Nonsmooth critical point theory and nonlinear boundary value problems:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
Chapman & Hall/CRC Press
2005
|
| Series: | Mathematical analysis and applications
8 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XIII, 775 S. |
| ISBN: | 1584884851 |
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Record in the Search Index
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| adam_text | Titel: Nonsmooth critical point theory and nonlinear boundary value problems
Autor: Gasiński, Leszek
Jahr: 2005
Series in Mathematical Analysis and Applications
Edited by Ravi P. Agarwai and Donal O Regan
VOLUME 8
NONSMOOTH CRITICAL
POINT THEORY AND
NONLINEAR BOUNDARY
VALUE PROBLEMS
LeszekGasihski
Nlkolaos S. Papageorgiou
CHAPMAN HALL/CRC
A CRC Press Company
Boca Raton London New York Washington, D.C.
Contents
Mathematical Background 1
1.1 Sobolev Spaces 2
1.1.1 Basic Definitions and Properties 2
1.1.2 Embedding Theorems 6
1.1.3 Poincare Inequality 7
1.1.4 Dual Space 8
1.1.5 Green Formula 9
1.1.6 One Dimensional Sobolev Spaces 10
1.2 Set-Valued Analysis 14
1.2.1 Upper and Lower Semicontinuity 15
1.2.2 /i-Lower and /i-Upper Semicontinuity 18
1.2.3 Measurability of Multifunctions 21
1.2.4 Measurable Selections 22
1.2.5 Continuous Selections 24
1.2.6 Convergence in the Kuratowski Sense 28
1.3 Nonsmooth Analysis 32
1.3.1 Convex Functions 32
1.3.2 Fenchel Transform 38
1.3.3 Subdifferential of Convex Functions 42
1.3.4 Clarke Subdifferential 48
1.3.5 Weak Slope 62
1.4 Nonlinear Operators 67
1.4.1 Compact Operators 68
1.4.2 Maximal Monotone Operators 70
1.4.3 Yosida Approximation 81
1.4.4 Pseudomonotone Operators 83
1.4.5 Nemytskii Operators 87
1.4.6 Ekeland Variational Principle 89
1.5 Elliptic Differential Equations 93
1.5.1 Ordinary Differential Equations 93
1.5.2 Partial Differential Equations 99
1.5.3 Regularity Results 112
1.6 Remarks 120
vn
Vlll
Critical Point Theory 123
2.1 Locally Lipschitz Functionals 123
2.1.1 Compactness-Type Conditions 123
2.1.2 Critical Points and Deformation Theorem 128
2.1.3 Linking Sets 136
2.1.4 Minimax Principles 138
2.1.5 Existence of,Multiple Critical Points 145
2.2 Constrained Locally Lipschitz Functionals 149
2.2.1 Critical Points of Constrained Functions 149
-2.2.2 Deformation Theorem 151
2.2.3 Minimax Principles 154
2.3 Perturbations of Locally Lipschitz Functionals 159
2.3.1 Critical Points of Perturbed Functions 159
2.3.2 Generalized Deformation Theorem 162
2.3.3 Minimax Principles 166
2.4 Local Linking and Extensions 171
2.4.1 Local Linking 171
2.4.2 Minimax Principles 181
2.4.3 Palais-Smale-Type Conditions 184
2.5 Continuous Functionals 187
2.5.1 Compactness-Type Conditions 187
2.5.2 Deformation Theorem 188
2.5.3 Minimax Principles . 195
2.6 Multivalued Functionals 197
2.6.1 Compactness-Type Conditions 197
2.6.2 Multivalued Deformation Theorem 199
2.6.3 Minimax Principles 200
2.7 Remarks 203
Ordinary Differential Equations 207
3.1 Dirichlet Problems 208
3.1.1 Formulation of the Problem 208
3.1.2 Approximation of the Problem 212
3.1.3 Existence Results 225
3.1.4 Problems with Non-Convex Nonlinearities 231
3.2 Periodic Problems 233
3.2.1 Auxiliary Problems 233
3.2.2 Formulation of the Problem 239
3.2.3 Approximation of the Problem 240
3.2.4 Existence Results 249
3.2.5 Problems with Non-Convex Nonlinearities 258
3.2.6 Scalar Problems 259
3.3 Nonlinear Boundary Conditions 268
IX
3.4 Variational Methods 283
3.4.1 Existence Theorems 284
3.4.2 Homoclinic Solutions 308
3.4.3 Scalar Problems 317
3.4.4 Multiple Periodic Solutions 343
3.4.5 Nonlinear Eigenvalue Problems 353
3.4.6 Problems with Nonlinear Boundary Conditions . . . . 356
3.4.7 Multiple Solutions for Smooth Problems 365
3.5 Method of Upper and Lower Solutions 372
3.6 Positive Solutions and Other Methods 389
3.6.1 Positive Solutions 389
3.6.2 Method Based on Monotone Operators 403
3.7 Hamiltonian Inclusions 414
