Young measures and compactness in measure spaces
type: Item
instance of: scholarly article
zbMATH Open document ID: 1259.28002
DOI: 10.1515/9783110280517
author: Liviu C. Florescu
author: Christiane Godet-Thobie
publication date: 2012-07-13T00:00:00Z
Mathematics Subject Classification ID: 28-01
Mathematics Subject Classification ID: 28-02
Mathematics Subject Classification ID: 60-01
Mathematics Subject Classification ID: 28A33
Mathematics Subject Classification ID: 28C15
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46N10
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 60B05
Mathematics Subject Classification ID: 60B10
zbMATH DE Number: 6055390
zbMATH Keywords: Young measures
zbMATH Keywords: measure space
zbMATH Keywords: compactness
zbMATH Keywords: stable convergence
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1515/9783110280517
OpenAlex ID: W613392573
title: Young measures and compactness in measure spaces
review text: This well-written monograph is a self-contained book which gathers theoretical aspects related to Young measures (measurability, disintegration, stable convergence, compactness). We quote the authors in relation to the applications of Young measures: ``In recent years, the technological progress created a great need for complex mathematical models. Many practical problems can be formulated using optimization theory and they hope to obtain an optimal solution. In most cases, such a optimal solution cannot be found. So, non-convex optimization problems (arising, e.g., in variational calculus, optimal control, nonlinear evolutions equations) may not possess a classical minimizer because the minimizing sequences have typically rapid oscillations. This behavior requires a relaxation of notion of solution for such problems; often we can obtain such a relaxation by means of Young measures.'' With regard to some of the areas in which these relaxed solutions find applications, the authors mention non-convex variational problems and differential inclusions, non-linear homogenization problems, micro-magnetic phenomena in ferro-magnetic materials, Nash equilibrium in game theory, Gamma-con\-vergence, different phenomena in continuum mechanics, optimal design and shape optimization problems.NEWLINENEWLINEThe layout of the book is as follows. The first chapter covers background material on measure theory in an abstract frame. In the second chapter, measure theory on topological spaces is presented. The compactness results from the first two chapters are used to study Young measures in the third chapter. One of the strengths of this book is the fact that all results are accompanied by full demonstrations and for many of these results even different proofs are given. Chapter 1 presents the main properties of the measure spaces and of the space of integrable functions. This chapter is useful, per se, for those interested in the theoretical foundations of measure theory, since it pro…
reviewed by: Humberto Rafeiro
description: scientific article; zbMATH DE number 6055390