Nonsmooth variational problems and their inequalities. Comparison principles and applications

403 Pages·2007·1.574 MB·English

by  Carl S., Le V.K., Motreanu D.

S M M
pringer onographs in athematics
Siegfried Carl
Vy Khoi Le
Dumitru Motreanu
Nonsmooth Variational
Problems and Their
Inequalities
Comparison Principles and
Applications
Siegfried Carl Vy Khoi Le
Institut für Mathematik Department of Mathematics and
Martin-Luther-Universität Statistics
Halle-Wittenberg University of Missouri-Rolla
D-06099 Halle Rolla,MO 65409
Germany U.S.A
[email protected] [email protected]
Dumitru Motreanu
Département de Mathématiques
Université de Perpignan
66860 Perpignan
France
[email protected]
Mathematics Subject Classifications (2000):(Primary) 35B05,35J20,35J85,35K85,35R70,
47J20,47J35,49J52,49J53;(Secondary) 35J60,35K55,35R05,35R45,49J40,58E35
Library ofCongress Control Number:2006933727
ISBN-13:978-0-387-30653-7 e-ISBN-13:978-0-387-46252-3
Printed on acid-free paper.
© 2007 Springer Science+Business Media,LLC
All rights reserved.This work may not be translated or copied in whole or in part without the
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Preface
Nonsmooth variational problems have their origin in the study of nondiffer-
entiable energy functionals, and they arise as necessary conditions of critical
pointsofsuchfunctionals.Inthisway,variationalinequalities arerelatedwith
convex energy or potential functionals, whereas the new class of hemivaria-
tional inequalities arise in the study of nonconvex potential functionals that
are,ingeneral,merelylocallyLipschitz.Thefoundationofvariationalinequal-
ities is from Fichera, Lions, and Stampacchia, and it dates back to the 1960s.
Hemivariational inequalities were first introduced by Panagiotopoulos about
twodecadesagoandarecloselyrelatedwiththedevelopmentofthenewcon-
cept of Clarke’s generalized gradient. By using this new type of inequalities,
Panagiotopoulos was able to solve various open questions in mechanics and
engineering.
This book focuses on nonsmooth variational problems not necessarily re-
lated with some potential or energy functional, which arise, e.g., in the study
of boundary value problems with nonsmooth data and/or nonsmooth con-
straints such as multivalued elliptic problems with multifunctions of Clarke’s
subgradient type, variational inequalities, hemivariational inequalities, and
their corresponding evolutionary counterparts. The main purpose is to pro-
vide a systematic and unified exposition of comparison principles based on
a suitably extended sub-supersolution method. This method manifests as an
effective and flexible technique to obtain existence and comparison results
of solutions. Moreover, it can be employed for the investigation of various
qualitative properties such as location, multiplicity, and extremality of solu-
tions. In the treatment of the problems under consideration, a wide range of
methods and techniques from nonlinear and nonsmooth analysis are applied;
a brief outline of which has been provided in a preliminary chapter to make
thebookself-contained.Thebookisanoutgrowthoftheauthors’researchon
the subject during the past 10 years. A great deal of the material presented
here has been obtained only in recent years and appears for the first time in
book form.
vi Preface
The materials presented in our book are accessible to graduate students
in mathematical and physical sciences, researchers in pure and applied math-
ematics, physics, mechanics, and engineering.
ItisourpleasuretoacknowledgeadebtofgratitudetoDr.VioricaMotre-
anuforhercompetentanddedicatedhelpduringthepreparationofthisbook
at its various stages. Finally, the authors are grateful to the very professional
editorial staff of Springer, particularly to Ana Bozicevic and Vaishali Damle
for their effective and productive collaboration.
Halle Siegfried Carl
Rolla Vy K. Le
Perpignan Dumitru Motreanu
September 2005
Contents
Preface ........................................................ v
1 Introduction............................................... 1
2 Mathematical Preliminaries................................ 11
2.1 Basic Functional Analysis ................................ 11
2.1.1 Operators in Normed Linear Spaces ................. 11
2.1.2 Duality in Banach Spaces .......................... 15
2.1.3 Convex Analysis and Calculus in Banach Spaces ...... 20
2.1.4 Partially Ordered Sets ............................. 27
2.2 Sobolev Spaces.......................................... 28
2.2.1 Spaces of Lebesgue Integrable Functions.............. 28
2.2.2 Definition of Sobolev Spaces ........................ 30
2.2.3 Chain Rule and Lattice Structure ................... 34
2.2.4 Some Inequalities.................................. 36
2.3 Operators of Monotone Type ............................. 39
2.3.1 Main Theorem on Pseudomonotone Operators ........ 39
2.3.2 Leray–Lions Operators............................. 41
2.3.3 Multivalued Pseudomonotone Operators.............. 45
2.4 First-Order Evolution Equations .......................... 49
2.4.1 Motivation ....................................... 50
2.4.2 Vector-Valued Functions ........................... 53
2.4.3 Evolution Triple and Generalized Derivative .......... 55
2.4.4 Existence Results for Evolution Equations ............ 59
2.4.5 Multivalued Evolution Equations.................... 62
2.5 Nonsmooth Analysis ..................................... 63
2.5.1 Clarke’s Generalized Gradient....................... 63
2.5.2 Some Calculus .................................... 68
2.5.3 Critical Point Theory .............................. 73
2.5.4 Linking Theorem.................................. 77
viii Contents
3 Variational Equations...................................... 81
3.1 Semilinear Elliptic Equations ............................. 81
