Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization

Variational analysis is the subject of this self-contained guide, which provides a detailed presentation of the most important tools in the field, as well as applications to geometry, mechanics, elasticity, and computer vision. This second edition introduces significant new material on several topics, including: quasi-open sets and quasi-continuity in the context of capacity theory and potential theory; mass transportation problems and the Kantorovich relaxed formulation of the Monge problem; and stochastic homogenization, with mathematical tools coming from ergodic theory. It also features an entirely new and comprehensive chapter devoted to gradient flows and the dynamical approach to equilibria, and extra examples in the areas of linearized elasticity systems, obstacles problems, convection-diffusion, semilinear equations, and the shape optimization procedure. The book is intended for PhD students, researchers, and practitioners who want to approach the field of variational analysis in a systematic way. INDICE: Preface to the second edition; Preface to the first edition; 1. Introduction; Part I. Basic Variational Principles: 2. Weak solution methods in variational analysis; 3. Abstract variational principles; 4. Complements on measure theory; 5. Sobolev spaces; 6. Variational problems: some classical examples; 7. The finite element method; 8. Spectral analysis of the Laplacian; 9. Convex duality and optimization; Part II. Advanced Variational Analysis: 10. Spaces BV and SBV; 11. Relaxation in Sobolev, BV, and Young measure spaces; 12. ?-Convergence and applications; 13. Integral functionals of the calculus of variations; 14. Applications in mechanics and computer vision; 15. Variational problems with a lack of coercivity; 16. An introduction to shape optimization problems; 17. Gradient flows; Bibliography; Index.

• ISBN: 978-1-61197-347-1
• Editorial: Society for Industrial and Applied Mathematics
• Encuadernacion: Cartoné
• Páginas: 800
• Fecha Publicación: 08/01/2015
• Nº Volúmenes: 1
• Idioma: Inglés