Young measures and compactness in measure spaces
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 provides a complete set of classical and recent compactness results in measure and function spaces. One can find classical results of weak compactness (e.g., the Vitali-Hahn-Saks, Radon-Nykodym and Dunford-Pettis theorems) as well as more recent results such as the Brooks-Chacon bitting lemma or even Rosenthal's lemma.NEWLINENEWLINE Chapter 2 is concerned with measures defined on topological spaces. It starts by studying several types of regularity, then going into some properties of Polish and Suslin sets, especially the property of being Radonian. The authors proceed with the introduction of the narrow topology on the measure space and a study of the different properties of semicontinuity. The compactness theorem of Prohorov is given as well as an application of it, namely, the introduction of the Wiener measure as the limit of a sequence of measures. It is also introduced the metric of Dudley and the metric of Lévy-Prohorov on the space of positive Radon measures.NEWLINENEWLINEChapter 3 is the main chapter studying Young measures based on the results of compactness obtain in the previous chapters. In this chapter, one finds the definition and several characteristics of the Young measure as well as some examples. A study of the stable topology on the Young measure space and the properties of its trace on the subspace of measurable functions is given, where the main results are the fact that the trace of the stable topology in the subspace of measurable functions is the topology of convergence in measure and that this subspace is dense in the space of Young measures. Another result in the chapter is Prohorov's theorem of compactness for the Young measure. Other results regarding compactness are proved, like the general result of Saadoune-Valadier and the bitting lemma, even in the case of an unbounded set of measurable functions. Other results given are, e.g., the fiber product lemma, several applications of Young measure theory in the study of strong compactness in \(L^p\), aspects of relaxed variational calculus and the Kinderlehrer-Pedregal's characterization of gradient Young measures. Finally, some results on the existence of solutions in a relaxed variant of variational calculus are presented. Many of the results are presented in the general framework of regular Suslin spaces although sometimes the results are given in the particular cases of Polish spaces or even of Euclidean spaces. The useful bibliography includes 182 items.
scientific article; zbMATH DE number 6055390