principal eigenvalue for the Monge-Ampere operator. The existence of a principal eigenvalue for the Monge-Ampere operator was first established (using a different technique) by P. L. Lions [9].
302
3
J. Jacobsen
Global bifurcation
In this section we consider the nonlinear Dirichlet problem
det D2u = g(u), x E fZ,
lu=0,
XEBft,
(3.1)
where the nonlinear term g : R -- R+ satisfies: 1. g is continuous, 2. g = o(juI") as Jul -' 0,
3. U -'ooas For example we may take g(u)=I8ul for any p > n and 6 E R. We seek nontrivial solutions to (3.1). The approach used is to embed (3.1) into the one parameter family of problems
fdet D2u = IAuln + g(u), x E cl,
1u=0,
xEOfl,
(3.2)
and consider the behavior of global bifurcation continua. That is, we shall first prove the existence of a global bifurcation branch of solution pairs (A, u) for the problem (3.2), then use a priori bounds to show the branch must cross the axis A = 0 nontrivially. By "crossing the A = 0 axis nontrivially" we shall mean the continuum crosses the A = 0 axis at a point (0, u) with tIul 10 0.
We associate to Equation (3.2) the solution operator H : R x E -' E defined by H(A, u) = z where f det D2z = IAuI" + g(u), x E 11,
z=0,
xEM.
By Theorem 1.1, H is well defined. It can also be shown that H is completely continuous [8]. Define h : R x E - E by h(A, w) = w - H(A, w) and consider the equation h(A,u) = 0.
(3.3)
Equation (3.3) has the trivial solution u = 0 for all A E R. We establish the existence of a global branch of nontrivial solutions bifurcating from the trivial branch at A = Ao, where AO is the principal eigenvalue of the Monge-Ampere operator associated with the domain fl. If (A, u) is a zero of (3.3) then u is a convex function which satisfies the equation (3.2). The existence of a global bifurcation branch for (3.3) will follow from the following Krasnosel'ski-Rabinowitz type bifurcation theorem:
Global Bifurcation for Monge-Ampere Operators
303
Theorem 3.1. (Global Bifurcation, [14]) Let F : R x E - E be completely continuous such that F(A, 0) = 0, for all A E R. Let there exist constants a, b E R, with a < b, such that (a, 0) and (b, 0) are not bifurcation points for the equation
u - F(A, u) = 0.
(3.4)
Furthermore, assume that
d(id - F(a, ), B,.(0), 0) # d(id - F(b, .), Br (0), 0), where B,.(0) is an isolating neighborhood of the trivial solution. Let S = {(A, u) : (A, u) solves (3.4) with u
0} U ([a, b] x 10}),
and let C be the component of S containing [a, b] x {0}. Then either
1. C is unbounded in R x E, or 2. C fl [(R\[a, b]) x {0}] 54 0.
We shall apply Theorem 3.1 to the operator H after we collect some lemmas. More detailed proofs of the lemmas below may be found in [8]. Lemma 3.2. Let 11 be a bounded, convex domain in R". Let {uk } C C(12) be a sequence o f convex functions with uk I80 = 0 for all k = 1, 2, 3, ... . Furthermore, suppose IIukII oo. Then Iukl - oo uniformly on compacta, that is uniformly on compact subsets of !Q.
Lemma 3.3. Let {vk} C C(St) be a collection of admissible solutions of the Dirichlet problem
fdet D2vk = gk, vk = 0,
x E S2,
xE
where gk : St - R form a collection of nonnegative continuous functions. If gk - oo, uniformly on compacta, then IIukII - 00. Lemma 3.4. If (A, u) solves (3.3), then IAI < Ao.
Proof. Suppose (A, u) is a solution of (3.3) with IAI > IA0I. Let uo be an eigenfunction corresponding to the principal eigenvalue for the MongeAmpere operator, i.e., uo satisfies f det DZuo = IAouoI'
uo = 0,
,
x E n, x E 49R.
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J. Jacobsen
By scaling if necessary, we may assume u(x) < uo(x) for all x E fl. Let b' > 0 be maximal such that u - 5*uo < 0 in Q. Let L be the operator defined by n
L = E Fi,(D2w)Di3, i,.t=1
where F = detlln, Fi3 = OF/Oui3, and w = 5'uo. It follows from the admissibility of uo that L is an elliptic operator. We compute
L(u - 5*uo) > F(D2u) - F(D2w) = 1IAul" +9(u)]1/n = IAIIuI I1 + g(u)
]
1/n
[IAo5*uoIn]1/n
- IA0II5*uol
IAIIul - IAoIIa uoI > 0,
since IAI > IAol and 0 < I8'uol < Jul for all x E 12 (the first inequality above follows from the concavity of the operator F on the admissible functions, see [2]). This implies, by the maximum principle, u = b*uo for all x E 12. Therefore, det D2u = det Dew, or IAuln
+
g(u)
= IAobuoln =
IAouln
Hence IAIn-IAoIn=-I9uI)
<0,
0
a contradiction, as A > Ao.
Lemma 3.5. If (A, u) is a zero of (3.3), with A E I, for some compact interval I C R, then there exists M > 0, sufficiently large such that IIul1 S M. M is independent of A. Proof. If not, there exists a sequence {(Ak, uk)}, with IIukII --+ oo. Without
loss of generality we may assume Ak - p E I. In other words, p is an asymptotic bifurcation value for (3.3). Let vk = uk/IIukII Then IIukII = 1 and the functions {vk} satisfy detD2Vk = IAkVkln +
9(uk) > 9(uk) IlUkll n
We may rewrite the right-hand side of (3.5) as 9(uk) = 9(uk)IVkln. IIukIIn
IukIn
IIukIl n
(3 . 5)
Global Bifurcation for Monge-Ampere Operators
305
Let SY CC Q. On St' the functions Vk = uk/IIukII are bounded away from 0 and by Lemma 3.2, the functions uk satisfy lukl -+ oo uniformly. Therefore "` oo uniformly on ST as k -+ oo. Combining equation (3.5) with ukn Lemma 3.3 we may conclude I Ivk I I - oo, which yields a contradiction as
IIukII=1. Using the same techniques one can readily prove:
Lemma 3.6. (0, 0) is not a bifurcation point for (3.3). We are ready to establish the theorem:
Theorem 3.7. (Global Bifurcation) (p, 0), is a bifurcation point for (3.3) if and only if I µI = Ao. Furthermore there exists a bounded continuum of nontrivial solutions to (3.3), which bifurcates from (Ao, 0) and lies in the strip {(A, u) : -Ao < A < Ao), which connects (- A0, 0) to (AO, 0).
Proof. [Necessity] Suppose (µ, 0) is a bifurcation point for (3.3). Then there exists a sequence (Ak, uk) - (µ, 0) such that I Iuk1154 0 for all k, and h(Ak, uk) = 0, i.e., the components of (Ak, uk) solve the equation
detD2uk = IAkukln +g(uk).
(3.6)
We consider again the sequence of unit vectors defined by vk = uk/IIukII Dividing (3.6) by IIukII' we see that the function vk solves f detD2uk = IAkvkI" + h(uk)IvkI'
,
x E ci,
xE8l,
vk=0, where h is defined by IEl° , 10,
if
0,
otherwise.
The condition g(u) = o(Iuln) as Jul -+ 0 implies h is bounded and continuous. Thus, by the complete continuity of H, we may obtain a convergent subsequence of {vk }, with vki - v and v 0, as I Ivkl I = 1 for all k. The condition IIukII -+ 0 implies the functions Uk -+ 0 uniformly on Sl, hence as k -' oo, the functions h(uk) also tend to 0 uniformly on Q. Since the functions IVkln are uniformly bounded, the term g(uk) = h(uk)Ivkltt
Iluklln
must also tend to 0 uniformly as k -+ oo. By taking subsequences if necessary, we may arrange it so that the sequences {uk}, {Ak}, and {vk, } converge together. Hence, letting k -, oo in (3.7), we see that v satisfies
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J. Jacobsen
the equation det D2v = I µvI",
lv=0,
x E SZ,
xE00.
(3.8)
Equation (3.8), together with Theorem 2.2, imply I141 = Ao. Thus ±Ao are the only possible bifurcation points for (3.3). Proof. [Sufficiency] The proof of sufficiency will follow from Theorem 3.1. It follows from arguments similar to those in the proof of necessity above that, for any fixed constant µ with Iµ1 # Ao, u = 0 is an isolated solution of
(3.3) and we may homotope out the perturbation term g(u) and preserve the degree. Hence, it suffices to find constants a, b such that d(id - H(a, ), B,.(0) 54 d(id - H(b, ), B,.(0), 0).
We may choose a = 0 and note that d(id - H(a, ), B,.(0), 0) = d(id, B,.(0), 0) = 1. The bound on A from Lemma 3.5 implies that for any constant b > AO we have d(id - H(b, ), B,.(0), 0) = 0,
hence by Theorem 3.1 we may conclude there exists a continuum C of solution pairs to (3.3) which is either 1. unbounded in R x E, or 2. C n [(R\[0, Ao]) x {0}] 36 0.
However, Lemmas 3.5 and 3.6 above imply the continuum is bounded in R x E, hence must connect to another bifurcation point (µ, 0), with µ V [0, Ao]
Finally, we may conclude that the Dirichlet problem (3.1) has at least one nontrivial solution. Theorem 3.8. Let S2 C Rn be bounded and strictly convex. Let g : R -, R+ be continuous such that
1. g = o(Iuln) as Jul -, 0, 2.
Then there exists a nontrivial solution to Equation (3.1).
Proof. By Theorem 3.7 there exists a bounded continuum of solutions connecting (-Ao, 0) with (Ao, 0). By Lemma 3.6, this continuum must cross the A = 0 axis nontrivially. See Figure 1.
Global Bifurcation for Monge-Ampere Operators
307
Figure 1. Bifurcation Continuum for (3.3) As an application, we obtain a nontrivial solution to the equation J det D2u = IbuIP,
u=0,
X E 11,
XEi,
for all 5 E R, p > n. Theorems of this type (for b = 1, 1 a ball) were first considered in [16, 17, 31, where radially symmetric solutions were established via variational methods. One can also consider lower order perturbation terms g that behave like 1uIP with p < n. In this case results for global asymptotic bifurcation apply, and one can establish similar theorems. For example, the following theorem follows from these considerations:
Theorem 3.9. ([8]) Let 11 C R" be bounded and strictly convex. Then the Dirichlet problem
Jdet D2u = IbujP,
X E 11,
lu=0,
xE09n,
(3.10)
admits a negative solution for all p > 0, p ¢ n and b E R. This theorem includes a result due to Oliker [12] in the case 6 = "li and p = 1 for S2 a bounded, strictly convex domain in R", n > 2. In [12], Oliker is concerned with evolution problems for nonparametric surfaces with speed depending on the Gaussian curvature. In looking for self-similar solutions, the author is led to the static Dirichlet problem above with the given value for J. This solution is then used to analyze the asymptotic behavior for the evolution problem.
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J. Jacobsen
Note that in contrast to the analogous problem for the Laplacian, here we have no critical exponent for (3.10). More explicitly, it was first established by Pohozaev [13] that the Dirichlet problem
1-Du = up, x E Sl, u>0, xESt, u=0, xEOf
(3.11)
has no solution if p > and H C III", n > 3, is bounded and starshaped. Finally, let us remark that the Laplace and the Monge-Ampere operator
are the first and the last k-Hessian operators, respectively. Thus it is of interest to consider the question, for which values of k does the k-Hessian operator have a critical exponent? The reader is referred to [17, 8, 4] for more information.
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[7] P. Guan and X. J. Wang, On a Monge-Ampere equation arising in geometric optics, J. Differential Geom. 48 (1998), 205-223. [8] J. Jacobsen, On Bifurcation Problems Associated with Monge-Ampere Operators, Ph.D. thesis, University of Utah, 1999.
[9] P. L. Lions, Two Remarks on Monge-Ampere equations, Ann. Mat. Pure Appl. 142 (1985), 263-275.
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[10] C. Loewner and L. Nirenberg, Partial differential equations invariant under conformal or projective transformations. In: Contributions to Analysis (A Collection of Papers Dedicated to Lipman Bers), Academic Press, New York, 1974, 245-274. [11] L. Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math. VI (1953), 337-394.
[12] V. Oliker, Evolution of nonparametric surfaces with speed depending on curvature, I. The Gauss curvature case, Indiana Univ. Math J. 40 (1991), 237-257. [13] S. L. Pohozaev, Eigenfunctions for the equation Du+af (u) = 0, Soviet Math. Dokl. 6 (1965), 1408-1411.
[14] K. Schmitt, Analysis Methods for the Study of Nonlinear Equations, University of Utah, 1995. [15] N. S. Trudinger, Weak solutions of Hessian equations, Comm. Partial Differential Equations 22 (1997), 1251-1261.
[16] K. Tso, On a real Monge-Ampere functional, Invent. Math. 101 (1990), 425-448.
[17] K. Tso, Remarks on critical exponents for Hessian operators, Ann. Inst. Henri Poincard 7 (1990), 113-122. [18] J. Urbas, Complete noncompact self-similar solutions of Gauss curvature flows I. Positive powers, Math. Ann. 311 (1998), 251-274.
Jon Jacobsen
Department of Mathematics 233 JWB, University of Utah SLC UT 84112 U.S.A. jacobsenOmath.utah.edu
Remarks on Boundedness of Semilinear Oscillators M. Kunze 1
Statement and proof of the results
There has been much interest during recent years in boundedness questions linked to forced oscillator equations of the type i + x + f(x) = p(t).
(1.1)
Here the case we are interested in is a bounded nonlinearity f : R -' R for which the limits f (oo) = 1im= . f (x) and f (-oo) = limZ, f (x) both exist. Moreover, p is a 27r-periodic forcing, and we ask for sharp conditions which allow us to decide whether all solutions of (1.1) are bounded in the (x, x)-phase plane. First we are going to explain what kind of condition is to be expected, without discussing the regularity of f and p which is going to play an important role later. From an old paper of Lazer and Leech [2] it is known that if additionally f (-oo) < f (x) < f (oo) on R, then 21r
f p(t)e.e dtI < 2[f (00) - f (-oo)]
(1.2)
0
is a necessary and sufficient condition for periodic solutions to exist. Thus if (1.2) is violated, then there is no bounded solution of (1.1), since otherwise there must as well exist a periodic solution, by Massera's theorem, see [5].
Hence the natural question arises whether (1.2) is also sufficient for all solutions of (1.1) to be bounded. This found a positive answer in the paper [4] of B. Liu, and the purpose of these notes is to explain this proof which does require only (1.2), but no smallness assumption on p as was needed before. The proof will also be simplified considerably using insights of R. Ortega that he explained during his lectures at the autumn school; see [7], also for many more references on the whole subject which are not repeated here. So the credits for the ideas in the present paper are exclusively due to both these persons. Nevertheless we had the feeling that writing a clear and short proof that makes use of the full technical machinery available at the moment could be useful, especially to non-specialists in this field.
312
M. Kunze
At the end, boundedness of all solutions will be obtained by application of a suitable version of Moser's twist theorem. This theorem particularly will guarantee the existence of a sequence of invariant curves I'n for the Poincare-map (time 27r return map), with "I,, - oo", i.e., every fixed ball in the phase plane is encircled by r'n for n large. Thus the time evolution up to time 27r of all trajectories starting inside and on rn do form arbitrarily large invariant cylinders in (x, x, t)-geometry (periodically repeated in tdirection), thus confining the solutions to stay inside. See the nice figure in [7].
Before we can apply the twist theorem we have to go through a series of changes of variables. First, introducing the primitive F(x) = fo f (x) dx of f , (1.1) is the Hamiltonian system
dx8Ho _ dy _ with
8Ho 8x '
Oy ' dt dt Ho (x, y, t) = (x2 + y2) + F(x) - xp(t) .
z
It is more convenient to work with polar coordinates in the plane, which we take as x = r1/2 cos 0 and y = r1/2 sin 9 for r > O and 9 E T1 (the unit circle). We obtain dr
9h
dO
8h
dt=a9' dt= - Or' with h(r, O, t) = 2Ho(x, y, t) = r + 2F(r1/2 cos 9) - 2r1/2 cos 9p(t) ; (1.4)
the factor 2 is necessary to get the new system Hamiltonian, since (r, 9) i(x, y) has Jakobian determinant 2 The main new ingredient in the proof of Liu was the performance of an averaging over the function f (resp. F), but for this purpose, the variable
9 in (1.4) has to be interpreted as time, and time t takes the role of the new angle variable. This trick was also used in [3] and it is useful when f is not smooth but only continuous in x, e.g., f (x) = sgn(x). The change
of variables 9 - t and t
9 works more precisely as follows. Since f is bounded, F(x) is linearly bounded. Hence for large and given h > 0 and fixed t, 9 E T1, the equation r + 2F(r1/2 cos 9) - 2r1/2 cos 9 p(t) = h
(1.5)
Cn+l,m,n+1 , has a unique solution r = r(h, t, 9), and this solution is of class if f E Cn and p E Cm. Moreover, r -+ oo as h - oo uniformly in t, 9 E T and vice versa. Thus we arrive at the system
dh_Or dt__Or
with Hamiltonian r(h, t, 9)
.
(1.6)
ah , at ' d9 Now we can pass to a system averaged in the new time 0 E V. To d9
do this, we introduce the notation I (r, O) = 2F(r1/2 cos 9) and J(r) =
Remarks on Boundedness of Semilinear Oscillators
313
fo' F(r1/2 cos 9)d9. In principle we would like just to replace I(r, 9) by
J(r) in the Hamiltonian e.g., in (1.4), but to again obtain a Hamiltonian system; also the radius and angle variables (h, t) have to be transformed in the right way to some (p, r). This works by means of a so-called generating function S; see [1]. In general, if we let
p=h-
, r=t+ aPs
and
H(p,r,9)=r(h,t,9)-
F9
at for "radius" p and "angle" r is again Hamiltonian. then the new system Taking S(p, 9) = fo [J(p) - I (p, 0)] d¢ independent of 'r, we hence have p = h and arrive at
dpOH drOH To = 79
Fp ,
8r ,
with
H(p, r, 9) = r(h, t, 9) + I (p, 9) - J(p) . (1.7)
Note that here t = t(p, r, 9) = r - fa [J'(p) - 37 (p, 0)] dO depends on all three variables (p, r, 9).
