Elements of Nonlinear Analysis

Description

Cover Title page Preface Chapter 1. Some Physical Motivations 1.1. An elementary theory of elasticity 1.2. A problem in biology 1.3. Exercises Chapter 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 Chapter 3. Elliptic Linear Problems 3.1. The Dirichlet problem 3.2. The Lax-Milgram theorem and its applications 3.3. Exercises Chapter 4. Elliptic Variational Inequalities 4.1. A generalization of the Lax-Milgram theorem 4.2. Some applications 4.3. Exercises Chapter 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 Chapter 6. A Regularity Theory for Nonlocal Variational Inequalities 6.1. Some general results 6.2. Applications to second order variational inequalities 6.3. Exercises Chapter 7. Uniqueness and Nonuniqueness Issues 7.1. Uniqueness result for local nonlinear problems 7.2. Nonuniqueness issues 7.3. Exercises Chapter 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 Chapter 9. Minimizers 9.1. Introduction 9.2. The direct method 9.3. Applications 9.4. The Euler Equation 9.5. Exercises Chapter 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 Chapter 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 Chapter 12. Nonlinear Parabolic Problems 12.1. Local problems 12.2. Nonlocal problems 12.3. Exercises Chapter 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 Bibliography Index