Elements of Nonlinear Analysis

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.

1. Some Physical Motivations.- 1.1. An elementary theory of elasticity.- 1.2. A problem in biology.- 1.3. Exercises.- 2. A Short Background in Functional Analysis.- 2.1. An introduction to distributions.- 2.2. Integration on boundaries.- 2.3. Introduction to Sobolev spaces.- 2.4. Exercises.- 3. Elliptic Linear Problems.- 3.1. The Dirichlet problem.- 3.2. The Lax-Milgram theorem and its applications.- 3.3. Exercises.- 4. Elliptic Variational Inequalities.- 4.1. A generalization of the Lax-Milgram theorem.- 4.2. Some applications.- 4.3. Exercises.- 5. Nonlinear Elliptic Problems.- 5.1. A compactness method.- 5.2. A monotonicity method.- 5.3. A generalization of variational inequalities.- 5.4. Some multivalued problems.- 5.5. Exercises.- 6. A Regularity Theory for Nonlocal Variational Inequalities.- 6.1. Some general results.- 6.2. Applications to second order variational inequalities.- 6.3. Exercises.- 7. Uniqueness and Nonuniqueness Issues.- 7.1. Uniqueness result for local nonlinear problems.- 7.2. Nonuniqueness issues.- 7.3. Exercises.- 8. Finite Element Methods for Elliptic Problems.- 8.1. An abstract setting.- 8.2. Some simple finite elements.- 8.3. Interpolation error.- 8.4. Convergence results.- 8.5. Approximation of nonlinear problems.- 8.6. Exercises.- 9. Minimizers.- 9.1. Introduction.- 9.2. The direct method.- 9.3. Applications.- 9.4. The Euler Equation.- 9.5. Exercises.- 10. Minimizing Sequences.- 10.1. Some model problems.- 10.2. Young measures.- 10.3. Construction of the minimizing sequences.- 10.4. A more elaborate issue.- 10.5. Numerical analysis of oscillations.- 10.6. Exercises.- 11. Linear Parabolic Equations.- 11.1. Introduction.- 11.2. Functional analysis for parabolic problems.- 11.3. The resolution of parabolic problems.- 11.4. Applications.- 11.5.Exercises.- 12. Nonlinear Parabolic Problems.- 12.1. Local problems.- 12.2. Nonlocal problems.- 12.3. Exercises.- 13. Asymptotic Analysis.- 13.1. The case of one stationary point.- 13.2. The case of several stationary points.- 13.3. A nonlinear case.- 13.4. Blow-up.- 13.5. Exercises.