Up to now almost nothing has happened, but we are already close to applying the twist theorem. Since we only have to find invariant curves with large "radius" p, we scale p = 3-2u-1, where u belongs to some fixed interval [1/a, a] (with a > 0 to be determined later) and b - 0. This finally yields in new variables (u, r, 9) the system du = -d2u28H (a-2u-1, r, 0),
dr
d9
d9
ar
= - 8H (a-2u-1, r, 0).
(1.8)
49P
A reader familiar with those boundedness results might wonder why we did not care about estimates on remainders up to this point. This is due to an idea of Ortega mentioned at the beginning that allows us to improve the presentation of the technical part of those arguments. We intend to find invariant curves for the Poincare-map of (1.7) at "p = oo", corresponding to b = 0 for (1.8). Thus we consider (1.8) as a differential equation in z = (u, r) and time 9 E T1 that depends on some parameter d, and we will derive the Taylor expansion in b of the solution up to order 1. The underlying abstract result about solutions to differential equations depending on some parameter S E [0, d.] is
Lemma 1.1. Let F : [0, 9.] x D x [0, a.] - RN be of class Co,M+1,M+1, for some open connected D C RN. Assume there exists a compact K C D such that for every initial value zo E K the solution z(9; zo, b) of z' = .F'(9, z, a), z(0) = zo, exists for all 9 E [0, 0.]. Then
z(9;zo,S)=z(9;zo,0)+aas(9;zo,0)+SR(9;zo,a), sup eE (0,9. 1
-40, d-*o+.
314
M. Kunze
Proof. Easily obtained with a Taylor expansion integral remainder term; see [7, Prop. 11], also for further notes. Thus for (1.8) we take 9. = 27r, N = 2, and .F(9, (u, r), b)
62u2 aH
(b-2,u-1 ,
T, e), _ aH (b-2u-1 , T, 9)1 P
Also let D = [1/a, a] x [0, 27r] and K = [2/a, a - 1] x [0, 27r]. Then by the arguments to follow, solutions starting in K remain in D up to time 9 = 2ir if b is sufficiently small, since the limiting equation for u from (1.8) as 6 -. 0 is = 0. Note also that F is of class Cn+l,n,m-1 in (9, u, ,r) for
f E C' and P E C'. Because later an estimate for the remainder 9Z in Cb is needed, we thus have to suppose f E C6 and p E C7. Then F is Cr' with respect to 6 E]0, 60), and we have to discuss now the regularity at 8 = 0. This requires two technical lemmas under the additional assumption
xkf(k)(x)-+0
as
IxI -+oo,
k=1,...,6.
(1.9)
Lemma 1.2. If (1.9) holds, then the derivatives of .F with respect to 9 up to order six can be continued to b = 0. Proof. From (1.5) we obtain (1+r-1/2 cos 9 [f (x)-p(t)]) 8r = 2x1/2 cos 9p(t), and therefore from _ WF, ali m
(_ 52u2 .H (b-2u-1 , T, 9))
= 0.
(1.10)
Similarly, (1 + r-1/2 cos 9 [f (x)
- p(r)])
Or
= 1.
Moreover, aI = P-1/2 cos Of (x) = 0(5), 5p
and by Lemma 1.3 below, also J'(p) = 0(p-1/2) = 0(5). Next, using 92, (1.9), YP-T = 0(p-3/2), and by Lemma 1.3, J"(p) = 0(p-3/2). Therefore 8 = 0(p-3/2) = 0(53), and hence _ + ar 8t + 8' - Y(p) yields also
(
P (6-2U-1,7-,9)) = -1.
For the first derivative, we have 09
as
aH
aP
(a-2u-1 T, 9)) = 26 3u' 1
2 19p2
(8-2u-1, T, B),
(1.11)
Remarks on Boundedness of Semilinear Oscillators
315
and
82H aP2
_ 82r
ah2 + 2
82r at
ar a2t 92r (at 12 ahat aP + at aP2 + ate l aP
021
(1.12)
ape - J, (P) .
Differentiating (1.5) twice with respect to h we get an expression for e , 823 p-312 cos 9 [ f (x) - p(t)] - p-1 cost 9 f' (x) turning out to behave as 2 821 for p -+ oo. The other principal parts in (1.12) are P = Z 2p-1 cost Of, (x) 2 p-3/2 cos Of (x) and J"(p) -p 3/2 [f (oo) - f (-oo)], cf. Lemma 1.3. Thus
-
ah2
p3/2 C! [f(00) - f(-oo)] - cos 9 p(t))
+ ape - J"(P)
.
Concerning the other terms in (1.12), with arguments analogous to the
above, it may be seen that they are O(p-2) = 0(54). E.g., p-1/2cos9p(t), hence MN
= conclude from p-3/2 = 53u3/2 that
ai
O(p-1/2p 3/2)
= O(p-2). Therefore we
ap (d-2u-1, r, o)) = u1/2 (!OO )
ab (
ah
f (-oo)] - cos O P(T)) . (1.13)
Next,
8 \-
= 2u2 -8
b2u2
a
(6-2u-1, ' 0)/
+6-1,u-1 92H (6-2u-1,T,9)
(b-2u-1, T, 0)
.
Then OH
Or
z22 aT
and
apar
=at
2p1/2 cos 9P(t)
- hat + atr a = P
1/2
cos 9 p(t) + O(52).
This yields as all i!
( -b2u2
(b-2u-1, T, 9)) _ -2u3/2 cos 9 p(T) .
(1.14)
The proof that the higher derivatives have a limit as 5 -- 0+ is carried out in the same way, although the calculations are rather tedious (but elementary).
0
M. Kunze
316
Lemma 1.3. Under the above assumption (1.9) we have that, for J(p) _
fo' F(pi/2 cos 9)d9, x _1
k
P-00 u
k = o, ... , 7 .
Jiki (P) _ Z [f (oo) - f (-oo)) ,
( dpkP1/2 J
2,r
P3/2J"(P)
2P1/2J1(P) + _ _
1/2
fo
f'(p1/2 cos 0) cos2 0 d8.
As a consequence of xf'(x) -+ 0 as IxI - co, we infer from the dominated convergence theorem that the second term vanishes as p - oo. Hence for k = 2 it is sufficient to prove the claim for k = 1, and in a similar way the higher derivatives are handled. Concerning k = 1, observe that again by the dominated convergence theorem 1
Pl/2J,(P)
= 27r (
\
pa/2
+ r3x/2 +
a/2
- _ JO
f (Pl/2 cos 8) cos 8 d8
'r/2 J
Ja/2
J0
f2A
2n
3,r/2
f (oo) cos 0 d8 +
J
f (-oo) cos 9 d9 + J
f (oo) cos 0
3ir/2
[f(oo) - f(-oo)1
_
Finally, the claim for k = 0 follows from de L'Hospital's rule.
0
After this technical intermezzo we can return to the application of Lemma
1.1 to (1.8) and derive the expansion of the solutions of (1.8) in J. With z = (u, T) in the notation of Lemma 1.1 we conclude for an initial value zo = (uo, To) from (1.10) and (1.11) that
u(O;zo,0)=uo,
T(9;zo,0)=ro-0.
Moreover, differentiating the right-hand sides of (1.8) with respect to 5 and setting b = 0 yields by means of (1.14) and (1.13) that e
T&
(8; zo, 0) = -2u0/2 f cos 0p(TO - 0) dp,
87-
(e; ZO, 0) = u 88
0 e
/2
(_[f(oo)
- f (-00)1 - cos o p(ro - o)J dO.
0
Then application of Lemma 1.1 and also changing 0 to -9 implies for
Remarks on Boundedness of Semilinear Oscillators
317
the solution the expansion
e(![f(oo) - f (-oo)] - cos 0 p(TO + O)) d1
T (9; Z0, 6) = 7-0 + 9 - 5uo12 I o
+ 5Rl (9; z0i 6)
,
e
u (9; zo, b) = u0 +
f
cos
(TO + 4) d0+ 6R2(9; zo, b) ,
-0 in C5(K) uniformly in 9 as b - 0+- The 6-dependent
with
Poincar6-map of (1.8) is P : (To, uo) '- (7-1, ul) = (T(2ir; zo, 6), u(21r; zo, 6)).
Hence it takes the form P5 :
r1
=To +21r+611(ro,uO)+6R1,
u1 =uo+612(TO,uo)+5A2, (1.15)
with
_
11(7-0, uo)
u'12 (2[f (oo) - f (-oo)] -U01/2'Y(TO)
cos9p(TO + 9) d9 0
/
(1.16)
,
2w
= 2u'2 fcos9z(ro+9)d9,
12(7-0, uo)
and 7.j(ro, uo, b) --+ 0, C5-uniformly in (7-0, uo) E TI x K = T1 x [2/a, a-1] as 6 -+ 0+. Now (1.15) does not have the standard form of a map with small twist, since the angular twist 11 (TO, uo) depends both on "angle" ro and "radius" u0, and not only on uO. There is, however, a version of the twist
theorem by Ortega [6J that allows us to deal with such situations. Recall that the fixed parameter a > 0 is still free to be chosen appropriately, and we are interested in finding invariant curves of P = P5 for all b > 0 small, because this corresponds to p -+ oo. Applied to the above situation, the appropriate twist theorem reads as follows. Lemma 1.4. Suppose that 11(7-0i uo) < 0 and 81 (TO, uO) < 0 for (7-o, uo) E ROL
T1 x K. Moreover, assume 11 E C6(T' x K), 12 E C5(Tl x K), and R; E C5'0(Tl x K x [0, boJ). Let there exist a function V E C6(T' x K) such that (7-0, uo) > 0 and 11(T0, u0)
0
(7-0,U,0) + 12(TO, u0)
(TO, u0) = 0,
(To,uo) E T1 x K. (1.17)
Then, if there are a, b with 2/a < a < b < a - 1 and
V(2/a) < V(a) < V(a) < V(b) < V(b) < V(a - 1),
(1.18)
where V (uo) = max,.0ET1 V(TO, uo) and V (uo) = min.I,ETI V(TO, uo), then
for 6 sufficiently small, P5 has an invariant curve.
318
M. Kunze
Note that P6, coming from a Hamiltonian system, has the intersection property, i.e., for every closed curve r in the cylinder Tl x K that is homotopic to {uo = const} we have P8(r) n r # 0. The assumptions of Lemma 1.4 are now easily verified for (1.15), with 1/2 V(ro,uo) = ro , the main observation being the following, finally using (1.2). We have
J
2x
max
roETI
J0
cos 0 p(ro + 9) d9 =
(1.19)
P(B)eie d9 0
as may be seen writing p(O) = co + Ek 1(c, cos k9 + dk sin k9) as a Fourier series. Indeed, the right-hand side equals r cl + 1, whereas the left-hand side reads 7r maxroET1 [cl cos ro - dl sin rol. The latter expression can be made equal to the former by choosing the vector (cos ro, sin To) E R2 in the direction of (c1, d1) E R2. Hence (1.19) is verified, and thus in particular
minroETI y(ro) > 0 by (1.2); see (1.16). Consequently, ll < 0 and
2,u < 0. Moreover, - = -2T > 0. Then
_
= 2r yields (1.17). It
remains to verify (1.18) for suitably chosen a, a, and b. Since V < V, we have to check by definition of V that (2/a)1'2
miny
<
al/2
maxy
-<-, and al/2
b1/2
min -y
maxy
61/2
mm-Y
<
(a - 1)1/2
maxy
1 R' but this is easy to satisfy, since we can take e.g., b = 1, a = 2(maxry
2, ,
and a sufficiently large. Hence Lemma 1.4 applies to (1.15). Having now found invariant curves of P6 for all d small, we can undo all the coordinate changes and conclude that there are arbitrarily large invariant cylinders in (x, ±, t)-geometry for (1.1). So we have completed the proof of the
Theorem 1.5. Assume p E C7 is 27r-periodic, f E C6, and that (1.9) and (1.2) hold. Then all solutions of (1.1) are bounded.
Acknowledgements: I'm most grateful to the organizers of the autumn school for their generous support. I also want to thank the referee of the paper for pointing out a mistake in an earlier version. REFERENCES [1] V.I. Arnold, Mathematical Methods of Classical Mechanics, 2nd edition, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1989. [2] A.C. Lazer and D.E. Leach, Bounded perturbations of forced harmonic oscillators at resonance, Ann. Math. Pura Appl. 82 (1969), 49-68.
Remarks on Boundedness of Semilinear Oscillators
319
[3] M. Levi, Quasiperiodic motions in superquadratic time-periodic potentials, Comm. Math. Phys. 143 (1991), 43-83. [4] B. Liu, Boundedness in nonlinear oscillations at resonance, J. Dif. Eq. 153 (1999), 142-174.
[5] J.L. Massera, The existence of periodic solutions of systems of differential equations, Duke Math J. 17 (1950), 457-475. [6] R. Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theorem, Proc. London Math. Soc. 79 (1999), 381-413.
[7] R. Ortega, Twist mappings, invariant curves and periodic differential equations, pp. 79-112.
M. Kunze Mathematisches Institut der Universitat Koln Weyertal 86 D-50931 Koln Germany mkunzecmi.uni-Koeln.de
The Dual Variational Method in Nonlocal Semilinear Tricomi Problems Daniela Lupo Kevin R. Payne ABSTRACT We describe the results obtained in [15] and [16) concerning the use of the dual variational approach in order to prove the existence and multiplicity of solutions for a nonlocal variational Tricomi problem.
1
Introduction and setting of the problem
In this work, we provide a survey of the results obtained in [15] and [161 on the use of variational methods to establish the existence and multiplicity of nontrivial solutions, in a suitably generalized sense, to the nonlocal semilinear Tricomi problem
f Tu - -yut2 - uyy = R f (u)
u=0
in 0,
onACUa,
(NST)
where p E R with 0 < p < 1, T = -y82 - 8y is the Tricomi operator on R2, R is the reflection operator on L2(1l) induced by composition with the map : R2 -+ R2 defined by y) = (-x, y), and 11 is a bounded region
in R2 that is symmetric with respect to the y-axis and has a piecewise smooth boundary O of the classical Tricomi form. That is, 80 consists of a smooth arc o in the elliptic region y > 0, with endpoints on the x-axis at A = (-xo, 0) and B = (xo, 0), and two characteristic arcs AC and BC for the Tricomi operator in the hyperbolic region y < 0 issuing from A and B and meeting at the point C on the y-axis. One knows that
AC: (x+xo)-3(-y)312=0 and BC:
(x-xo)+3(-y)312=0.
Finally, f (u) indicates a nonlinear term associated to an f E C°(R) with an asymptotically linear or sublinear growth at infinity and such that f (0) = 0; we note that this last hypothesis implies that u - 0 is always a solution of
(NST).
322
D. Lupo, K.R. Payne
First of all, let us specify in which sense we are looking for generalized solutions. We denote by Cr (St) the set of all smooth functions on St such that u - 0 on r, by Wtl, the closure of Cr (U) with respect to the W1,2(n) norm, and by Wr 1 the dual of Wrl. which can be shown to be the norm closure of L2(0) with respect to the norm w
f(w,W)L21 W, = 0#,Wr SUP hGllWr
(1 1 )
In this way, one obtains rigged triples of Hilbert spaces with inclusion chains such as
WWCL2(Sl)CWr1. The space WACUo is a Hilbert space in which one can find solutions u to Tu = f for all f E L2(St) in a strong sense as is clarified in Section 2.
Definition 1.1. One says that u E Wa' cuo is a generalized solution of (NST) if Tu = Rf (u) in L2(Sl),
(1.2)
and there exists a sequence {uj } C CacuQ (H) such that
lim lim IlTu,, - R (f (u)) IIw-i (1.3) -00 flu, - ufiwlACUO = 0 and j-pp Bcuo = 0. The form of the problem (NST) arises from various considerations. In a general sense, we are interested in the use of variational methods for boundary value problems involving mixed (elliptic-hyperbolic) type partial differential equations. Our motivation is twofold. On the one hand, there are interesting physical problems such as transonic potential flow past profiles which are modeled by nonlinear mixed type boundary value problems which admit variational characterizations (cf. section 4 of [4]). On the other hand, global variational treatments of mixed type problems would improve the
basic understanding as to why solutions to such problems should exist at all. Even for linear problems, the techniques employed often involve pasting together solutions found independently in elliptic and hyperbolic regions; with notable exceptions such as works based upon the positive symmetric systems technique of Friedrichs (cf. [11], [181). Variational tools would provide another approach which is independent of type, with some
added ability to interpret the results. The results presented here can be thought of as a first step towards understanding the obstructions to variational formulations for mixed type problems and proposing a possible method for their resolution; they represent the first variational treatment of a nonlinear Tricomi problem. More precisely, we consider a mixed type problem whose linear part is the most
Nonlocal Semilinear Tricomi Problems
323
well understood, the seminal linear Tricomi problem [24]
fTu= f in n, u=0 onACUa,
(LT)
and proceed to add to it the mildest kind of nonlinear structure, namely a semilinear term. The problem (LT) has long been connected through the pioneering work of Frankl' [10] to the problem of transonic nozzle flow, in which, through the hodograph transformation, the elliptic behavior of the operator in H+ = Il n{(x, y) I y > 0} describes the subsonic part of the flow, while the hyperbolic behavior in 11- = Stn{(x, Y) I y < 0} describes the supersonic flow. The term T icomi problem refers to the placement of the boundary data on only the portion AC U a of the boundary; such a boundary condition is chosen because the presence of a hyperbolic region will overdetermine the
problem for classical solutions if one attempts to place data on the entire boundary (cf. [2], for a maximum principle argument). More precisely, if one considers any portion of the boundary r D AC U a, then the problem of finding classical solutions u E C2(91) nC°(i) to (LT) is overdetermined,
that is
f Tu = 0
u=0
in St,
=u=0 on Q.
on I' On the other hand, under some restrictions on a, there is a wealth of results on the linear Tricomi problem (LT), including the existence of unique strong solutions in Hilbert spaces well adapted to this boundary condition. However, it should be noted that despite some 70 years of study, basic information remains unknown; for example, to our knowledge, the only established spectral result is the existence of one positive eigenvalue for the Tricomi operator supplemented with the Tricomi boundary condition (cf. [12]).
