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

This volume is an excellent guide for anyone interested in variational analysis, optimization, and PDEs. It offers a detailed presentation of the most important tools in variational analysis as well as applications to problems in geometry, mechanics, elasticity, and computer vision.

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

Hedy Attouch is a professor in the Institut de Mathématique et de Modélisation de Montpellier II, where he also has been director of the Laboratory of Convex Analysis and of ACSIOM. His research focuses on variational analysis, convex analysis, continuous optimization, semialgebraic optimization, gradient flows, the interaction among these fields of research, and their applications. He has published more than 100 articles in international journals and has written 6 books. He serves as editor for several journals on continuous optimization and is responsible for several international research programs.

Giuseppe Buttazzo is a professor in the Department of Mathematics at the University of Pisa. He has been a keynote speaker at many international conferences and workshops on the fields of calculus of variations, nonlinear PDEs, applied mathematics, control theory, and related topics. He is the author of more than 180 scientific publications and 20 books, and he serves as an editor of several international journals.

Gérard Michaille is a professor at the University of Nîmes and member of the UMR-CNRS Institut de Mathématique et de Modélisation de Montpellier. He works in the areas of variational analysis, homogenization, and the applications of PDEs in mechanics and physics.