3.1.1 Comparison Principle .............................. 82
3.1.2 Directed and Compact Solution Set.................. 84
3.1.3 Extremal Solutions ................................ 91
3.2 Quasilinear Elliptic Equations............................. 93
3.2.1 Comparison Principle .............................. 94
3.2.2 Directed and Compact Solution Set.................. 97
3.2.3 Extremal Solutions ................................103
3.3 Quasilinear Parabolic Equations...........................105
3.3.1 Parabolic Equation with p-Laplacian.................110
3.3.2 Comparison Principle for Quasilinear Equations .......112
3.3.3 Directed and Compact Solution Set..................116
3.3.4 Extremal Solutions ................................122
3.4 Sign-Changing Solutions via Fuˇcik Spectrum ...............123
3.4.1 Introduction ......................................124
3.4.2 Preliminaries .....................................125
3.4.3 Main Result ......................................130
3.5 Quasilinear Elliptic Problems of Periodic Type ..............134
3.5.1 Problem Setting...................................134
3.5.2 Sub-Supersolutions ................................136
3.5.3 Existence Result ..................................138
3.6 Notes and Comments ....................................141
4 Multivalued Variational Equations.........................143
4.1 Motivation and Introductory Examples.....................143
4.1.1 Motivation .......................................144
4.1.2 Comparison Principle: Subdifferential Case ...........146
4.1.3 Comparison Principle: Clarke’s Gradient Case.........149
4.2 Inclusions with Global Growth on Clarke’s Gradient .........155
4.2.1 Preliminaries .....................................157
4.2.2 Comparison and Compactness Results ...............160
4.3 Inclusions with Local Growth on Clarke’s Gradient ..........167
4.3.1 Comparison Principle ..............................167
4.3.2 Compactness and Extremality Results ...............176
4.4 Application: Difference of Multifunctions ...................180
4.4.1 Hypotheses and Main Result........................181
4.4.2 A Priori Bounds ..................................182
4.4.3 Proof of Theorem 4.36 .............................186
4.5 Parabolic Inclusions with Local Growth ....................190
4.5.1 Comparison Principle ..............................191
4.5.2 Extremality and Compactness Results ...............201
4.6 An Alternative Concept of Sub-Supersolutions ..............208
4.7 Notes and Comments ....................................209
Contents ix
5 Variational Inequalities ....................................211
5.1 Variational Inequalities on Closed Convex Sets ..............213
5.1.1 Solutions and Extremal Solutions above Subsolutions ..214
5.1.2 Comparison Principle and Extremal Solutions.........226
5.2 Variational Inequalities with Convex Functionals ............234
5.2.1 General Settings—Sub- and Supersolutions ...........235
5.2.2 Existence and Comparison Results...................238
5.2.3 Some Examples ...................................242
5.3 Evolutionary Variational Inequalities.......................246
5.3.1 General Settings ..................................247
5.3.2 Comparison Principle ..............................249
5.3.3 Obstacle Problem .................................255
5.4 Sub-Supersolutions and Monotone Penalty Approximations ...257
5.4.1 Hypotheses and Preliminary Results .................258
5.4.2 Obstacle Problem .................................260
5.4.3 Generalized Obstacle Problem ......................262
5.5 Systems of Variational Inequalities.........................267
5.5.1 Notations and Assumptions.........................268
5.5.2 Preliminaries .....................................269
5.5.3 Comparison Principle for Systems ...................272
5.5.4 Generalization, Minimal and Maximal Solutions.......274
5.5.5 Weakly Coupled Systems and Extremal Solutions .....275
5.6 Notes and Comments ....................................277
6 Hemivariational Inequalities ...............................279
6.1 Notion of Sub-Supersolution ..............................281
6.2 Quasilinear Elliptic Hemivariational Inequalities.............285
6.2.1 Comparison Principle ..............................286
6.2.2 Extremal Solutions and Compactness Results .........290
6.2.3 Application.......................................293
6.3 Evolutionary Hemivariational Inequalities ..................299
6.3.1 Sub-Supersolutions and Equivalence of Problems ......301
6.3.2 Existence and Comparison Results...................303
6.3.3 Compactness and Extremality Results ...............310
6.4 Notes and Comments ....................................316
7 Variational–Hemivariational Inequalities ...................319
7.1 Elliptic Variational–Hemivariational Inequalities.............319
7.1.1 Comparison Principle ..............................320
7.1.2 Compactness and Extremality ......................328
7.2 Evolution Variational–Hemivariational Inequalities...........336
7.2.1 Definitions and Hypotheses .........................338
7.2.2 Preliminary Results................................340
7.2.3 Existence and Comparison Result ...................343
7.2.4 Compactness and Extremality ......................351
x Contents
7.3 Nonsmooth Critical Point Theory .........................355
7.4 A Constraint Hemivariational Inequality....................362
7.5 Eigenvalue Problem for a Variational–Hemivariational
Inequality ..............................................368
7.6 Notes and Comments ....................................375
List of Symbols ................................................379
References.....................................................381
Index..........................................................393