The main difficulty in using variational methods for semilinear Tricomi
problems is a manifest asymmetry in the operator T that results from placing the boundary conditions on only a portion of the boundary. In cleanest terms, T does not map WACuo into its dual, but rather into the dual of the adjoint problem (LT)*, in which vanishing data is placed on BC U a. Our approach involves symmetrizing the linear operator T by first assuming that SZ is symmetric and then by composing T with the reflection R, which induces an isometric isomorphism between the adjoint boundary spaces. In this way, RT does map WACU. into Wacuo, and hence (NST) will admit a variational structure. One could consider a direct variational approach to the symmetrized problem, but this remains problematic since crucial information on the linear Tricomi operator remains unavailable for
the study of the direct functional. The use of dual variational methods allows us to take advantage of the compactness of the inverse of the linear operator. In fact, the linear operator RT does possess a priori estimates
324
D. Lupo, K.R. Payne
with the loss of one derivative and hence it admits an inverse (RT) -1 which is compact on LZ(1l).
Furthermore, given the present knowledge of the linear operator, the necessity of obtaining the continuity of the Nemistkii operator associated to the nonlinearity constrains one to consider nonlinearities with at most an asymptotically linear growth. In order to treat superlinear growth cases, one would need an appropriate LP theory for the linear Tricomi operator, which is not present in the literature. A few additional remarks on the kind of boundary conditions one could consider can be found in [15].
Finally, while we cannot say that the presence of the nonlocal effect in (NST) results directly from physical reasoning for example, there are reasons to believe that the problem is sound. Not only does it possess a variational structure, but it is possible that the corresponding problem without the reflection possesses only the trivial solution. This is, in fact, the case for sublinear increasing nonlinearities that are C', as follows from the uniqueness theorems of [21] (our nonlinearities need not be Lipschitz). These considerations are analogous with the problem of finding nontrivial solutions for a nonlocal semilinear O.D.E. problem whose linear part is the simplest second order ordinary differential operator, which we consider on a bounded interval with homogeneous Cauchy conditions at one endpoint. This non self-adjoint problem can be symmetrized by composition with a reflection operator and the same methods apply in the same way. The analog suggests the robustness of the phenomena described and allows for additional comparative remarks on the nonlocal (with R) versus nonlocal (without R) forms of the problem in terms of uniqueness of the trivial solution.
2
The linear results
In this section, we recall the main results for the linear problem (LT) and its adjoint problem (LT)*. The main tools for treating such problems are the classical (a, b, c)-integral method of Friedrichs and the theory of spaces of positive and negative norms in the sense of Leray and Lax, as developed by Berezanskii [3] and Didenko [8]. Denoting by TAC and TBC the unique continuous extensions of T relative to the dense subspaces CACuo(?!) and Cacuo(!i) respectively, one gets the
continuity estimate IITAcuII
ClllullwA,,,,,,
u E WACu,,
(2.1)
with an analogous result for TBC. In order to obtain solvability results for the problem (LT) and its adjoint
problem (LT)', in which the formal transpose T' of T is again T and the
Nonlocal Semilinear Tricomi Problems
325
boundary conditions are placed on BC U or, one exploits the fact that a priori estimates with the loss of one derivative are often possible to establish for such a mixed type differential operator. We encode this principle into the following definition.
Definition 2.1. A Tricomi domain Q will be called admissible if there exist positive constants C2 and C3 such that u E WA'CUa
IIUIIL2'<
C2IITACUIIW_1
and IIvIIL2
< C3IITBCVIIW_i ACua ,
v E WBCua
(2.3)
Such L2 - WAcua and L2 - WBCua a priori estimates were first obtained by Didenko [8]. Moreover, it is shown that it is possible to choose admissible domains which are symmetric with respect to the y-axis and that the admissibility holds for any piecewise C2 graph a that obeys an explicit bound on IJullci (cf. [15]). Standard functional analysis then gives the following solvability result.
Proposition 2.2.
(cf. [8], [15]) Let S2 be an admissible T^icomi domain.
Then for every f E L2(SZ) there exists a unique strong solution u E WA'CUa to the problem (LT) in the following sense: there exists a sequence {uj } C
Cacua(cl) such that lim Iluj - ullwlACUo = 0 and j00
= 0. lim IlTuj - fllw-1 BCUO
j--.oo
An analogous statement holds for the adjoint problem (LT)*. For an admissible lYicomi domain, one may then define W = {u E WACua : TACU E L2(S2)}
(2.4)
which is a dense subspace of L2(1l) as it contains Co (11). One easily verifies that TAC I w admits a continuous left inverse
TAC : L2(Sl) -- W C Wacua
L2(St),
(2.5)
such that TAC is a compact operator on L2(Sl). It is this compactness property on L2(Q) which suggests the use of a dual variational method for the nonlinear problem. However, for a generic Tricomi domain, an application of the divergence theorem yields the fundamental identity (Tu, v)L2
= f,,(yuZvx + uyvy) dxdy (2.6)
(u,Tv)L2,
U E CACuo(SZ), V E
CBCuot)
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D. Lupo, K.R. Payne
This identity clearly demonstrates a manifest asymmetry that results from the imposition of the boundary conditions on only a portion of the boundary. The closest statements to TAC being symmetric that one can make here are (TACU, V) BC = (u, TBCV)AC,
u E WACuo9 v E WBCUo
(TAC' ,v)L2 = (U,TBCV)L3,
u E WACuo, V E WBCU,,
(2.7)
or
where WJCUQ and WBCUo are W2'2(11) norm closures of CACuo(SI) and CecUo(Sl) respectively (cf. [9] for (2.7) and [3] for (2.8)). In order to circumnavigate this asymmetry, we will consider from now on symmetric admissible 7hicomi domains. On such symmetric domains, we will introduce a reflection operator in the obvious way which effectively symmetrizes the Tricomi operator TAC for use in the variational method.
We consider the linear map ' : R2 --, R2 with '(x, y) = (-x, y), and the induced operator
R : L2(St) -, L2(S2) with Ru = u o
(2.9)
Clearly, R is a norm preserving, self-adjoint automorphism on L2(Q) such that R2 = Id. Moreover, by restriction, R establishes an isometric isomorphism between WACuo and WACuo ; thus one can easily verify the following proposition.
Proposition 2.3. (cf. [15]) Let 11 be a symmetric admissible 7hicomi domain and let R be the reflection operator defined by (2.9). Then the operator gotten by composing TAC with R, RTAC : W C WACUO - L2(S2),
(2.10)
D(TAC) = W = {w E WACuo : TACW E L2(SZ)},
(2.11)
where
satisfies the following properties:
(a) RTAC is a closed, densely defined operator which admits a continuous left inverse (RTAC)-1 : L2(SZ) _ W C WACuo.
(b) (RTAC)-' : L2(l) - L2(Sl) is a compact operator. (c) RTAC is symmetric in the sense that (RTACU, v)L2 _ (u, RTACV)L2,
u, v E W = D(RTAC).
(2.12)
Nonlocal Semilinear Tricomi Problems
327
Since K = (RTAC)-1 : L2(0) -4 L2(f2) is a compact linear operator which is injective, non-surjective, and self-adjoint, one knows that the spectrum of K consists of {0} U {Jib}, where pj is a sequence of eigenvalues, of finite multiplicity, whose only possible accumulation point is zero. We denote by At the positive eigenvalues of K, written in non-increasing order, counting their multiplicities, and similarly we denote by Fib the negative eigenvalues written in non-decreasing order, counting their multiplicity. The operator K is not positive definite on L2(j) and hence may admit both positive and negative eigenvalues. In fact, there are an infinite number of each sign, as one can show using the variational characterization of eigenvalues (cf. [7] and [171).
Proposition 2.4. (cf. [161) If 12 is a symmetric admissible Tricomi domain, then the inverse of the reflected 71^icomi operator K = has an infinite number of positive eigenvalues and an infinite number of (RTAC)-'
negative eigenvalues.
We conclude this section with a few additional observations which follow
from straightforward considerations. We will denote by et, j E N, the L2 (RTAC)-' with L2 norm equal to one that is associated to eigenvectors of the eigenvalues At,, where, in fact, ej E WA C WACU,
Corollary 2.5. If S2 is a symmetric admissible Tricomi domain, then the reflected Tricomi operator RTAC : WA C WACu, I L2 (Q) admits infinitely many positive and negative eigenvalues at = (ju )-1 with associated eigenfunctions {et}BEN where A -- ±oo as j - +00.
Remark 2.6. The eigenfunctions e E WACUo C W','(9) must lie in LQ(S2) for all q E (1, +oo) by the Sobolev imbedding theorem. In addition, since L2(Q) is separable and K is self-adjoint, {e }5EN forms a complete orthonormal basis of L2(Sf). We will denote by a((RTAC)-') the spectrum of (RTAC)-l.
Remark 2.7. In all that follows, S2 will be a symmetric admissible Tricomi domain so that all of the results of Section 2 will apply.
3 A sketch of the dual variational formulation The rough idea behind the use of a dual variational method is the following.
Let f be an invertible function and denote by g = f -' its inverse. Then 3uo E WACu, such that uo 54 0 and TACUO = Rf(uo) in L2(S2)
(3.1)
is equivalent to 3uo E WACu, such that uO 36 0 and RTACUO = f (uo) in L2(1).
(3.2)
D. Lupo, K.R. Payne
328
Hence if we are able to find a vo E L2(St) such that vo 0 0 and g(vo) = Kvo,
(3.3)
where K = (RTAC)-1, then uo = Kvo is a nontrivial solution of our problem. Indeed
f (g(vo)) = f (Kvo) a vo = f (uo) a RTACUO = K-' uo = f (uo) Therefore our goal will be to set up a variational formulation of the dual problem. To this end, we define J : L'(11) --+ R as
J(v) =
Jn
G(v) dxdy - 1
2Jn
vKv dxdy,
where G(v) = fo g(t)dt denotes the primitive of g such that G(0) = 0. Then it is easy to check that if g satisfies the growth condition (cf. [1)) Ig(t)l < Kl + K21t1,
(3.4)
then J E C' (L2 (0), R),
J'(v)[w] = f1 g(v)w dxdy - J wKv dxdy, and hence critical points of J will be weak solutions of the dual problem. Let us remark explicitly that the key point in proving the second statement is the symmetry of K.
4 The asymptotically linear case In this section, we will consider two different situations. First we suppose that the nonlinearity is asymptotically linear at infinity, but sublinear at zero, while in the second case we suppose that the nonlinearity is asymptotically linear (with different slopes) both at zero and at infinity. In both of these cases we will be able to show the existence of a nontrivial solution of (NST), which in the first case will be the preimage of a mountain pass critical point of the dual functional, while the second one will be the preimage of a linking critical point. Let us start with the first case. Let Q be a symmetric admissible Tricomi domain and suppose that the nonlinear term in (NST) satisfies the following set of hypotheses.
(fl) f E CO(R, R) and is strictly increasing
Furthermore, suppose that the nonlinearity f is written as f (s) = s/a + f ,,.(s), where a is a real number such that
Nonlocal Semilinear Dicomi Problems
(f2) aVa((RTAC)-1) and 0
329
0,
where the perturbation f,, E C°(R, R) satisfies
(f3) Iff1 <-c1+c2lsIP,with c1>0,c2>0and 0
04) IWs)I
C31SIp, with C3 > 0.
Our main result is the following theorem.
Theorem 4.1. Let S2 be a symmetric admissible Tricomi domain and as-
sume that f satisfies (f 1), (f 2), (f3) and (f4). Then (NST) admits at least one nontrivial generalized solution.
To begin, we note that by using the hypotheses (fi), i = 1, ... , 4 one can check that the continuous inverse g = f -1 satisfies:
(gl) g E C°(R, R) and is strictly increasing, g(t)t > 0 for t # 0;
(g2) g(t) = at + g. (t); and the perturbation g... E C°(R, R) satisfies (g3) Igao(t)I < C1 +C21tIp, with C1 > 0, C2 > 0,
(g4) Igoo(t)I > C31tl', with C3 > 0.
Now, let us remark that (g2) and (g3) imply that g satisfies Ig(t)I < C1 + C21t1 t E R with C1 > 0, C2 > 0,
(4.1)
and hence J E C1(L2, R).
Denote by Vj = span{ei , ... , e,+} and note that, by Remark 2.8, every v E L2 can be written as v = z + w where z E V, and w E V j; furthermore one has
- f wKwdxdy>-µ+1IIWI12 - J n zKz dxdy > -i
IIzIILz,
- fn zKz dxdy < -,
IIzIft2.
and
Lemma 4.2. Let fl be a symmetric admissible Tricomi domain and assume that g satisfies (gl), (g2) and (g3). Then J satisfies the Palais-Smale condition on L2(1l).
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D. Lupo, K.R. Payne
Proof. (cf. [161, Lemma 4.3). One can show that: a V a(K) implies that any Palais-Smale sequence is bounded;
the continuity of the Nemistkii operator f : L2 - L2 implies that the PS sequence must converge. The following lemma provides a geometrical structure suitable for constructing a mountain pass critical point for the dual functional J.
Lemma 4.3. Let Il be a symmetric admissible 7hcomi domain and assume that g satisfies (gl), (g2), (g3) and (g4). Then there exists a p > 0 and an a > 0 such that J(v) > a for every v E L2 with IIvII = p. Further-
more, J(tei) - -oo for t -+ +oo. Proof. (cf [161, Lemma 4.2) By (f 2), there exists a k such that Pk+1 < a < ilk. Then the decomposition L2 = Vk ® Vl and the inequality (4.2) imply the existence of a positive constant C > 0 such that
J(v)
>C
>C +
f
Jn
IvIp+i dxdy + 2a
Jn
Iz + wln+1 dxdy +
r
v2 dxdy +
a 2u1
k+1 2
- J vKv dxdy
fw2 dxdy
(4.5)
fz2dxdY.
We want to show that there exist p > 0 and a > 0 such that J(v) > a > 0, for every v E L2 such that IIvII = p. We note that in (4.5) the last term is negative, and hence we want to show that, near zero, the positive "subquadratic" term (given by an Lp+1-norm to the power p + 1 with p < 1) dominates the negative "quadratic" term (given by the square of an L2 norm); an argument by contradiction can be given. On the other hand, by (g2) and (g3), one has for t > 0,
J(tei) < C5 + C4tn+1 L Iei Ip+1 < Cb +
C6tp+1
+a
and hence J(tei) -- -oo fort
dxdy +
2t2
jetKet dxdy
tt t2,
+oo since p + 1 < 2 and a - pi < 0.
Proof of Theorem 4.1. If f satisfies (f 1), (f 2), (f 3) and (f4), then g = f -1 satisfies (gl), (g2), (g3) and (g4), thus the functional J will admit, by Lemmas 4.2 and 4.3, a mountain pass critical point vo such that J(vo) > 0 and hence vo # 0 in L2. Such critical points of J are weak solutions to the dual equation and uo = Kvo E Wacu. is a nontrivial generalized solution
to (NST).
Nonlocal Semilinear Tricomi Problems
331
We consider now the case in which the nonlinearity f is asymptotically linear both at zero and at infinity, with two different slopes. More precisely, we suppose that the nonlinear term in (NST) satisfies the following set of
hypotheses: (f 1) and it is possible to write f (s) = s/a+f,,.(s) = s/b+fo(s), where
(f2)' the numbers a and b such that a V a((RTAc)-1) and µj++l < b <
and the perturbations
fo E C°(R,R) satisfy
(f3)' f,,. is bounded and
fo(s)/s = 0,
(f4)' -s2/b < sfo(s) < 0, s E R. We obtain the following Theorem.
Theorem 4.4. Let Sl be a symmetric admissible Thicomi domain and as-
sume that f satisfies (f 1), (f 2)', (f 3)', and (f4)'. Then, if a > µj+, the problem (NST) admits at least one nontrivial solution. The solution is found as the preimage of a linking critical point (cf. [16]) of the dual action functional J. To show that there is a nontrivial linking critical point, one needs J to satisfy an appropriate geometrical situation (cf. Lemma 4.5) and the Palais-Smale compactness condition. We remark that the totality of conditions imposed above will require that a > b, and that the compactness condition is given in Lemma 4.3. However, the linking
structure here (Lemma 4.5) does exploit a > At > b. To begin with, one can check that the hypotheses (f 1), (fi)', i = 2, 3, 4 imply that the inverse g = f -1 satisfies
(gl)' g E C°(R, R), g is invertible g(t) = at + ge(t) = bt + go(t), where a and b are as in (f2)' and the perturbations go and goo belong to C°(R, R) and satisfy
(g2)' g. is bounded and limt..o go(t)/t = 0,
(g3)' tgo(t) > 0, t E R. In this situation, the dual action functional J can be decomposed, depending on the situation, into
J(v) =
Jn
2 Jn v2 dxdy - 12Jn vKv dxdy G,o(v) dxdy + a lv2 dxdy - 1 r vKv dxdy, Go(v) dxdy + b
2
2
n
where Go and G,,. satisfy, for every t E R, IG,,.(t)l < Mjtj and 0 < Go <
2
(a - b)2t2 + MItI.
(4.6)
D. Lupo, K.R. Payne
332
Denoting by 8BP(Vj) = {v E Vj I IIvii = p} and by
8QR(Vl (D Rej)
= {w + ae I a > 0 and U{w E 1 '
I
11u; + ae II = R}
IIwII <_ R},
the following lemma provides a geometrical structure suitable for construct-
ing a linking for the dual functional J.
