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 evalu