Elements of Nonlinear Analysis

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