Lemma 4.5. Let SZ be a symmetric admissible Tricomi domain and as-
sume that g satisfies (gl)', (g2)', and (g3)'. If a > pt, then there exist p > 0 and R > p such that the following inequality holds, sup
J(v) <
8Bn(V,)
J(v).
inf
(4.8)
8Qn(Vjl(DRe, )
Proof. (cf. 116), Lemma 4.5.) The inequality (4.4) applied to arbitrary v E Vj yields
J(v) = <
2 J v2 dxdy + J Go(v) dxdy - 2 n
2
n
(b - p )IIvII2 +
Jn
vKv dxdy
(4.9)
JGO(v)dxdY.
Since it is possible to show that ulimo
IIvII2dxdy fn Go()
= 0,
vE V
we obtain the claim that there exists a p > 0 such that sup J(v) < 0. BB,,(V,)
(4.10)
On the other hand, to estimate the inf from below, we begin by observing
that the inequality (4.2) and the lower bound in (4.6) yield, for arbitrary
WEVil, J(w) ? fo Go(w) dxdy + 2 fn w2 dxdy
- 2i&+i IIwII2 > 2 (b - IA +i)IIwil2, (4.11)
which is nonnegative for IIwII < R. Furthermore, for arbitrary w + aet E Red , the inequalities (4.2), (4.3) together with the upper bound on
Nonlocal Semilinear Tricomi Problems
G,,. from (4.6) yields, setting R2 = 11w + aet 1I2 =
IIw112
333
+ a2, taking into
account that a > µ,+ ? Aj+l,
J(w +
(a - µ )R2 - M1S211/2R.
(4.12)
It is clear that it is possible to2 choose R sufficiently large to make the right-hand side of (4.12) strictly positive, which together with (4.10) and (4.11) completes the lemma. 0
Proof of Theorem 4.4. If f satisfies (f 1), (f 2)', (f3)' and (f4)', then g = f -1 satisfies (gl), (g2)', and (g3)', thus the functional J will admit, by Lemma 4.5, a linking geometrical structure. Furthermore, the (PS) condition holds, and hence there will exist a nontrivial critical point vo 54 0. In fact, the Linking Theorem provides two distinct critical levels (cf. [14]) and thus a nontrivial one; hence uo = Kvo E WACuo is a nontrivial generalized solution to (NST).
5
The sublinear case
We suppose that the nonlinearity f satisfies (f 1) and (f2)" If (s)I < cl + c2IsIp, s E R, for some p E (0, 1), with cl > 0, c2 > 0, (f3)" If (s)I > c3IsIP - c4i s E R for some p E (0, 1), with c3 > 0, c4 > 0.
Remark 5.1. The hypothesis (f 2)" implies that f induces a continuous Nemitski operator from LP+1(0) into L(P+')/P(Sz) for p E (0,1).
In this situation the dual action functional will be J : L(P+1)/1'(fl) - R defined by
J(v) = fn G(v) dxdy - 1
J
2 n
vKv dxdy,
(5.1)
which is C1 and whose critical points are weak solutions to the dual equation (3.3). In this case, the operator K is given by K = j o (RTAC)-1 o i: L(v+1)"(S1) --, Lp+1(0),
where i and j are the inclusion maps
L(P+1)/r(l) `, L2(Q)
Lp+1(&))
which are well defined and continuous since p + 1 < 2 < (p + 1) /p and St is bounded.
The hypotheses on the nonlinearity f imply that the inverse function g satisfies: (gl),
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D. Lupo, K.R. Payne
(g2)" Ig(t)I < C1 +C2ItI1/P, t E R with C1 > 0, C2 > 0,
(g3)" Ig(t)I
C3ItI1/11
- C4, t E R with C3 > 0, C4 > 0.
Utilizing (gl), (g2)" and (g3)", it is easy to see that J is still weakly lower semicontinuous and coercive; thus, by a classical argument (cf. [231),
J is bounded from below and attains its minimum. The existence of a nontrivial generalized solution of (NST) can be shown since it is possible
to construct a function v' such that J(v*) < 0 and hence the minimum must be nontrivial.
Theorem 5.2. Let f2 be a symmetric admissible Thcomi domain and assume that f satisfies (f 1), (f2)" and (f3)". Then (NST) admits a generalized nontrivial solution WACUo
On the other hand, if one also utilizes the eigenvalue properties of Section
2 together with a nonlinearity f that satisfies (in addition to (f 1), (f 2)", (f 3)") the hypothesis
(f4)" f is odd, f E C1(R \ {0}) and lim,.o f'(s) = +oo, a much stronger result is true.
Theorem 5.3. Let fI be a symmetric admissible Tricomi domain and assume that f satisfies (f 1), (f 2)", (f3)" and (f4)". Then (NST) admits infinitely many generalized solutions in WACU.
Hypothesis (f4)" implies that the inverse g will also satisfy
(g4)" g is odd, g E C'(R) and g'(0) = 0. In this case, one has J E C2(L(P+1)/P(fl)(cZ),R), J" (0) [w1 J (w21 = - fn w1 Kw2 dxdy i.e.,
J(v) =
J(0)) + J'(o)[v] + J"(o)[v][vl + o(IIvII(P+1)/P))
- J vKv dxdy + o(IIvII(P+1)/P)). n
(5.2)
Example 5.4. The function f (s) = sign(s)IsIP with p E (0, 1) satisfies (f1), (f2) (f3)" and (f4)". It is clear that the growth condition will imply that the functional J will still satisfy Lemma 4.2 on L(P+1)/P(9)
Proof of Theorem 5.3. J is an even functional, and thus we may apply Theorem 8 of [61 to get our result. More precisely, denote by S(L(P+1)1P(SI))
Nonlocal Semilinear Tricomi Problems
335
the set of A C L(p+1)/p(cl) \ {0} which are symmetric with respect to the origin, and let
r,, = {A I A closed, A E S(L(p+1)1"(0)), 7(A) > m}, where 7(A) denotes the Krasnoselski genus (cf. [23] or [6] for definition and properties), and finally denote by
cm = inf sup J(v). AErm
A
From -oo < c, < 0, it follows that cm is a critical level, and hence it suffices to produce, for every m E N, a set A E rm such that J(v) < 0 for every v E A. To this aim, let us consider V. = span {e+ , ... , e,+n}, where, as in Section (RTAC)-1 associated to the positive eigenval1, et are L2 eigenvectors of _> µ,+n. These eigenvectors belong to L(p+1)/p(cl) by ues pi > µ2 > Remark 2.6. Consider the (m - 1)-dimensional norm spheres Am,e = {v E Vm I IIvII(p+l)/p = e}; it is known that ry(A,n,,) = m. Then, by (5.2) one gets, for every v E A,,,,,,
J(v) < _Cµ+IIvII(p+l)/p + o(IIvII(p+l)/p), where the constant C > 0 comes from the fact that V,n C L(p+1)/p(n) is finite dimensional (all norms being equivalent on finite dimensional spaces). By choosing c small enough, we get the desired result for A = Am,,. Note = Cm+k for some k > 1, we still get infinitely many distinct that if cn =
critical points. In fact, in such a case, denoting by Ku,,, the set of critical points at level cn < 0, one has that 0 0 Kim and furthermore, since J is even, Kc,,, E S(L(p+l)/p(1l)). Hence, properties of the Krasnoselski genus give ry(Kcm) > 2, which provides infinitely many distinct critical points at level c,n.
The infinitely many distinct critical points vj E L(p+1)/p(0) which result give rise to infinitely many nontrivial generalized solutions uu E WACuo where u3 is found as in the proof of Proposition 3.4 of [15]. 0
6 An ODE analog In this section we consider an analogous nonlocal semilinear problem for a second order ordinary differential operator with Cauchy conditions whose manifest asymmetry can be resolved by composition with a reflection operator, which results in a variational structure amenable to the very same dual variational methods we have used for (NST). Moreover, in this simplified setting some additional remarks concerning the role of the reflection operator with respect to existence of nontrivial solutions follow easily.
336
D. Lupo, K.R. Payne
We consider the question of finding nontrivial solutions u = u(x) to the problem u" = R f (u) on I = (-a, a), 1 u(-a) = u'(-a) = 0,
(NSO)
where R is the reflection operator about 0 in I and f will be a continuous strictly increasing nonlinearity with f (0) = 0. Just as in the case of (NST), u - 0 will always be a solution of (NSO). Moreover, the operator T = -d2/dx2 will fail to be symmetric on natural subspaces of L2(I) associated to the boundary conditions; for example, W?a defined as the W2'2(I)-norm closure of C°Q(7) = {u E CO°(I) : u(-a) = u'(-a) = 0}. One can easily prove (cf. [16]) the obvious analogs of Theorems 4.1, 4.2, 5.2, and 5.3 for the problem (NSO) by following the established lines used for the problem (NST). The only differences being that: 1) the solutions have some added a priori regularity (they lie in W?a) that results from the ellipticity of T = -d2/dx2, 2) the spectrum of the linear part RT_a consists only of positive eigenvalues, and 3) the corresponding eigenfunctions also have higher regularity. On the other hand, if one considers the problem (NSO) without the reflection R, that is
u" = f (u) on I = (-a, a), j u(-a) = u'(-a) = 0,
(SO)
one can ask if there are still nontrivial solutions. Here it is important to note
that the problems (NSO) and (SO) are not equivalent, since one cannot
say that Rf = f for some function f; that is, Rf(u(x)) = f(u(-x)) = f (u(x))? For f E Lip(R), one certainly has the uniqueness of the trivial solution u - 0 for (SO), but, for example, f (s) = sign(s) IsIP with p E (0, 1) fails to be Lipschitz near the origin, and hence (SO) may admit the Peano phenomenon of an infinite number of solutions. However, the problem (SO) can be transformed into an initial value problem for a Hamiltonian system for which there is an isolated equilibrium at the origin in the phase plane, and hence one obtains uniqueness of the trivial solution for the unreflected problem (SO) (cf. [16], Prop. 5.1.). Therefore, one can say that the presence of the nonlocal effect in (NSO), as represented by R, not only yields a variational structure but allows for nontrivial solutions as well. For the problem (NST), similar uniqueness considerations may well hold, although the lack of regularity in the nonlinearity f does not allow one to apply known results such as those of [13] and [21] to conclude that the unreflected problem has the unique trivial solution.
Nonlocal Semilinear Tricomi Problems
337
REFERENCES [1] A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, Cambridge University Press, Cambridge, 1993.
[2] S. Agmon, L. Nirenberg and M. H. Protter, A maximum principle for a class of hyperbolic equations and applications to equations of mixed elliptic-hyperbolic type, Comm. Pure Appl. Math. 6 (1953), 455-470.
[3] Y. M. Berezanskii, Expansions in Eigenfunctions of Selfadjoint Operators, Trans. Math. Monographs, Vol. 17, Amer. Math. Soc., Providence, R.I., 1968. [4] H. Berger, G. Warnecke and W. Wendland, Finite elements for tran-
sonic potential flows, Numerical Methods for Partial Differential Equations 6 (1990), 17-42. [5] H. Brezis, Analyse Ftinctionelle, Masson, Paris, 1983.
[6] F. H. Clarke, Periodic solutions of Hamilton's equations and local minima of the dual action, Trans. Amer. Math. Soc. 287 (1985), 239-251.
[7] R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 1, Interscience, New York, 1953. [8] V. P. Didenko, On the generalized solvability of the Tricomi problem, Ukrain. Math. J. 25 (1973), 10-18. [9] V. P. Didenko, A variational problem for equations of mixed type, Differential Equations 13 (1977), 29-32.
[10] F. I. Frankl', On the problems of Chaplygin for mixed sub- and supersonic flows, Isv. Akad. Nauk. USSR Ser. Mat. 9 (1945), 121143.
[11] K. O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math. 11 (1958), 338-418.
[12] N. N. Gadai, Existence of a spectrum for Tricomi's operator, Differential Equations 17 (1981), 20-25. [13] D. K. Gvazava, On uniqueness of solution of the Tricomi problem for a class of nonlinear equations, Soviet Math. Dokl. 11 (1970), 65-69.
[14] D. Lupo and A. M. Micheletti, Multiple solutions for Hamiltonian systems via limit relative category, J. Comp. Appl. Math. 52 (1994), 325-335.
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D. Lupo, K.R. Payne
[15] D. Lupo and K. R. Payne, A dual variational approach to a class of nonlocal semilinear Tricomi problems, NoDEA Nonlinear Differential Equations Appl. 6 (1999), 247-266. [161 D. Lupo, A. M. Micheletti and K. R. Payne, Existence of eigenvalues
for reflected Tricomi operators and applications to multiplicity of sultions for sublinear and asymptotically linear nonlocal Tricomi problems, Advances in Dif. Equations 4 (1999), 391-412.
[17] A. Manes and A. M. Micheletti, Un'estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital. (4) 7 (1973), 285-301. [181 C. S. Morawetz, Non-existence of transonic flow past a profile I, II, Comm. Pure App!. Math. 9 (1956), 45-68; Comm. Pure Appl. Math. 10 (1957), 107-131.
[19] C. S. Morawetz, The Dirichlet problem for the Tricomi equation, Comm. Pure Appl. Math. 23 (1970), 587-601.
[201 K. R. Payne, Interior regularity for the Dirichlet problem for the Tricomi equation, J. Math. Anal. App!. 199 (1996), 271-292. [21] J. M. Rassias, On three new uniqueness theorems of the Tricomi problem for nonlinear mixed type equations. In: Mixed Type Equations, Teubner-Texte Math., Vol. 90, Leipzig, 1986, 269-279.
[221 C. Rebelo, Periodic solutions of nonautonomous planar systems via the Poincare_Birkhoff theorem, Ph.D. Dissertation, Faculade de Ciencias da Universidade de Lisboa, Lisboa, 1996. [23] M. Struwe, Variational Methods, Springer-Verlag, Berlin, 1990. [24] F. G. Tricomi, Sulle equazioni lineari alle derivate parziali di secondo
ordine, di tipo misto, Atti Acad. Naz. Lincei Mem. Cl. Fis. Mat. Nat. (5) 14 (1923), 134-247.
D. Lupo and K.R. Payne Dipartimento di Matematica Politecnico di Milano Piazza Leonardo da Vinci, 32, 20133 Milano Italy danlupOmate.polimi.it,paynekomate.polimi.it
Symmetry Properties of Positive Solutions of Nonlinear Differential Equations Involving the p-Laplace Operator Filomena Pacella 1
Statements
In the last twenty years, much attention has been devoted to the study of symmetry properties of positive solutions of nonlinear partial differential equations whose model problem is
-Du = f (u)
lu=0
in Sl,
onOSl,
where Sl is a domain in RN, N > 2, which is invariant under some symmetry and f is usually taken as a locally Lipschitz continuous function. The main
approach to this question is through the, by now classical, moving plane method of Alexandrov and Serrin which is essentially based on maximum principles ([19]). Using this method, many symmetry results were obtained, starting with
the famous paper [12] by Gidas, Ni and Nirenberg where, among other things, it is proved that, if Sl is a ball, solutions of (1.1) are radially symmetric and strictly radially decreasing. More recently, several mathematicians have started to address the same kind of questions for positive solutions of problems similar to (1.1) but with the Laplace operator replaced by the, so called, p-laplacian. More precisely, we consider the problem
I-apu=f(u)
u>0
in Q, in St,
U=0
on an,
where Apu = div(IDuIp-2Du), p > 1. In this case, the solutions can be considered only in a weak sense since, generally, they belong to the space Cl,'(Sl) ([11], [21]). Anyway, this is not
340
F. Pacella
a real difficulty because the moving plane method as well as other symmetry tools can be adapted to weak solutions of strictly elliptic problems in divergence form ([41, [51).
The main obstacle in dealing with problem (1.2), for p 2, is that the p-laplacian is degenerate in the critical points of the solutions, so that comparison principles (which could substitute the maximum principles in applying the moving plane method when the operator is not linear) are not available in the same form as for p = 2. Actually, counterexamples both to the validity of comparison principles and to the symmetry results are available for any p in correspondence to different degrees of regularity of f ([17], [2]).
Here we will review some of the results recently obtained about problem (1.2).
When S2 is a ball, a first partial result is obtained in [1], where it is proved that the solutions of (1.2) are radially symmetric, assuming that their gradient vanishes only at the origin 0. In this case, the solutions are of class C2 in S2 \ {0} and there the equation is uniformly elliptic; therefore the application of the moving plane method does not present much difficulty.
In [17] it is shown, by a suitable approximation procedure, that isolated solutions with nonzero index, in suitable function spaces, are symmetric. A different approach was used in [18] where, combining symmetrization techniques and the Pohozaev identity (as done in [16]), it is proved that if p = N, Sl is a ball and f is merely continuous, but f (s) > 0 for s > 0, then u is radially symmetric and strictly radially decreasing. Using a new rearrangement technique, called the continuous Steiner symmetrization, Brock ([2] and [3]) has recently obtained the symmetry result
in the ball, without any assumption on the critical set of the solutions, with the regularity of f depending on the exponent p. For other symmetric domains he shows that solutions are "locally symmetric" in a suitable sense.
A first step towards extending the moving plane method to solutions of problems involving the p-laplacian operator has been done in [6]. In this paper, the author proves some interesting weak and strong comparison principles for solutions of differential inequalities involving the p-laplacian. Some of them can be summarized as follows. Let 11 be a domain in RN and assume that u, v E C1(St) weakly solve
-Opu < f (u)
in SZ,
l-upv > f (v)
in Il,
with f : R - R locally Lipschitz continuous. For any set A C SZ we define
MA = MA(u,v) = sup(IDul + IDvI), A
and denote by IAA its Lebesgue measure.
Symmetry Properties for Differential Equations with p-Laplace Operator
341
Theorem 1.1. (Weak comparison principle.) Suppose that 11 is bounded,
1 < p < 2, u, v E C' (S2). Then there exist a, M > 0, depending on p, IS2I, Mn and the LOO norms of u and v such that: if an open set Sl' C 0 satisfies S2' = Al U A2r IA1 n A2I = 0, IA1I < a, MA, < M then u < v on 8S2' implies u < v in 0'.
