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Full Description
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.
Contents
Preface; 1. Introduction; Part I. First PartPrinciples; 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.
