Librería Portfolio

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.

Suitable for anyone who wants to approach the field of variational analysis in a systematic way
Contains a substantial amount of new material on a range of significant modern topics
Presents powerful applications to problems in geometry, mechanics, elasticity, and computer vision



Table of Contents

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. G-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.