Theorem 1.2. (Strong comparison principle.) Suppose that 1 < p < 00 and define Zv = {x E S2 : Du(x) = Dv(x) = 01. If u < v in S2 and there exists xo E S2 \ Zv with u(xo) = v(xo), then u = v in the connected component of 0 \ Z,, containing x0. As a consequence of these comparison principles, Damascelli proves in [6] a symmetry result in bounded domains which relies on the assumption that the set of the critical points of u does not disconnect the caps which are constructed by the moving plane method. To be more precise we need some notation. Let v be a direction in RN, i.e., v E RN and IvI = 1. For a real number A and a bounded domain 0, we define
T' ={xERN:x.v=A}, S2a=IX ERN:x.v
x` A
(i.e., Ra is the reflection through the hyperplane T'), and
a(v) = inf x v. xEf2
If A > a(v) then SZa is nonempty, thus we set (SILKY = Ra(QA)
We observe that if S2 is smooth and A > a(v) , with A - a(v) small, then the reflected cap(SZa)' is contained in 11 and will remain in it, at least until one of the following occurs: (i) (fl")' becomes internally tangent to OS2 at some point not on Ta; or (ii) TX' is orthogonal to 8S2 at some point.
Let A1(v) be the set of those A > a(v) such that for each u E (a(v),A) none of the conditions (i) and (ii) holds and define ,\I (v) =supA1(v). Next we define
A2(v) = {A > a(v)
:
(Sµ)' C 11 for any µ E (a(v),,\)}
F. Pacella
342
and if A2(v) # 0, A2(v) = sup A2(v).
We observe that, if S1 is smooth, then 0 j4 A1(v) C A2(V)-
If a(v) < A < A2(V), x E D,uEC'(St)weset uA(x) = u(x\),
Za = Z, \(u) = {x E 1
Du(x), = Du'(x) = 0}
Z = Z(u) = {x E Q: D(u(x) = 0). Finally, we define
Ao(v) = {A E (a(v), A2(v)] : u < uµ in 11' for any µ E (a(v), a]}
and set Ao(v) = sup Ao(v).
Obviously, we have Ao(v) < A2(v).
The symmetry result of [6] comes from this slightly more general theorem whose proof can be found in [6], [8].
Theorem 1.3. Let u E C' (1i) be a weak solution of (1.1), with 1 < p < 2 and SZ bounded. For any direction v such that A2(v) 96 0 we have that Ao(v) # 0 and, if \o(v) < \2(v), then there exists at least one connected component C' of SZIX0(V) \ such that u = uao(,,) in C" Applying the previous theorem to a bounded domain Sl convex and sym-
metric with respect to the hyperplane To = {x E R" : x v = 0}, we get
Corollary 1.4. ([6]) If the set ZX (resp. Z;') does not disconnect any connected component of 11', (resp. 1l '), for any ,\ < 0, then u is symmetric with respect to the hyperplane To and is nondecreasing in the v direction
for Subsequently, in [8] the authors succeed in proving a complete symmetry
result in smooth domains without making any hypothesis on the critical set of the solution. To be more precise they get the following Theorem 1.5. Let !Q be a bounded smooth domain and u E C' (S2) a weak
solution of (1.1) with 1 < p < 2. For any direction v and for A in the interval
we have
u(x) < u(xa)
for any x E Q'.
Symmetry Properties for Differential Equations with p-Laplace Operator
343
Moreover,
a (x) > 0
for any x E
Z,
where Z = {x E 11 1 Du(x) = 0}.
Corollary 1.6. If, for a direction v, the domain Il is symmetric with respect to the hyperplane To = {x E RN I x v = 0} and Al (v) = a, (-v) = 0, then u is symmetric, i.e., u(x) = u(xo) for any x E fl, and decreasing in the v direction in S1o. Moreover > 0 in no \ Z. The result of Theorem 1.5 is based on Theorem 1.3. In fact, this theorem asserts that, once we start the moving plain procedure, we must reach the maximal possible position (i.e., A2(v)) unless the set Z of the critical points of u creates a connected component C of the set where Du 0, which is symmetric with respect to the hyperplane Tao(,,) and where u coincides with the symmetric function ujTherefore, the proof of Theorem 1.5 is based
in showing that such a set C cannot exist. To do that, a new argument is presented in [8], which consists in simultaneously moving hyperplanes orthogonal to directions close to v. In order to be efficient, this procedure of moving nearby hyperplanes needs a certain continuity of the minimal and maximal position of the hyperplanes Ta with respect to v. To ensure this continuity property we assume !Q is smooth. Note also that in a later work [9] we have improved the results of [8] by weakening the regularity of f requiring it to be locally Lipschitz continuous only in (0, +oo). Finally, let us consider the analogous problem in RN, N > 2
1-Apu = f (u)
in RN,
u>0, u(x) --> 0
as fix) -- oo.
In the case p = 2, the radial symmetry of solution of (1.2) was proved first in [13] under certain assumptions on the behaviour of the solutions near infinity and for a class of C' nonlinearities. Later, it was extended by Li ([14]) to fully nonlinear elliptic equations and by Li and Ni [15] to general ground state solutions under the condition f(s) < 0 for s small. In all these papers, the Alexandrov-Serrin moving plane method is used.
Regarding the case p # 2, in [1] it is proved that if f E C' ([0, oo)), f'(0) < 0 and the solution u has only one critical point 0 E RN, then it is radially symmetric about 0, for any p > 1. Recently, Serrin and Zou ([20]), using the moving plane method, got the same result but requiring f to be only locally Lipschitz continuous in (0, +oo) and nonincreasing near zero. Moreover, they consider C' nonnegative solutions of more general quasilinear equations and in particular they study a class of quasilinear elliptic operators having a singularity at the origin which includes as a special case the p-laplacian, 1 < p < 2. In this case they get symmetry results under the weaker assumptions that the set {x E RN I IDu(x)I > 0} is connected. In
344
F. Pacella
[2] and [3], using again the continuous Steiner symmetrization, the author proves some interesting symmetry results in RN, without any assumption on the critical set of the solutions. The same is proved in [7], but using the moving plane method. More precisely we have
Theorem 1.7. Assume that (HI) f is locally Lipschitz continuous in (0, oo),
(H2) there exists xo > 0 such that f is nonincreasing on (0, so).
Then any solution u E C1(RN) fl W',P(RN) of (1.2), for 1 < p < 2, is radially symmetric about some point xo E RN, i.e., u = u(r), with r = Ix - xoI and u'(r) <0 for all r > 0. The proof of this theorem is based on a weak comparison principle for solutions of differential inequalities in unbounded domains which extends Theorem 1.1.
Finally, we would like to remark that in [7] as well as in [2], [3], the solutions are assumed to be in C1(RN)f1WlrP(RN) while in [20] the authors consider nonnegative solutions only in Cl(RN). In a paper in preparation,
([10]), the authors remove the WIM-assumption on the solution and get a symmetry result under some a priori hypotheses on the decay of the solution at infinity (as done also in [2], [3]). REFERENCES
[1] M. Badiale and E.Nabana, A note on radiality of solutions of plaplacian equations, Applicable Anal. 52 (1994), 35-43. [2] F. Brock, Radial symmetry for nonnegative solutions of semilinear elliptic equations involving the p-Laplacian. In: Proceeding of the Conference, Calculus of Variations, Applications and Computations, Ponth-Mousson, 1997.
[3] F. Brock, Continuous rearrangement and symmetry of solutions of elliptic problems, habilitation thesis, Leipzig, 1998.
[4] E.N. Dancer, Some notes on the method of moving planes, Bull. Austral. Math. Soc. 46 (1992), 425-434. [5] L. Damascelli, Some remarks on the method of moving planes, Differential Integral Equations 11:3 (1998), 493-501.
[6] L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic equations, Ann. Inst. H. Poincare 15 (1998), 493-516.
[7] L. Damascelli, F. Pacella and M. Ramaswami, Symmetry of ground states of p-Laplace equations via the moving plane method, Arch. Rat. Mech. 148 (1999), 291-308.
Symmetry Properties for Differential Equations with p-Laplace Operator [8]
[9]
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L. Damascelli and F. Pacella, Monotonicity and symmetry of solutions of p-Laplace equations, 1 < p < 2 via the moving plane method, Ann. Scuola Norm. Sup. Pisa XXVI:4 (1998), 689-707.
L. Damascelli and F. Pacella, Monotonicity and symmetry results for p-Laplace equations and applications, Advances in Differential Equations 5 (2000), 1179-1200.
[10] L. Damascelli and M. Ramaswami, Symmetry of C1-solutions of pLaplace equations in RN, preprint. [11] E. Di Benedetto, Cl+« local regularity of weak solutions of degenerate elliptic equations, Nonlinear Analysis T.M.A. 7 (1983), 827-850. [12] B. Gidas, W.M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209-243.
[13] B. Gidas, W.M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in RN, Math. Anal. Appl., Part A, Advances in Math. Suppl. Studies 7A (1981), 369-403.
[14] C.M. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Comm. Partial Differential Equations 16 (1991), 585-615. [15] Y. Li and W.M. Ni, Radial symmetry of positive solutions of nonlinear elliptic equations in RN, Comm. Partial Differential Equations 18 (1993), 1043-1054.
[16J P.L. Lions, Two geometrical properties of solutions of semilinear problems, Applicable Anal. 12 (1981), 267-272.
[17] M. Grossi, S. Kesavan, F. Pacella and M. Ramaswami, Symmetry of positive solutions of some nonlinear equations, Top. Meth. Nonlinear Analysis 12 (1998), 47-59. [18] S. Kesavan and F. Pacella, Symmetry of positive solutions of a quasilinear elliptic equations via isoperimetric inequalities, Applicable Anal. 54 (1994), 27-37.
[19] J. Serrin, A symmetry problem in potential theory, Arch. Rat. Mech. 43 (1971), 304-318. [20] J. Serrin and H. Zou, Symmetry of ground states of quasilinear elliptic equations, Arch. Rat. Mech. 148 (1999), 265-290.
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[211 P. Tolksdorff, Regularity for a more general class of quasilinear elliptic
equations, J. Dif. Eq. 51 (1984), 121-150.
Filomena Pacella Dipartimento di Matematica Univ. degli Studi di Roma "La Sapienza" Piazzale Aldo Moro, 2 1-00185 Roma Italy pacellacmercurio.mat.uniromal.it
A Maximum Principle with Applications to the Forced Sine-Gordon Equation Aureliano M. Robles-Perez 1
Introduction
In this note I report some work done in collaboration with R. Ortega. The details and a list of references can be seen in [3]. Our objective is to study the forced sine-Gordon equation. In particular we want to study properties of solutions of the problem j utt (t, x) - uxx (t, x) + cut (t, x) + a sin u(t, x) = f (t, x), u doubly periodic (same period in t and x). As we can see, the sine-Gordon equation is a P.D.E. that is very similar to the O.D.E. called the forced pendulum equation
u(t) + cu(t) + a sin u(t) = f (t), u periodic. As we all know, there are many results for the forced pendulum equation. For example in [2] Mawhin obtained results for this equation using the method of upper and lower solutions.
So we try to generalize those results to equation (1.1). In order to do this, it is known that it is necessary to have a maximum principle for the associated linear equation. This is our first objective: we are going to see when a maximum principle holds and some properties of it.
At the end of the note we are going to go back to the sine-Gordon equation to see how the method of upper and lower solutions applies to it.
2
Some examples
The linear equation associated to the sine-Gordon equation is the telegraph equation. It is given by the operator
2au = Utt - uxx + cut - .\u,
348
A.M. Robles-Perez
acting on doubly periodic functions (with the same period in t and x). Loosely speaking, we understand that the operator .Ca has a maximum principle if it satisfies the condition
u>0.
.Cau= f, f >0
Also we can consider this principle as a positivity principle for the inverse operator of ,Ca.
It is interesting to begin with some easy examples in order to have an idea about what to expect. First, what happens when we have no friction, that is, when c = 0? 2.1
No maximum principle for c = 0
We consider the equation
.C,,u=utt-uxx-Au=f. If A = 0, then we integrate over a fundamental region 11 and we have that fn f = 0 is a necessary condition for existence of doubly periodic solutions. But this is impossible if f > 0. So there is no solution if f is positive. If A # 0 we can build a counter-example. Let A < 0 be a fixed constant (if A > 0 then the reasoning is similar). We take u(t, x) = 1- cos t cos x (an eigenfunction of the wave operator). Neither this function nor its image by the operator .C,, changes sign. The minimum of u is equal to zero. So we
try to perturb it with another function w and a small parameter e such that the new function u. = u + ew changes sign but its image £ ,u. does not. In order to do this we can make an easy computation. Well, we have no maximum principle when c = 0. Now what happens when we have friction, that is, if c # 0? 2.2
Towards the maximum principle for c > 0
For c > 0 we are going to consider that u depends only on one variable, x or t. If u = u(x) we have the second order operator
fau=-
Z - Au.
There is a classical maximum principle. Such a principle is
Maximum Principle for lau a -A > 0. If u = u(t) we have the second order operator Lau
_
d 2 u
du
- .1u. dt Now we cannot apply classical tools. Some authors speak of an anti-maximum principle (see for instance [11). In this case dt2
+c
Maximum Principle with Applications to the Forced Sine-Gordon Equation
349
Maximum Principle for Lau 4* 0 < -A < 4 + .1
Moreover if -A > 4 + 4, then we can find f positive such that u changes sign.
With these results in mind, one could think that for the telegraph equation one should have
Maximum Principle for ,Ca 4* 0 < -A < 4 + 4, but this is not true. In fact, we are going to prove that there exists A' such that we have a maximum principle if and only if A* < A < 0. Moreover
a < -A* < 4 + 4 and, in general, -A* # 4 + 4.
3
Concept of solution and regularity
Now we have an approximate idea about what we can get. Therefore we can
start with our study. But first we make an identification of spaces which will be very useful. We work with doubly periodic functions. So, if we know a function over a fundamental domain ([0, 2ir] x [0, 2ir] for example) then we know it over the whole plane by periodicity. We define the torus .11.2 = (R/27rZ) x (R/27rZ). We project R2 onto T2: II
:
R2 (x, y)
'-' =
T2 ,
(-t, !), x + 27rZ,
y+21rZ. In this way if we consider spaces on the torus, then we speak about functions
on the torus or doubly periodic functions on the plane. R2
II
u doubly periodic
I T2
J' U
For example we have LP(T2), C(T2) (continuous functions), Z(T2) _ CI(T2) (test functions), Z(T2) (distributions), etc. We are ready to give rigorous ideas. First the concept of a solution. Consider the telegraph equation with f E L'(T2), P"\ U(t, x) = utt(t, x) - uxx(t, x) + cut(t, x) - Au(t, x) = f (t, x), u doubly periodic (same period in t and x). We say that u E L'(T2) is a solution of (3.1) if and only if it verifies
f u(2"0 - A) =
f
f-0,
d-0 E Z(T2),
(3.1)
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A.M. Robles-Perez
where Cu = utt - uxs + cut and £' is the formal adjoint of Z. Notice that we consider a weak concept of a solution. We can get as a first result the regularity of the solution.
Theorem 3.1. If a is not a real eigenvalue of £, then (3.1) has a unique solution. Moreover
1. if f E L1(T2) then u is continuous;
2. if f E LP(T2) then u is a-Holder continuous with a = 1 - p. We want to comment on this last result. When we work with parabolic equations we gain regularity, but for hyperbolic ones this is false in general. However the doubly periodic solutions of the telegraph equation gain regularity. In some sense we can consider that our problem is similar to parabolic problems. Before going any further, a brief remark about the spectrum of Z. When there is friction, £ is not selfadjoint and it has a complex spectrum. But we are interested only in real eigenvalues. So we are going to speak about the real spectrum of £ as the set of real eigenvalues. In order to calculate it, with Fourier analysis we obtain Anm = m2 - n2 + icn
(m E N, n E Z).
(3.2)
Observe that if c = 0 then £ is selfadjoint. So we have only real eigenvalues. We can explain this fact if we do "c -+ 0" in (3.2): the eigenvalues collapse over the real axis. If A = 0 we have an infinite-dimensional eigenspace and if A # 0 we have finite-dimensional eigenspaces.
4 Maximum principle We use two definitions:
1. £a satisfies the maximum principle if
a¢aR(£), f E P (TI), f > 0
a.e. R2
u(t, x) > 0 `d(t, x) E R2.
2. £a satisfies the strong maximum principle if a
ae(£),
f E Ll (T2), f ? 0 a.e. R2, fT, f > 0 = u(t, x) > 0 d(t, x) E R2. Notice that u is continuous in both cases. So it is defined on the whole plane. We have the following result.
Maximum Principle with Applications to the Forced Sine-Gordon Equation
351
Theorem 4.1. There exists a function v : (0, oo) - (0, CO),
c F-4 v(c)
such that C,\ satisfies the maximum principle if and only if -A E (0, v(c)j.
Moreover the maximum principle is always strong and the function v satisfies 42
< V /(C) G 4
C2
v(c) -
+ 41
as c
/ +00,
where jo is the first positive zero of the Bessel function Jo.
(Observe that jo verifies that 07 < 1) 4
Now we give a sketch of the proof. It is divided into three steps. In the first and second steps we give the tools. In the third step we see the conclusions.
5.1
Step 1: Green's function
Suppose that u is the solution of our problem. Then we have the integral expression
u(t, x) = (G *
f)(t,x)=f G(t-'r,x-l;)f(T,l;)drdd.
That is, u is the convolution of G, the Green's function, and f. Then it is clear that we have characterizations for the maximum principle and for the strong maximum principle, which are given in terms of the positivity of G.
Proposition 5.1. 1. The maximum principle holds if and only if G > 0. 2. The strong maximum principle holds if and only if G > 0 almost everywhere.
