This book is a short introductory text to variational techniques with applications to differential equations. It presents a sampling of topics in critical point theory with applications to existence and multiplicity of solutions in nonlinear problems involving ordinary differential equations (ODEs) and partial differential equations (PDEs).
Five simple problems in ODEs which illustrate existence of solutions from a variational point of view are introduced in the first chapter. These problems set the stage for the topics covered, including minimization, deformation results, the mountain-pass theorem, the saddle-point theorem, critical points under constraints, a duality principle, critical points in the presence of symmetry, and problems with lack of compactness. Each topic is presented in a straightforward manner, and followed by one or two illustrative applications.
The concise, straightforward, user-friendly approach of this textbook will appeal to graduate students and researchers interested in differential equations, analysis, and functional analysis.
Author(s): David G. Costa
Series: Birkhäuser Advanced Texts / Basler Lehrbücher
Edition: 1
Publisher: Birkhäuser Boston
Year: 2007
Language: English
Pages: 151
Tags: Математика;Дифференциальные уравнения;
Cover......Page 1
An Invitation to Variational Methods In Differential Equations......Page 4
Copyright - ISBN: 0817645357......Page 5
Dedication Page......Page 6
Table of Contents......Page 8
Preface......Page 10
Some Notations and Conventions......Page 12
1 Five Illustrating Problems......Page 14
1 Basic Results......Page 20
2 Application to a Dirichlet Problem......Page 24
3 Exercises......Page 28
1 Preliminaries......Page 32
2 Some Versions of the Deformation Theorem......Page 33
3 A Minimum Principle and an Application......Page 37
4 Exercises......Page 40
1 Critical Points of Minimax Type......Page 42
2 The Mountain-Pass Theorem......Page 44
3 Two Basic Applications......Page 45
4 Exercises......Page 50
1 Preliminaries. The Topological Degree......Page 52
2 The Abstract Result......Page 54
3 Application to a Resonant Problem......Page 55
4 Exercises......Page 59
1 Introduction. The Basic Minimization Principle Revisited......Page 62
2 Natural Constraints......Page 63
3 Applications......Page 65
4 Exercises......Page 75
1 Convex Functions. The Legendre-Fenchel Transform......Page 76
2 A Variational Formulation for a Class of Problems......Page 79
3 A Dual Variational Formulation......Page 80
4 Applications......Page 83
1 Introduction......Page 88
2 The Lusternik-Schnirelman Theory......Page 89
3 The Basic Abstract Multiplicity Result......Page 91
4 Application to a Problem with a Z_2-Symmetry......Page 96
1 A Geometric S¹-index......Page 100
2 A Multiplicity Result......Page 103
3 Application to a Class of Problems......Page 105
4 A Dirichlet Problem on a Plane Disk......Page 108
1 Introduction......Page 112
2 Two Beautiful Lemmas......Page 113
3 A Problem in R^N......Page 116
1 (PS)_c for Strongly Resonant Problems......Page 128
2 A Class of Indefinite Problems......Page 131
3 An Application......Page 134
1 Ekeland Variational Principle......Page 138
References......Page 144
Index......Page 148
