Elements of Nonlinear Analysis

"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

Contenu

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.