So we must determine the sign of G. In order to do this, we calculate an explicit expression of G. If we apply Fourier analysis, then 00
G(t, x) =
1
4rr2
et(nt+mx) 1: m2 n2 \ + icn mm=-oo 1
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A.M. Robles-Perez
This expression is very useful for regularity results but it is not so useful to determine the sign of G. We must take another way. Let us try with fundamental solutions. Consider d = -A - a > 0. Then the function U, given by U ( t , x) _
2e-fit Jo (/d(t2 - x ))
jxj < t,
,
0,
otherwise
is the fundamental solution of Cau = bo in D'(R2) (bo E M(R2) such that < 0, bo >= 0(0, 0), V E 1)(R2)). We are looking for periodic solutions. So we make copies of U, we translate them and we add them. In this way we find the Green's function for our problem, arriving at the expression
G(t, x) =
U(t + 27rn, x + 21rm)
in T2.
(5.1)
(n,m)EZ2
This double sum converges and so G is well defined. Moreover, G is continuous except in the characteristic lines (the family of lines C = {x ± t = 27rN, N E Z}). With the expression (5.1) of G, we can study the sign in a much better way than with the previous Fourier series. There is one special case in which we can improve the situation. Indeed, if d = 0 then Jo(O) = 1 and U and G have easy expressions. In particular,
G(t, x) _
2i 1
ee
a-it,
: a,
if (t X) E Dio, if (t, x) E Dol.
(We let Ds, denote the connected component of V = R2 - C with center at the point (i7r, j7r), where i + j is an odd number). With this explicit expression we easily deduce the following properties.
Proposition 5.2. 1. Jumps in discontinuities are known (and independent of d). 2. G is analytic in TI-0- and UO-1.
3. G is not identically zero in Dlo and Dol. 4. Z = { (t, x) E R2 / G(t, x) = 0} has measure zero. So whenever there is a maximum principle it is strong. All these properties can also be proved for d > 0.
We must point out that we can get G in another way. If we consider A = - a and make the change u = e-ct/2v, then we arrive at the wave equation and we know its explicit solution (D'Alembert formula). Then we
Maximum Principle with Applications to the Forced Sine-Gordon Equation
353
make the inverse change and we have an explicit integral expression of the solution of the telegraph equation. Straightforward computations lead us to G.
5.2
Step 2: Linear positive operators
In this step we are going to review some facts about linear positive operators. Let us consider a Banach space X and a closed cone C in X. Then X is an ordered Banach space with the ordering
x,yEX, x>yax-yEC. Given an operator A over X, we say that A is positive if A(C) C C,
A is strongly positive if A(C - {0}) C C (C 54 0). If A is compact and strongly positive, then we can apply Krein-Rutman's theory and we obtain the following result.
Theorem 5.3. There exists a unique positive Ao which is the spectral radius of A and such that its eigenfunction u is strictly positive.
As a corollary we have a sufficient condition in order to find positive solutions of certain linear equations.
Corollary 5.4. Consider the system
av =Acp+f, f>0, with A in the same conditions as before. If A > A0 = p(A), then there exists a unique solution cp that is positive.
In order to apply all these results we use the notation X = C(T2), C = {u E X : u > 0 in R2} and Aa = .Ca 1 (where A is not a real eigenvalue of C and A is such that the strong maximum principle holds for La). 5.3 1.
Step 3: Conclusions (a) For A _
d=0).
-42,
.Ca has a maximum principle (remember the case
(b) By the theory of linear positive operators, we know that if .Ca. satisfies a maximum principle, then .La satisfies a maximum principle for each A E [a 0). So, taking A 4 , we have a maxi-
mum principle in [- 4 , 0).
A.M. Robles-Perez
354
(c) If A It au(2) and Ga > 6 > 0 (this is equivalent to essinfT, Ga > 0) then there exists Eo such that .Ca+, verifies the strong maximum principle if IeI < co. Since ess inf. Ga > 0 for A = - a , then we have a maximum principle for some A < _. (d) We define v = - inf{A E (-oo, 0) : C,, satisfies the maximum principle}. We can prove that this infimum is a minimum.
Moreover, we know that v depends on c (v = v(c)) and that v(c) < `1. So we have a maximum principle for .Ca when A E [-v, 0). We must remark that ess infr. G = 0 for -v(c). 2. In order to prove the asymptotic results:
(a) When c tends to zero we take a < 0 fixed and u. (t, x) = 1
-
cos t cos x + ew(t, x) (remember that we have used this function before). This function changes sign and f = (u. )tt - (u. )sx - Au.
is positive. If c is small, then g = Cj,u. = f + c(u.)t is positive too. Therefore the maximum principle does not hold for that A. (b) When c tends to infinity we use the estimates I of tG(t, x) -
2
Jo (
d(t2
- x2)) 15 kle-"r
Ief (t+2,,)G(t, x) - 2 {Jo (v"dl(t + 27r)2
if (t, x) E Dio,
- (x - 27r)21)
+ Jo ( d[(t + 2ir)2 - x2j) }I < k2e-07 if (t, x) E Dol.
And with this we finish the "proof". We go on with the last part of this note.
6
Upper and lower solutions
We return to the nonlinear equation .Cu = utt - uxs + cut = F(t, x, u)
in V(T2),
(6.1)
where F : T2 x R - R satisfies Caratheodory conditions. We say that u. is a lower solution if and only if
U. E L°O(T2) and Cu. < F(t, x, u.) in V(T2). We say that u' is an upper solution if and only if it verifies the reversed inequality. Again we are using this concept in a weak sense, but in this case we must take positive test functions. We have a result of classical style about the existence of solutions.
Maximum Principle with Applications to the Forced Sine-Gordon Equation
355
Theorem 6.1. Let u', u. be upper and lower solutions of (6.1) satisfying
u. < u`
a. e. R2.
In addition, assume
F(t, x, u2) - F(t, x, ul) > -v(u2 - u1)
(6.2)
for a.e. (t,x) E R2 and every ul, u2i with
u.(t,x) < ul < u2 < u'(t,x). (The constant v = v(c) was defined by Theorem 4.1). Then (6.1) has a solution u E C(T2) satisfying
u.
1. We want to apply the maximum principle. So we need that the associated linear equation satisfies F(t, x, u) > -v(c). For this we impose the condition (6.2).
2. In this case the upper and lower solutions are weak. So we need a maximum principle for measures.
Lemma 6.2. If A E [-v,0) and C,,u > 0 in V(T2) for u E L'(T2) then u > 0 a. e. in R2.
7
Applications to sine-Gordon equation
At last we will see two results for the sine-Gordon equation, one qualitative and another one quantitative. In the first one, we consider a free parameter s, but the mean value of f is equal to zero,
utt - uxx + cut + a sin u = f (t, x) +s.
(7.1)
The problem is equivalent to (1.1). Remember that f = j+ f with fz, f = 0
and4'fj.,f=f.
Following Mawhin's ideas in his result for the forced pendulum (see [2]) we have a necessary and sufficient condition for existence of solutions.
Theorem 7.1. If Ian < v(c), there exists an I C R nonempty closed interval such that (7.1) has solutions if and only ifs E I.
356
A.M. Robles-Perez
For the quantitative result we forget the parameter s and suppose that
fT,f =0: utt - uyy + cut + a sin u = f (t, x).
(7.2)
We need to solve the auxiliary problem
Lu = f (t, x),
JT7
u = 0 (with f E
L'(T2)).
For this problem we have that there exists a unique solution U E C(T2). Moreover if I I U II oo
< 2 , then u. = U - 2 and u' = U + a are lower and
upper solutions of (7.2).
Theorem 7.2. If IIUIIOO < i and 0 < a < v, then (7.2) has a doubly periodic solution u such that IIu - UIIao 5 i Acknowledgment. I thank R. Ortega for helping me with the English. REFERENCES [1] Ph. Clement and A. Peletier, An anti-maximum principle for secondorder elliptic operators, J. Differential Equations 34 (1979), 218-229. [2]
J. Mawhin, Periodic oscillations of forced pendulum-like equations. In: Lecture Notes in Mathematics, Vol. 964, Springer-Verlag, Berlin, 1982, pp. 458-476.
[3] R. Ortega and A.M. Robles-Perez, A maximum principle for periodic solutions of the telegraph equation, J. Math. Anal. Appl. 221 (1998), 625-651.
Aureliano Robles-Perez Departamento de Matematica Aplicada, Facultad de Ciencias Universidad de Granada Spain aroblesCgoliat.ugr.es
Lipschitzian Regularity Conditions for the Minimizing Trajectories of Optimal Control Problems Andrei V. Sarychev Delfim F. Marado Torres ABSTRACT We survey some conditions for Lipschitzian regularity of minimizers in various problems of the calculus of variations and optimal control theory. Some recent results obtained by the authors are presented as well.
1
Introduction
First optimality conditions and first existence results in the calculus of variations have been separated in time by more than a century. The formalism based on the Euler-Lagrange equation deals with a given (local) minimizer whose existence need not be established. Minimizers which appeared in the classical examples of the variational problems were either smooth or piecewise smooth and the question of validity of the optimality conditions for them has been solved rather easily.
It was only at the end of the nineteenth century that the existence issue for the problems of the calculus of variations has been addressed. In 1915, Leonida Tonelli established the first general existence result for the basic problem of the calculus of variations in the class W1,1 of absolutely continuous functions. It is not difficult to construct an example where a minimum in the class of piecewise-smooth or even Lipschitzian functions is not attained - an absolutely continuous minimizer exists but has an unbounded derivative. Another problem arises here: necessary optimality conditions (different forms of the Euler-Lagrange equation) may cease to be valid for the minimizers with an unbounded derivative. These minimizers
may exhibit other weird properties: for example it may happen that they can not be approximated (by the value of the functional) by a sequence of piecewise smooth or Lipschitzian functions, since the infimum over the class of absolutely continuous functions is strictly less than the one over the class of Lipschitzian functions (Lavrentiev Phenomenon). An approach
358
A.V. Sarychev, D.F. Marado Torres
to overcoming these difficulties could be in finding the conditions (classes of integrands) for which respective minimizers are Lipschitzian. For them, the Euler-Lagrange equation is valid and often (under some additional assumptions) one can establish even more regularity, like piecewise C' or C2 differentiability. Therefore obtaining conditions of Lipschitzian regularity is a pertinent problem. The bibliography regarding Lipschitzian regularity is vast. Due to the restrictions on the volume of the paper we can not provide it here. We refer the readers to the book [1] where further references can be found. First regularity results for the basic problem of the calculus of variations belong to L. Tonelli. Various contributions have been made by C. Morrey and more recently by F. H. Clarke, R. B. Vinter and others. Less has been done for the problem with high-order derivatives, and for the Lagrange problems of optimal control the results are scarce. Here we develop a new approach to establishing Lipschitzian regularity. It is based on a transformation of the initial problem into a time-optimal control problem and applying to the latter the Pontryagin Maximum Principle. This approach allows us to obtain conditions of Lipschitzian regularity for a broad class of Lagrange problems with a control-affine dynamics. When applying these results to the particular case of the basic problems of the calculus of variations or to problems with high-order derivatives, we manage to obtain new conditions which were not previously known. The work on obtaining conditions for general optimal control problems is in progress and the results will appear elsewhere.
2
Problem (P) - Lagrange problem of optimal control with control-affine dynamics
We will be concerned with the Lagrange problem of optimal control with nonlinear control-acne dynamics. We look for an integrable control
and the corresponding absolutely continuous trajectory
satisfying the differential equation (t) = (P (t, x (t) , u (t)) := f (t, x (t)) + g (t, x (t)) u (t)
with boundary conditions x (a) = xa, x (b) = xb,
Minimizing Trajectories of Optimal Control Problems
359
such that they provide minimal value for the integral functional
f
b
L (t, x (t) , u (t)) dt.
This problem is denoted by (P). We assume C1-smoothness of all data of the problem: L : [a, b] x R" x Rm -* R, f :
[a, b] x 1Rn
R", g : la, b] x Rn -- R"'.
Also there are no constraints on the control values: u E R'. Two major issues related to the problem are existence of minimizers and minimality conditions.
3
Existence theorem for (P)
For our problem (P) we have the following existence theorem, which pro-
vides conditions under which the problem has a solution in the class of integrable controls:' If
(coercivity) there exists a function 0 : RI -+ R, bounded from below, o such that lim r-.+oo
9-(r)
r
= +oo, L (t, x, u) > 9 ((lu11) for all (t, x, u)
(convexity) L (t, x, u) is convex with respect to u for every (t, x); Ilf (t, x)II <_ 0 (t) + c Ilxll
,
IIg (t, x)II < i1 (t) + c Ilxll,
V(t,x)E[a,b]xR', 0,0G!0, 0EL1, GEL,,; and provided that there exist at least one admissible pair (x(.), then 0 problem (P) has an absolute minimum in the space E L1. We call this result a Tonelli-type theorem because in the particular case of the basic problem in the calculus of variations, that is when dynamics
is determined by the equation i = u, we may choose 0(t) = 0, c = 0, ,p(t) = 1 and obtain the classical existence theorem proved by Leonida Tonelli in 1915. 1 L. Cesari, Optimization Theory and Applications, Springer, 1983, pp. 392-393
360
4
A.V. Sarychev, D.F. Marado Torres
Necessary optimality condition for (P) Pontryagin Maximum Principle
-
Now we formulate the necessary optimality condition for the problem (P).
This is the celebrated Pontryagin Maximum Principle - a fundamental result of optimal control theory: If (x u is a minimizer of our problem (P) and u E Lam, then 3 (/'o,' 0, where 1P E W1,1 and 00 is a constant less than or equal to zero, such that (x u satisfies: o, 0
the Hamiltonian system
x= 8H 8'
8H
Ip
=- 8x'
with the Hamiltonian
H = 1' L (t, x, u) + (u', f (t, x)) + (0, g (t, x) u), the maximality condition H (t, x (t) , 1'o,
(t) , u (t)) = sup H (t, x (t) , uERm
'
o,
' (t) , u)
for almost all t E [a, b].
0
Of course we should like to have necessary conditions at our disposal to identify minimizers predicted by existence theory. Here we are assuming that is not merely integrable, but a measurable and bounded function.
5
Gap between existence and optimality results
Analyzing the hypotheses of both necessary conditions and existence theorems, we come to the conclusion that there is a gap between `Existence Theory' and `Necessary Optimality Conditions':
For minimizers predicted by existence theory, necessary optimality conditions may fail to be valid. In the class of controls where classical necessary optimality conditions are valid (measurable bounded controls) existence is not guaranteed.
One way to attack this problem is to postulate conditions which assure that all minimizing controls are bounded and therefore standard necessary optimality conditions are applicable to them. We shall call these conditions Lipschitzian regularity conditions. But before making a survey of `Lipschitzian regularity conditions', we will illustrate this mismatch between existence and necessary conditions with an example.
Minimizing 'Trajectories of Optimal Control Problems
6
361
Optimal trajectories can be non-Lipschitzian
The following example was constructed by J. Ball & V. Mizel in 1985: 1
(1x3 - t212
x (t) = u (t)
,
IuI14
+ c Iu12) dt - min,
x(O)=O, x(l)=k,
s > 0.
It is a rather simple exercise to see that all hypotheses of Tonelli's existence theorem are satisfied. Nevertheless, its minimizer is an unbounded function. In fact, it has been proved by F. H. Clarke & R. B. Vinter that for certain choices of constants k and £, this problem has a unique integrable optimal control u (t) = a t-1/3 One can prove that the Pontryagin Maximum Principle (Euler-Lagrange equation in integral form) is not satisfied, since after substituting the minimizer into the right-hand side of the adjoint equation of the Pontryagin Maximum Principle we obtain
(t) = Lx (t, x (t) , i (t)) = Ct-4/3, which results in a divergent integral for calculation of ?P(.). Now we will present some `Lipschitzian regularity conditions' which implies that minimizing controls are bounded. This will also provide validity of the Pontryagin Maximum Principle for these minimizers.
7 7.1
Brief survey of existing results on Lipschitzian regularity Lipschitzian regularity conditions for the basic problem of the calculus of variations
Various conditions of Lipschitzian regularity are known for the basic problem of the calculus of variations
j
b
L (t, x (t) , u (t))
dt -mi,
(t) = u (t) .
They start with a condition obtained by Leonida Tonelli at the beginning of the century. Some recent contributions are due to Francis Clarke and Richard Vinter. Let us list some of these conditions: L. Tonelli - C. B. Morrey: IIL=II + IILuII 5 c ILI + r
(c > 0).
362
A.V. Sarychev, D.F. Marado Torres
S. Bernstein, n = 1; F. H. Clarke & R. B. Vinter, n > 1:
L> -j+a
Ilull'+A,
ILuu (L= - Lut - Lu1 u) 11 <_ c
(a,0>0),
(IIuII2+Q +
1)
,
(Luu > 0)
F. H. Clarke & R. B. Vinter:
* L autonomous: L = L(x, u), * ILtI < c ILI + k (t), (k E L1), (k (.),
* IIL=II
E L1).
R. B. Vinter: L (t, , ) convex for each t. Under each of these conditions the minimizing controls are bounded and satisfy the Pontryagin Maximum Principle.
7.2
Lipschitzian regularity for the problems of the calculus of variations with high-order derivatives
One may present problems of the calculus of variations with high-order derivatives in the form r6
L (t, x (t) , u (t)) dt - min,
± (t) = A x (t) + B u (t),
0
1
0
0
0
0
0
1
0
0
A=
B= 0
0
0 0
0 0
... ...
1
0
0
1
The conditions of Lipschitzian regularity for this type of problems are more scarce. A principal contribution was made by F. H. Clarke and R. B. Vinter (1990). They deduced a condition of the Tonelli-Morrey type lI L= I I < c (L + II ul l) + -y (t) r (x),
y(.) integrable,
r(.) locally bounded.
