Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization(6 MPS-SIAM Series on Optimization)

This self-contained book is excellent for graduate-level courses devoted to variational analysis, optimization, and partial differential equations (PDEs). It provides readers with a complete guide to problems in these fields as well as a detailed presentation of the most important tools and methods of variational analysis. New trends in variational analysis are also presented, along with recent developments and applications in this area. It contains several applications to problems in geometry, mechanics, elasticity, and computer vision, along with a complete list of references. The book is divided into two parts. In Part I, classical Sobolev spaces are introduced and the reader is provided with the basic tools and methods of variational analysis and optimization in infinite dimensional spaces, with applications to classical PDE problems. In Part II, BV spaces are introduced and new trends in variational analysis are presented.

Table of Contents:
Preface; 1. Introduction; Part I. First Part: 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. Second Part: Advanced Variational Analysis; 10. Spaces BV and SBV; 11. Relaxation in Sobolev, BV and Young measures spaces; 12. z-convergence and applications; 13. Integral functionals of the calculus of variations; 14. Application in mechanics and computer vision; 15. Variational problems with a lack of coercivity; 16. An introduction to shape optimization problems; Bibliography; Index.

About the Author :
Hedy Attouch is Professor of Mathematics at Université Montpellier II, France. He is Director of Laboratoire d'Analyse Convexe and of ACSIOM (the analysis, computational, and optimization component of the Institute of Mathematics and Modelization of Montpellier). He has published 100 articles and has supervised 23 theses in the fields of variational analysis and optimization. Giuseppe Buttazzo is Professor of Mathematics at Universit... di Pisa. He is an editor of several international journals and author of more than 130 scientific articles and 16 books. He has supervised 13 theses in the fields of calculus of variations, nonlinear PDEs, control theory, and related topics. Gérard Michaille is Professor at Centre Universitaire de Formation et de Recherche at Nîmes (CUFR) and a member of the UMR-CNRS I3S of the Mathematical Department at Université Montpellier II, France. He works in the fields of variational analysis and homogenization and their applications in mechanics.

Review :
'This book is a solid treatise on the (contemporary) calculus of variations. The material presented is quite extensive and slightly nontraditional. For example, the authors include a chapter on convex duality and subdifferential calculus. Often, books on the modern calculus of variations and books devoted to convex optimization have little if any overlap. I believe readers will appreciate the nontrivial overlap in the present text.' Rustum Choksi, Associate Professor of Applied and Computational Mathematics, Simon Fraser University 'The second part has some discussion of more advanced background material (such as BV and SBV functions) needed for work on many modern variational problems, as well as discussions of recent results on a variety of problems, including variational approaches to image segmentation, fracture mechanics, and shape optimization.' Robert Jerrard, Professor of Mathematics, University of Toronto