3.8 Remarks 448
Elliptic Equations 453
4.1 Problems at Resonance 454
4.1.1 Semilinear Problems at Resonance 455
4.1.2 Nonlinear Problems at Resonance 474
4.1.3 Variational-Hemivariational Inequality at Resonance . 485
4.1.4 Strongly Resonant Problems 495
4.2 Neumann Problems 500
4.2.1 Spectrum of (-Ap,W1
p
(fi)) 501
4.2.2 Homogeneous Neumann Problems 516
4.2.3 Nonhomogeneous Neumann Problem 526
4.3 Problems with an Area-Type Term 536
4.4 Strongly Nonlinear Problems 558
4.5 Method of Upper and Lower Solutions 586
4.5.1 Existence of Solutions 587
4.5.2 Existence of Extremal Solutions 598
4.6 Multiplicity Results 608
4.6.1 Semilinear Problems 608
4.6.2 Nonlinear Problems 640
4.7 Positive Solutions 661
4.8 Problems with Discontinuous Nonlinearities 683
4.9 Remarks 700
Appendix 707
A.I Set Theory and Topology 707
A.2 Measure Theory 714
A.3 Functional Analysis 718
A.4 Nonlinear Analysis · 724
X-
List of Symbols 727
References 735
Index 763
|
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| spelling | Gasiński, Leszek Verfasser (DE-588)1067201238 aut Nonsmooth critical point theory and nonlinear boundary value problems Leszek Gasiński ; Nikolaos S. Papageorgiou Boca Raton [u.a.] Chapman & Hall/CRC Press 2005 XIII, 775 S. txt rdacontent n rdamedia nc rdacarrier Mathematical analysis and applications 8 Includes bibliographical references and index Points critiques, Théorie des (Analyse mathématique) Problèmes aux limites non linéaires aCritical point theory (Mathematical analysis) aNonlinear boundary value problems Nichtlineares Randwertproblem (DE-588)4129830-5 gnd rswk-swf Kritischer Punkt Mathematik (DE-588)4207169-0 gnd rswk-swf Kritischer Punkt Mathematik (DE-588)4207169-0 s Nichtlineares Randwertproblem (DE-588)4129830-5 s DE-604 Papageōrgiu, Nikolaos 1958- Sonstige (DE-588)137051085 oth Mathematical analysis and applications 8 (DE-604)BV013405523 8 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012947968&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gasiński, Leszek Nonsmooth critical point theory and nonlinear boundary value problems Mathematical analysis and applications Points critiques, Théorie des (Analyse mathématique) Problèmes aux limites non linéaires aCritical point theory (Mathematical analysis) aNonlinear boundary value problems Nichtlineares Randwertproblem (DE-588)4129830-5 gnd Kritischer Punkt Mathematik (DE-588)4207169-0 gnd |
| subject_GND | (DE-588)4129830-5 (DE-588)4207169-0 |
| title | Nonsmooth critical point theory and nonlinear boundary value problems |
| title_auth | Nonsmooth critical point theory and nonlinear boundary value problems |
| title_exact_search | Nonsmooth critical point theory and nonlinear boundary value problems |
| title_full | Nonsmooth critical point theory and nonlinear boundary value problems Leszek Gasiński ; Nikolaos S. Papageorgiou |
| title_fullStr | Nonsmooth critical point theory and nonlinear boundary value problems Leszek Gasiński ; Nikolaos S. Papageorgiou |
| title_full_unstemmed | Nonsmooth critical point theory and nonlinear boundary value problems Leszek Gasiński ; Nikolaos S. Papageorgiou |
| title_short | Nonsmooth critical point theory and nonlinear boundary value problems |
| title_sort | nonsmooth critical point theory and nonlinear boundary value problems |
| topic | Points critiques, Théorie des (Analyse mathématique) Problèmes aux limites non linéaires aCritical point theory (Mathematical analysis) aNonlinear boundary value problems Nichtlineares Randwertproblem (DE-588)4129830-5 gnd Kritischer Punkt Mathematik (DE-588)4207169-0 gnd |
| topic_facet | Points critiques, Théorie des (Analyse mathématique) Problèmes aux limites non linéaires aCritical point theory (Mathematical analysis) aNonlinear boundary value problems Nichtlineares Randwertproblem Kritischer Punkt Mathematik |
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