Optimal controls for this class of problems exhibit some phenomena which do not occur in the basic problem of the calculus of variations. For example, the first of the authors proved (answering an open question posed by F. H. Clarke and R. B. Vinter) that autonomous (= time invariant) integrands may have minimizing controls which are unbounded. In fact it has been shown that a Lavrentiev gap can be present for autonomous problems (see [2])
Minimizing Trajectories of Optimal Control Problems
7.3
363
Lipschitzian regularity of trajectories in optimal control problems
Lipschitzian regularity conditions for general Lagrange problems of optimal
control are even a bigger rarity. We are only aware of a contribution due to F. H. Clarke and R. B. Vinter in 1990. They proved a Tonelli-Morrey type condition for the Lagrange problem of optimal control with linear autonomous dynamics: b
f L (t, x (t) , u (t)) dt - min, a
.t(t) =A x(t) +Bu(t) +d(t). This result is obtained using a transformation of the problem into a problem of the calculus of variations with high-order derivatives and then using Tonelli-Morrey type condition from the previous section.
8
Lipschitzian regularity of trajectories for the problem (P)
Here we address the more general problem (P),
f
b
i (t) = f (t, x (t)) + g (t, x (t)) u (t) and present new results on Lipschitzian regularity of minimizing trajectories for this problem. To deal with it we employ a different method. The result will be formulated as some growth condition imposed on (the derivatives of) L and W, which implies boundedness of any minimizing control and Lipschitzian regularity of the minimizing trajectory
Theorem. Under the hypotheses: g (t, x) has rank m for all t and x; (coercivity) 3 0: Ro --+ R and C E R s. t. for all (t, x, u), L (t, x, u) > 0 (IIuII) > C
,
and
r
lim
o e (r)
0;
(growth condition) 3 constants ry, ,Q, i and µ, with ry > 0, fl < 2 and
µ > max {p - 2, -2}, s.t. for all (t, x, u), (ILtI+IIL=II+IIL(pt- LtVII+IIL(v= - L.''II) IIuII'
<7L,6
+77;
all the minimizers u(.) of the Lagrange problem (P), which are not abnormal extremal controls, are essentially bounded on [a, b]. 0
364
A.V. Sarychev, D.F. Marado Tones
Corollary. Under the hypotheses of the theorem, all the minimizers of the problem satisfy the (normal or abnormal form of) Pontryagin Maximum Principle.
Sketch of the proof Here only the main ideas are represented. Details and complete proofs can be found in [3]. This approach will be summarized in four steps.
8.1
Reduction to a time-optimal problem
First we reduce the problem (P) to an autonomous time-optimal control problem. Given the coercivity condition, Lagrangian L is bounded from
below and since adding a constant to L, in the problem (P), does not change the minimizers, we may assume, without lost of generality, that L is strictly positive. We introduce the new time variable t
T (t) =
L (0, x(9), u(9)) dB,
f
t E [a, b],
which is a strictly monotonous absolutely continuous function of t. Then we consider t(T) (the inverse function of T(t)) and z(T) = x(t(T)) as components of the state trajectories, and v(T) = u(t(T)) as the control. The following time-optimal problem appears:
T - min, t( T )
1
_
L (t(T), z(T), V(T))
V: ,
Z(r) _ V (t(r), z(T), y(T)) L (t(T), z(T), v(T))
R -+ Rm,
t(0) = a, t(T) = b, z(O) = xa, z(T) = xb.
So far, the new control variable continues to take its values in Rm and, a priori, control v(.) can be unbounded. In the next step, compactification of the space of admissible controls will be done.
8.2
Compactification of the control set
Following an idea developed by R. V. Gamkrelidze,2 we proceed with com-
pactification of the set of control parameters. The point is that for each t and z, the set of all velocities becomes compact if the point 0, which corresponds to the infinite value of the control v, will be added. 2See Chapter 8 of R. V. Gamkrelidze's book "Principles of Optimal Control Theory", Plenum Press, 1978.
Minimizing Trajectories of Optimal Control Problems
365
We compactify the space Rm of admissible controls, adding the infinity S". The one-to-one correspondence between the Euclidean point: Rm space RRm and the sphere S' is established by means of the stereographic projection
The set
a:SmR-.
/
E R'"
l
is now compact and the
I ( L (t lz, v)' L (t, z, v)) : v I following extended autonomous optimal control problem is defined properly:
1
T - min
L (t, z, ir(w)) ,
z(
,wE
T) _ p (t, z, a (w))
= a, t(T) = b, S', t(O) z(0) = xa, z(T) = xb.
L (t, z, 7r(w))
It can be proved (see [3]) that this last problem is equivalent to the problem (P), in the following sense: to every admissible pair (x(.), u(.)) of the original problem, there corresponds an admissible triple (t(.), z(.), w(.)) of takes values different from the north the extended system, such that pole (point at infinity) almost everywhere, t(O) = a, t(T) = b, z(O) = xa,
z(T) = Xb, the transfer time T for this latter solution equals the value of the integral functional of the original problem associated with the pair (x(.), Moreover, every solution of the extended/compactified problem, with the control different from the north pole almost everywhere, results from this correspondence. does not To prove that is bounded is the same as to prove that take its values at the north pole. 8.3
Pontryagin Maximum Principle and Lipschitzian regularity
Assume for the moment that the Pontryagin Maximum Principle is applicable to the time-optimal control problem. The validity of this assumption will be discussed in the next step. Writing down the maximality condition for the compactified problem,
and from the fact that the maximized Hamiltonian is constant along a minimizer of the autonomous problem, we conclude that there exists c > 0
such that
0 < c = sup vERt
f (t, z) + g (t, z) v) L (t, z, v)
If v tends to infinity, then, by virtue of the coercivity condition, L has superlinear growth, and hence c must be zero: lim 11v11-+00
'6t + (0Z , f (t, z) + g (t, z) v) = 0 = c.
L (t, z, v)
366
A.V. Sarychev, D.F. Marado Torres
It occurs (see [3, Proposition 11) that this can only be an option if the minimizing control is an abnormal extremal control. Thus, for minimizers which are not abnormal extremal controls, there must be c > 0 and therefore v must be bounded.
8.4
Applicability of the Pontryagin Maximum Principle to the compactified problem
The growth conditions on L and W which guarantee Lipschitzian regularity will arise from the applicability of the Pontryagin Maximum Principle. Given the coercivity condition, the right-hand side of the compactified problem equals zero when w coincides with the north pole. So the right-
hand side is continuous on the entire sphere and to apply the Pontryagin Maximum Principle to the compactified problem, we only need to assure that the right-hand side is continuously differentiable with respect to the state variables t and z. The only problem is the continuous differentiability at the north pole. For it, the fulfillment of the following growth condition is sufficient:
3 y > 0, /3 < 2, p > max {/3 - 2, -2} and 77 E R, such that
(ILtI+IIL=II+IILwt-LtWII+IILw=-L.vII) IIuII" <7Lv+17.
9
Corollary for the basic problem of the calculus of variations
In this section we apply the main theorem to the basic problem of the calculus of variations and to the problem with high-order derivatives. It is straightforward to see that the dynamics corresponding to those problems is controllable and therefore there are no abnormal extremals. For the basic problem of the calculus of variations b
L (t, x, x) dt - min, a
E W1,1,
or in the optimal control notation, b
JL(t,x,u)dt -+ min, X = u,
E L1,
the growth condition takes form: 3 -y > 0, Q < 2, µ > max {/3 - 1, -1 }, such that
Minimizing Trajectories of Optimal Control Problems
367
* (ILtI + IIL=II) Ilull" < 7 L° +17.
(ILtI + IIL=II) II-+II' <- 7 Lp +17
This implies that all the minimizers x(.), predicted by Tonelli's existence theorem, are Lipschitzian:
U() E Lo,
10
X() E Wl,oo.
Corollary for the problem of the calculus of variations with high-order derivatives
The problems of the calculus of variations where the Lagrangian L depends on derivatives up to order m, m > 1, 3 b
L (t, x, x, ... , x(m)) dt -+ min, x() E Wm,1
can be formulated as in Section 7.2, b
Ia
L (t,
,
u) dt -- mi,
,
C=AC+Bu, E W1,1, u() E L1.
The growth condition (7 > 0, Q < 2, µ > max {,Q - 1, -1})
(ILtI+IIL.(i)II) llx(m)II" <7LQ+ 1 .. (ILtI +IILCII) Ilull" <7Lp+n
Vi E{0,...,m-1}
-
implies that all the minimizers x(.) belong to Wm,,,.: U() E L,, = x() E W,,,,oo.
11
Final remarks
The approach we suggest seems to be more effective in dealing with the Lagrange problems of optimal control - it covers a much broader class of problems. But even for the simplest case - the basic problem of the calculus 3We use the notation W,,,,p, m = 1, ... , 1 < p < oo, to represent the class of functions which are absolutely continuous together with their derivatives up to order m - 1 and have m-th derivative belonging to Lp.
368
A.V. Sarychev, D.F. Marado Torres
of variations - these conditions turn out to be new (as far as we know). For example the minimization problem
fl [(±4 + 1)3 e(i4+1) (t+j -arctan x)1 dt -+ min, with boundary conditions x(0) = xo, x(1) = x1, satisfies the hypotheses of Tonelli's existence theorem and the condition of Section 9 allows us to conclude that the minimizer is Lipschitzian while the regularity conditions of Section 7.1 fail.
REFERENCES
[1l F. H. Clarke, Methods of Dynamic and Nonsmooth Optimization, SIAM, Philadelphia, 1989.
[2] A. V. Sarychev, First and Second-Order Integral Functionals of the Calculus of Variations Which Exhibit the Lavrentiev Phenomenon, Journal of Dynamical and Control Systems 3 (1997), 565-588. [3] A. V. Sarychev and D. F. M. Torres, Lipschitzian Regularity of Minimizers for Optimal Control Problems with Control-Affine Dynamics, Applied Mathematics and Optimization 41 (2000), 237-254.
[41 D. F. M. Torres, Regularidade Lipschitziana dos Minimizantes no Calculo das Variacoes e Controlo bptimo, MSc. thesis, Univ. Aveiro, Portugal, 1997.
Andrei V. Sarychev, Delfim F. Marado Torres Departamento de Matematica Universidade de Aveiro 3810 - Aveiro Portugal ansarGmat.ua.pt,delfim@mat.ua.pt
Abstract Concentration Compactness and Elliptic Equations on Unbounded Domains Ian Schindler Kyril Tintarev 1
Abstract statements and proofs
The notion of concentration compactness, used in numerous applications (cf. P.-L.Lions [41,[5]), was formulated originally in terms of specific functional spaces. In fact, much of the method can be formulated in general terms of a non-compact group G of bounded operators on a Banach space E. We say that a bounded sequence Uk converges to zero weakly with concentration, if for any sequence 9k E G, gkuk converges weakly to zero. If G is a compact group, the concentrated weak convergence is equivalent to the weak convergence. We prove, under certain assumptions on the group G, that one can subtract from any bounded sequence Uk countably many terms of the form gkv (where v = w-limgk luk) so that the residual (on a renamed subsequence) will converge weakly with concentration. In applications one has then to interpret the concentrated weak convergence in analytical terms. For example, when E is the Sobolev space H1(RN) and G is the group of shifts, the concentrated weak convergence implies convergence in LP with p E (2,2*) (cf. [3]).
In this paper we consider the case when the space is D1.2(RN) (the closure of Co in the gradient norm) and the group G is the product of lattice shifts and dilations, in which case the concentrated weak convergence implies convergence in L2*, where 2* = NN2. Concentration compactness
combining the "whole space" and the "critical exponent" cases has been studied before (cf. [2], [8] and in particular [6], whose problem, on more general domains, is considered below). Let E be a separable reflexive Banach space and let G be an infinite mulResearch supported by an NFR grant M-AA/MA 10442-309.
370
I. Schindler, K. Tintarev
tiplicative group of bounded linear operators on E, satisfying the following properties: (P1) SUP9EG 11911E < 00-
(P2) 9k E G, u E E, 9ku - 9o(u) 4- go E G.
(P3) u E E \ {0}, gku
w # 0 = 9k has a strongly convergent subse-
quence.
We will say that gk E G is a dissipating sequence if there is a u E E \ {0} such that gku 0. In fact, if gk is a dissipating sequence, then for any v E E, gkv - 0. Indeed, otherwise, (P3) would imply that gk has a subsequence, strongly convergent to some go. Thus, there should exist a non-zero w E E such that gow = 0. However, by (P2), go E G and therefore it cannot vanish on w, a contradiction. We define now the "weak convergence with concentration" relative to the group G.
Definition 1.1. Let u, uk E E. We will say that uk converges to u weakly with concentration which we will denote as cw
Uk ` U,
if for all ,E Es, lira sup(g(uk - u), cp) = 0k 00 9EG
Definition 1.2. Let {qk} be a normalized basis for E'. Then we define the norm 1/2
IIuliG := sup 9EG
I(9u'Ok)I2 k=1
2k
(1.2)
We can now formulate the last assumption.
(P4) Let Uk E E be a bounded sequence and let w(n) E E and gk" E G, k, n E N, be such that the sequence
9ktti9kmi-1 -0 for m # n
(1.3)
9kn)uk - wl"1, n E N.
(1.4)
and
Then Ilw(n)IIG -0 asn -oo.
Abstract Concentration Compactness and Elliptic Equations
371
The heuristic meaning of this condition is to require that the lower bound of "masses" of the weak limit points of the "distinct character" (1.4) will be zero. By the "distinct character" we mean (1.3) which in the case of the shifts amounts to translations with pairwise distances going to infinity, and in the case of dilations corresponds to dilations on incomparable scales. By "masses" we mean the weak G-norm, although in some applications one can verify (P4) with the Hilbert norm as well. Theorem 1.3. Let Uk E E be a bounded sequence. Then there exists W(n) E E, gkn) E G, k, n E N and Nk E N U oo such that for a renumbered subsequence gkn)-lgkm)
0 for n 36m
and
N IIuk
9k(n)W(n) 1iG
-' 0.
n=1
In the proof of Theorem 1.3 we will use the following results.
Lemma 1.4. Let gk E G and gk - go. Then gk 1 E* . If uk uo then 9kuk - 90uo
90 1 and gk
-' go* in
We omit the proof of the lemma as elementary. Lemma 1.5. Let Uk E E be a bounded sequence. If gk and gk are such that
gk`)uk - w(') # 0 i = 1, 2
(1.7)
and 1)-1
9(k2)9(i
f 0,
then there is a renumbered subsequence g(2)9(1)-1 - go E G such that w(2)
= gow(l).
(1.9)
Proof. Relation (1.8) and (P3) imply the existence of the strong limit go. Let W E E*. Then (w(2)
- 9k2)uk, P)
- (9k2)9k1)-1 gk1)uk, cP) (w(2), gyp) - (9kl)uk, (9(2)9k1)-1)'W) (w(2), W)
(1.10) (1.11)
The left-hand side of (1.10) converges to 0 by (1.7). Then by Lemma 1.1, 9k2)9k-1) - goo. Thus using (1.10) we have (W(2), (p)
which yields (1.9).
- (u'(1), go* V) = 0,
(1.12)
372
I. Schindler, K. Tintarev
Proposition 1.6. If uj E E is a bounded sequence, then IIujtic -+ 0 a cw uj -0. Proof. Assume that IIu, - ul is - 0. Then sup9EC I (g(u - uj), Ok)I2 < 2ktlu - ujllc, so that convergence in G -norm implies concentrated weak convergence.
Conversely, for every e > 0 there exists N such that N 2 IIu-ujllc<1:gEC
I(9\u - uj), 4jk)l2
+
21c
(1.13)
so if (1.1) holds, then for all j sufficiently large, Ilu - ujll c < 2e.
O
Proof of Theorem 1.3. If uk - 0, the theorem is verified with w(") = 0 for all n. Otherwise there is a w(1) E E \ {0} and gk1) E G such that on a renumbered subsequence,
9kl)-luk
(1.14)
w(1).
By Lemma 1.4, applying gk1) to (1.14) one has vkl)
= uk - gkl)w(1) -, 0.
(1.15)
If vk1) - 0, the theorem is again verified with w(") = 0, n > 2. If not, there exists gk2) and w(2) # 0 such that, on a renumbered subsequence, (1.16)
gk2) -1 vk1) 7 0.
If gk2)-Igk1) does not dissipate, then on a renumbered subsequence,
-go E G.
(1.17)
9ow(1) + w(2).
(1.18)
9(2)-1gk1)
Then (1.16) and (1.15) imply -1
gk2)
By (1.14),
9(k1)-'
Uk
gk2)-1gk1)
(2)g(k2)-luk
we get w(1). Applying to this gk2)-luk - gow(1). Therefore using (1.18) we obtain w(2) = 0, a contradiction. Consequently, g(k2)-19(k1)
dissipates.
(1.19)
We define recursively (1) w (1) vk(n+1) = uk - gk
(n) w (n) - gk
.
(1.20)
Abstract Concentration Compactness and Elliptic Equations
373
If for some n, vkn+1) G 0, the statement is proved. Otherwise, there is gkn+1), and a w(n+1) # 0 such that, on a subsequence, we have (n+l)vk(n+l)
9k
w
(n)
(1.21)
An argument similar to the one above shows that g(i°)-19(q) dissipates for all p :A q.
(1.22)
Property (P4) implies (n) 11W
Let us estimate
Ilvkl) (n+ IIG. Let M > 0 and gk E G. (gkvkn+l),
j=1
(1.23)
IIG -n-m 0-
7)2
_<
2-M+1
sup gEG
2'
II9II sup Ilvkn+l)II
(gkvkn+l),0j)2
+
(1.24)
>j=1 If
gkvkn+l)
)
-s w (on a renumbered subsequence), the left-hand side of (1.24)
can be estimated by c(n)2-M + IIwiIc + fk(M) with limk,00 ek(M) = 0. This implies that (on a renamed subsequence) limsup (Ivkn+1)IIc < 2llw(n+l)IIG
k-oo
(1.25)
where w(n+1) E E \ {0} satisfies (1.21) with some gkn+1) E G satisfying (1.22). Let k(N), N = 1, 2.... be such that Ilv(n) llc <_ 3IIw(n) IIG for all k > k(N), n < N.
