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Full Description
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.
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
