Variational Analysis in Sobolev and BV Spaces Applications to PDEs and Optimization, Second Edition

Among the new elements in this second edition: the section of Chapter 5 on capacity theory and elements of potential theory now includes the concepts of quasi-open sets and quasi-continuity; Chapter 6 includes an increased number of examples in the areas of linearized elasticity system, obstacles problems, convection-diffusion, and semilinear equations; Chapter 11 has been expanded to include a section on mass transportation problems and the Kantorovich relaxed formulation of the Monge problem; a new subsection on stochastic homogenization in Chapter 12 establishes the mathematical tools coming from ergodic theory, and illustrates them in the scope of statistically homogeneous materials; Chapter 16 has been augmented by examples illustrating the shape optimization procedure; and Chapter 17 is an entirely new and comprehensive chapter devoted to gradient flows and the dynamical approach to equilibria.
Audience: The book is intended for Ph.D. students, researchers, and practitioners who want to approach the field of variational analysis in a systematic way.
Contents: Chapter 1: Introduction; Part I: Basic Variational Principles; Chapter 2: Weak Solution Methods in Variational Analysis; Chapter 3: Abstract Variational Principles; Chapter 4: Complements on Measure Theory; Chapter 5: Sobolev Spaces; Chapter 6: Variational Problems: Some Classical Examples; Chapter 7: The Finite Element Method; Chapter 8: Spectral Analysis of the Laplacian; Chapter 9: Convex Duality and Optimization; Part II: Advanced Variational Analysis; Chapter 10: Spaces BV and SBV; Chapter 11: Relaxation in Sobolev, BV, and Young Measures Spaces; Chapter 12: Convergence and Applications; Chapter 13: Integral Functionals of the Calculus of Variations; Chapter 14: Applications in Mechanics and Computer Vision; Chapter 15: Variational Problems with a Lack of Coercivity; Chapter 16: An Introduction to Shape Optimization Problems; Chapter 17: Gradient Flows