(1.26)
Since IIw(n)IIc - 0 by (P4), (1.27)
which implies (1.6) for a relabled subsequence.
0
We turn now to concretization of the abstract concentration compactness for the case of E = D1.2(RN), the closure of C0(RN) in the norm IIu112 _ fRN IDu12 dx. We define the group G as a direct product of the group of
shifts, u - u( + a), a E RN and the group of dilations
,Otu(x) := tu(tx), t E (0,00).
(1.28)
374
I. Schindler, K. Tintarev
with some a E RN
The elements of G can be represented as u -*
and t > 0. The group clearly satisfies (P1). Note that fl = 01/t = Q 1 which yields (P2). To get (P3) observe that /3t - oo if and only if t -> 00 or t -. 0 and Ot( + a) is bounded if and only if lal and t are bounded. To prove (P4) consider M Vkk
L
:= uk - [1 gk(")w(n)
(1.29)
n=1
with gkn), w(n) as in Theorem 1.3. Taking the square norm of viM), we get M
0
<
M
+ E IIw(n)112 - 2 E(9kn)
IIUkII2
n=1
uk, W(n))
n=1
+21
(9(,n)w("),9(km)w(m))
(1.30)
nom
Note that the last sum converges to zero, since every term in it can be written as (gkw(m), w(n)) with gk - 0. The limit of the third term in (1.30) on a sufficiently rarefied subsequence is zero, so that (1.30) yields M
E IIw(n)II2 < liminf Ilukll2
(1.31)
n=1
for every M, and (P4) follows. We formulate the following concretization of Theorem 1.3:
Theorem 1.7. Let uk be a bounded sequence in E and let Z be an Ndimensional lattice on RN. Then there exist v(1), V(2)'. ., w(1), w(2), E E \ {0} (finitely or countably many) and 1(k1), yk2) E Z, (a(1), tk1)), E RN X (0, 00) such that (a(2), t(2)),
and either or Ia(kP)
tikP) - ,pk9) I - oo
or
tkP) /tk9)
-' 0,
tkP)/tk9) -1 00,
- ak9) I -4 00, P # q,
(1.32)
and for every p, tkP)
-, oo or tkP) -+ 0,
(1.33)
where N-2
w(n)
= w - lim tk")
uk (tkn) 1('
- akn)) ),
v(m) = w - limuk(' - -Y( 1"), IIw(n)II2 +
IIv(m)II2 5 hminf IIuk1I2,
(1.34) (1.35)
(1.36)
Abstract Concentration Compactness and Elliptic Equations
375
N-2
the series E. tkn)
converges in E, and, on a renamed
w(") (t(11) (
subsequence,
_
N-2
/ Etkk)-T- W(n)(tkn)(' +akn))) + E V(m)(' +7k)) - 0.
uk n
(1.37)
m
Moreover, if, additionally, IIukIIL2(RN) < C, C > 0, then none of the
tkn)
will converge to zero.
Proof. If the set of nonzero w(n), v(m) is finite, the theorem easily follows from Theorem 1.3. For the sake of convenience we separate the weak limits v(m) obtained by sequences of gk with bounded tk and bounded 1/tk from
all the rest; without loss of generality we set the parameter tk in these dilation to 1. Also without loss of generality, the shifts 7k can be assumed
to be lattice points: for a general sequence of shifts ak, select 7k E Z so that Iak - 7k1 is bounded by the diameter of the lattice cell. Passing to a convergent subsequence, assume that ak - 7k - /3 and given w = w - lire uk( - ak) set v = w(. + /0). Then w - lim uk( - 7k) = v. The relation (1.36) is indeed (1.31). To prove (1.37) when the set of w(n), v(m) is infinite, it suffices to prove the convergence ofn gkn)w(n) + Em 9k v("') uniformly in k. Indeed, let y(P) stay for both w n) and v(m) and let M < K. Then K
K
9 P)y(P)II2 II
p=M
=
(1.38)
IIy(P)I12 + 2
p=M
M
The second sum in (1.38) can be controlled by choosing appropriate subsequences, namely for every p selecting and renumbering a subsequence of gkP) without changing gkq), q < p, so that I
(gkP)'g(q)y(g)1 y(P))I < 2-A: /P3' for q < p.
(1.39)
Then K
K
1` P==M
y(P))I
<
2-k(p - 1)/p3
(1.40)
P=M
< C2-k/vim.
(1.41)
Thus Ep 1 gkP)y(P) is a Cauchy sequence uniformly in k.
To prove the last assertion of the theorem observe that if tkn) - 0, the L2 norm of the sequence in the right-hand side of (1.23) converges to zero, which implies (using Fatou's Lemma) that IIw(n)IIL2(RN) = 0, a contradiction. 0
376
I. Schindler, K. Tintarev
0, then
Lemma 1.8. Let uk E H'(RN). If IIukIIHa(RN) < C and uk
IIukIILP -0 for23 and forallp>2 when N<2. Proof. For N < 2 and for N >_ 3 with 2 < p < 2`, this is essentially Lemma I.1 from [4]. Consider now the case N > 3 and p = 2` (in this case the statement is also almost immediate from [6]). Let tk -+ oo be such that I'akl2N/(N_2)
N-4
--+ 0.
(1.42)
fukj>tk Let vk be defined by
uk = tvk(tkx).
(1.43)
Then
f
N-2
u hl
IukI2 =
Jv4<1 IV 2 <
RN
Ivk12+*
0,
(1.44)
since IIvkIIc -' 0 and 2 < 2 + A < 2N/(N - 2). We should note that Gconvergence implies Go-convergence, where Go C G is the group of shifts on RN: Proposition 1.6 shows that concentrated weak convergence is independent of a choice of basis and the spaces D1"2 andH' have common bases. This in turn, for a sequence bounded in H1, implies L"-convergence, 2 < p < 2`. Relation (1.42) combined with (1.44) verify the lemma.
Lemma 1.9. Let H = H1(RN) and let Uk, v(m), 'ir , w(">, akn), and tkn> be as in Theorem 1.7. If F : R -+ R is a continuous function and IF(s)I < C(Islq + Isle) with 2 < q < p < 2* whenever N > 3, then
lim f F(uk) +
k-.oo
IukI2*
E fF(v(m)) +
N
Ef
+E f Iv(m)I2*.
1w
(n) I2.
(1.45)
M
Proof. Note that tk w(")(tk")( + a(k"))) - 0 in 1?, 2
uk -
v(m)( + ryk"'1)) - O in L7,
2
(1.46)
M
where
I%(n)
-
oo for n # m. One can easily see, using Lemma 1.8, 'Ykm) that, on an appropriate subsequence,
lim f NF(uk) _
k-.oo
M
f RN F(v(*+)).
(1.47)
Abstract Concentration Compactness and Elliptic Equations
377
A proof that luk I2') -'
E Iw(n) I2' + E Iv(m) I2' n
(1.48)
MEN
is also based on a conclusion due to Lemma 1.8: Uk
-
N-9 Etkn)-7- to(n)(tkn)
+a(kn)))
n
>v(m)(. +ry(m)) -+ 0 in L2' 1
(1.49)
m
with Irykn)--ykm)I -+ 0 and either tkn)/tkm) akm) I
oo or tkm)/t,((n)
- oo or Iakn)
- oo for all m # n, which leads to separation (on an appropriately
rarefied subsequence) of the contributions of the different terms to the value
of the integral. The details are left to the reader.
0
Definition 1.10. Let S1 C RN and let a = {ak}kEN be a sequence on an N-dimensional lattice Z. The following set will be called an asymptotics of S1 at infinity with respect to the sequence a : N --+ Z: en (1.50) S1° f + ak. := U I
I
nEN k>n
In other words, an asymptotic set consists of points x such that for every
j starting with some j (x), the point x - aj lies in Q. In particular, it is easy to see that if B is a bounded set, S21 \ B = S12 \ B and sequence a has no bounded subsequence, then S1i = 1l elementary properties of asymptotic sets:
.
One can easily verify further
Proposition 1.11. The following statements are true: (i) If a sequence a has no bounded subsequence and 11 is bounded, then SZ° = 0.
(ii) If a sequence a has no bounded subsequence and RN \ S1 is bounded,
then f1' = RN (iii) If Q is a subsequence of a, then S11 C SZQ.
(iv) If ao E RN, then S1°+
= S1° + ao.
(v) suprEn dist(x, RN \ Sl) < oo «`da, S1° # RN.
Lemma 1.12. If 11 C RN is an open set, Uk is bounded in H1(RN) and v("'), rykm) are as in Theorem 1.7, then {v(m) # 0} C sets of measure zero. Moreover, if h('Y(m)) then V(m) E Ho (S1).
RN \ (RN \ n7`-)),
(1.51)
378
I. Schindler, K. Tintarev
Definition 1.13. We shall say that an open set SZ C RN is quasiperiodic with respect to an N-dimensional lattice Z, if for every sequence a on Z, there exist a y E RN such that the set fl(a) defined by (1.51) is contained in 11 + y.
Note that any bounded set SZ is quasiperiodic by this definition since its asymptotic sets are intersections of finitely many S2 + y, y E Z (and empty
for any unbounded sequence). It is also easy to show that the following sets are quasiperiodic: a product of a bounded set with Rk, k = 1, ..., N; a set symmetric with respect to the lattice Z; an open set, with boundary of zero measure, satisfying conditions of Theorem 1.1 or Theorem 1.2 in [6]. We can also notice that if SZ is quasiperiodic and B is bounded, then SZ U B is quasiperiodic. Let
A0:=
j IDtI2.
inf f, , u2=1,uECo (SZ)
(1.52)
n
Theorem 1.14. Let 11 C RN, N > 3, be a quasiperiodic open set in the sense of Definition 1.13 with respect to some N-lattice and assume that A0 > 0. If A E (0, A0), then the problem
-Du - Au =
u2.-',
u E HOW),
(1.53)
has a positive solution.
The theorem holds for domains of asymptotically cylindric shape as in [6], but also, for example, for the following union of open unit balls: SZ1 = (UnEZ,k=1,...,NB1(nek)), where vectors ek form a basis in RN. It will also hold for any open set SZ2 D SZ1 such that for every e > 0 there is an R > 0 such that SZ2 \ BR(0) C SZ1 + BE(0).
Proof. Consider the variational problem c(f2, A) =
sup In Ivup-a f,, U2=1J n
.1 > 0.
(1.54)
The constant c(S2, A) is finite only for A < A0 To verify solvability of (1.53)
we will show that the maximum in (1.54) is attained. It is known ([1], Lemma 1.1, cf. also [5]) that c(91, A) > c(BE, A) > c(RN, 0) for all A > 0
(1.55)
where B, is an open ball of radius e > 0 contained in Q. We will use this relation below to exclude maximizing sequences for c(SZ, A) that involve dilations of a maximizer for c(RN, 0). The argument is as follows. Let Uk be a maximizing sequence for c(92, A) which we consider as a sequence of
Abstract Concentration Compactness and Elliptic Equations
379
elements in Hd(RN). We apply Theorem 1.7. Let QA(u) := f (IVu12-Au2). Then we will have, on a renamed subsequence,
>QA(v(m))+>Qo(w(n)) < 1 M
(1.56)
n
and
c(1l, A) = [: f Ivy"`?12,
+
f I(w(n)12*.
(1.57)
If any of W(n) is not a maximizer (up to a constant multiple) for c(RN, 0), then uk is not a maximizing sequence, since subtracting this w(n) from Uk and adding instead a sequence of suitable dilations of the Talenti function, multiplied by a smooth cut-off function supported on 0, will preserve the bound (1.56) while increasing the value off Iuk12' above c(O,.\). A simple subadditivity argument shows that one would also increase the value of f if one had more than one non-zero W(n), which we will denote as w(".) . If all V(-) = 0, then we would have c(0, A) = c(RN, 0), a contradiction to (1.55). A similar subadditivity argument wouldIu12 provide Iuk12.
will that at most one v("`), say, v("`O) will be non-zero, and that f not attain its maximal value unless w(no) = 0 and QA(v(mO)) = 1. Thus uk( - yk"`0)) - v(--) in Ho. Consequently, v('"O) is the maximizer for
0
c(St,
REFERENCES [1] H. Brezis and L. Nirenberg, Positive solutions of an elliptic equation with a nonlinearity involving critical Sobolev exponent, Comm. Pure Appl. Math. 36 (1983), 437-476.
[2] J. Chabrowsky, Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents, Calc. Var. 3 (1995), 493-512. [3] E. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math. 74 (1983), 441-448. [4] P.L. Lions, The concentration-compactness principle in the calculus of
variations. The locally compact case, part 2, Ann. Inst. H. Poincare Analyse Non Lineaire 1 (1984), 223-283. [5] P.L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, part 2, Revista Matematica Iberoamericana 1 (1985), 45-121.
380
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[6] M. Ramos, Z.-Q. Wang, and M. Willem, Positive solutions for elliptic
equations with critical growth in unbounded domains, in: A. Ioffe, S. Reich, I. Shaffrir (eds), Calculus of Variations and Differential Equations, Chapman & Hall/CRC Research Notes in Mathematics 410, 192-199, 2000. [7]
I. Schindler and K. Tintarev, Semilinear elliptic problems on unbounded domains, in: A. Ioffe, S. Reich, I. Shafrir (eds), Calculus of Variations and Differential Equations, Chapman & Hall/CRC Research Notes in Mathematics 410, 210-217, 2000.
[8] M. Willem, Minimax Theorems, Birkhauser, 1996.
Ian Schindler
University of Toulouse 1 ischindlQmath.univ-tlsel.fr
Kyril Tintarev Uppsala University
kyrilOmath.uu.se
Progress in Nonlinear Differential Equations and Their Applications Editor Haim Brezis D6partement de Math6matiques Universitd P. et M. Curie 4, Place Jussieu 75252 Paris Cedex 05 France
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Partial Differential Equations and Mathematical Physics: The Danish-Swedish Analysis Seminar, 1995 Lars Hdrmander and Anders Melin, editors
PNLDE 22
Partial Differential Equations and Functional Analysis: In Memory of Pierre Grisvard J. Cea, D. Chenais, G. Geymonat, and J.-L Lions, editors
PNLDE 23
Geometry of Harmonic Maps Yuanlong Xin
PNLDE 24
Minimax Theorems Michel Willem
PNLDE 25
Variational Methods for Discontinuous Structures Raul Serapioni and Franco Tomarelli
PNLDE 26
Algebraic Aspects of Integrable Systems: In Memory of Irene Dorfman A. S. Fokas and 1. M. Gelfand, editors
PNLDE 27 Topological Nonlinear Analysis II: Degree, Singularity, and Variations Michele Matzeu and Alfonso Vignoli, editors PNLDE 28
Models of Phase Transitions Augusto Visintin
PNLDE 29
Nonlinear Partial Differential Equations in Geometry and Physics: The 1995 Barrett Lectures Garth Baker and Alexandre Freire, editors
PNLDE 30
Parametrized Measures and Variational Principles
Pablo Pedregal PNLDE 31
Topics in Mathematical Modelling of Composite Materials Andrej Cherkaev and Robert Kohn, editors
PNLDE 32 Geometrical Optics and Related Topics Ferruccio Colombini and Nicolas Lerner, editors PNLDE 33
Entire Solutions of Semilinear Elliptic Equations 1. Kuzin and S. Pohozaev
PNLDE 34
Stability of Functional Equations in Several Variables Donald H. Hyers, George /sac and Themistocles M. Rassias
PNLDE 35 Topics in Nonlinear Analysis: The Herbert Amann Anniversary Volume Joachim Escher and Gieri Simonett, editors
PNLDE 36 Variational Problems with Concentration Martin Flucher PNLDE 37
Implicit Partial Differential Equations Bernard Dacorogna and Paolo Marcellini
PNLDE 38 Systems of Conservation Laws: Two-Dimensional Riemann Problems Yuxi Zheng
PNLDE 39
Linear and Nonlinear Aspects of Vortices: The Ginzburg-Landau Model Frank Pacard and Tristan Riviire
PNLDE 40 Recent Trends in Nonlinear Analysis: Festschrift Dedicated to Alfonso Vignoli on the Occasion of his Sixtieth Birthday Jiirgen Appell, editor PNLDE 41
Quasi-hydrodynamic Semiconductor Equations Ansgar Jiingel
PNLDE 42
Semigroups of Operators: Theory and Applications A. V. Balakrishnan, editor
PNLDE 43
Nonlinear Analysis and its Applications to Differential Equations M.R. Grosshino, M. Ramos, C. Rebelo, and L. Sanchez, editors
\onlinea- Anah sis and its :applications to Differential Equations M.R. (,rossinho, \1. Ranlos, C. Rebelo, L. Sanchci, Fditors
This cork. consisting of espositor% articles as sell as research papers, highlights recent desclopnlents in nonlinear analssis and differential equations. The material is largos an outgrossth of autumn school courses and seminars held at file Cnisersits of Lisbon and has been thoroughly refereed. -
Seseral topics in ordinary differential equations and partial differential equations are the focus of keN articles, including: periodic solutions of sNstems ssith p-I.aplacian thpe operators (.1. \las%hin)
bifurcation in sariational inequalities 1K. Schmitt) a geometric approach to ds namical sN stems in the plane s is mist theorems (R. Ortega)
ash mptotic behas for and periodic solutions for \asicr-Stokes equations (E. Feircisl)
mechanics on Riemannian manifolds (N M. Olisa) techniques of loser and upper solutions for ODE% (C. Dc Coster and P. Ilabets) number of related subjects dealing ssith properties of solutions. e.g.. bifurcations. symmetries, nonlinear oscillations, are treated in other articles.
This solume reflects rich and saried fields of research and sill be a useful resource for mathematicians and graduate students in file and I 'DE conununih.
ISBN 0-8176-4188-2
Kirkhdtiscr ISBN 0-8176-4188-2 slsss.hirl.hauscr.com
