Elements of Nonlinear Analysis

1. Front Matter

   Pages i-vii

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2. Some Physical Motivations

   • Michel Chipot

   Pages 1-14

3. A Short Background in Functional Analysis

   • Michel Chipot

   Pages 15-38

4. Elliptic Linear Problems

   • Michel Chipot

   Pages 39-48

5. Elliptic Variational Inequalities

   • Michel Chipot

   Pages 49-58

6. Nonlinear Elliptic Problems

   • Michel Chipot

   Pages 59-83

7. A Regularity Theory for Nonlocal Variational Inequalities

   • Michel Chipot

   Pages 85-94

8. Uniqueness and Nonuniqueness Issues

   • Michel Chipot

   Pages 95-103

9. Finite Element Methods for Elliptic Problems

   • Michel Chipot

   Pages 105-129

10. Minimizers

    • Michel Chipot

    Pages 131-143

11. Minimizing Sequences

    • Michel Chipot

    Pages 145-183

12. Linear Parabolic Equations

    • Michel Chipot

    Pages 185-205

13. Nonlinear Parabolic Problems

    • Michel Chipot

    Pages 207-220

14. Asymptotic Analysis

    • Michel Chipot

    Pages 221-250

15. Back Matter

    Pages 251-256

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The goal of this book is to present some modern aspects of nonlinear analysis. Some of the material introduced is classical, some more exotic. We have tried to emphasize simple cases and ideas more than complicated refinements. Also, as far as possible, we present proofs that are not classical or not available in the usual literature. Of course, only a small part of nonlinear analysis is covered. Our hope is that the reader - with the help of these notes - can rapidly access the many different aspects of the field. We start by introducing two physical issues: elasticity and diffusion. The pre­ sentation here is original and self contained, and helps to motivate all the rest of the book. Then we turn to some theoretical material in analysis that will be needed throughout (Chapter 2). The next six chapters are devoted to various aspects of elliptic problems. Starting with the basics of the linear theory, we introduce a first type of nonlinear problem that has today invaded the whole mathematical world: variational inequalities. In particular, in Chapter 6, we introduce a simple theory of regularity for nonlocal variational inequalities. We also attack the question of the existence, uniqueness and approximation of solutions of quasilinear and mono­ tone problems (see Chapters 5, 7, 8). The material needed to read these parts is contained in Chapter 2. The arguments are explained using the simplest possible examples.

• Calculus of Variations
• Distribution
• Euler–Lagrange equation
• Numerical analysis
• applied mathematics
• finite element method
• functional analysis
• partial differential equations

"This is a relatively formal book which starts out as a motivational chapter introducing two examples, elasticity and diffusion, of the application of nonlinear analysis. Aimed mainly at graduate students and nonspecialists, the material is then developed in a largely abstract fashion, introducing topics in existence, uniqueness, regularity and approximation for elliptic and parabolic problems."   --Aslib Book Guide

"This book covers some of the main aspects of nonlinear analysis. It concentrates on stressing the fundamental ideas instead of elaborating on the intricacies of the more esoteric ones…it encompass[es] many methods of dynamical systems in quite simple and original settings. I recommend this book to anyone interested in the main and essential concepts of nonlinear analysis as well as the relevant methodologies and applications."   --Mathematical Reviews

• Institut für Mathematik, Universität Zürich, Zürich, Switzerland

  Michel Chipot

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