Direct methods in the calculus of variations
type: Item
instance of: scholarly article
title: Direct methods in the calculus of variations
zbMATH Open document ID: 0703.49001
author: Bernard Dacorogna
published in: Applied Mathematical Sciences
publication date: 1993-06-05T00:00:00Z
review text: From the author's preface: ``In recent years there has been a considerable renewal of interest in the classical problems of the calculus of variations, both from the point of view of mathematics and of applications. Some of the most powerful tools for proving existence of minima for such problems are known as direct methods. They are often the only available ones, particularly for vectorial problems. It is the aim of this book to present them. These methods were introduced by Tonelli, following earlier work of Hilbert and Lebesgue. Although there are excellent books on calculus of variations and on direct methods, there are recent important developments which cannot be found in these books; in particular, those dealing with vector valued functions and relaxation of non convex problems. These two last ones are important in applications to nonlinear elasticity, optimal design.... In these fields the variational methods are particularly effective. Parts of the mathematical developments and of the renewal of interest in these methods find its motivations in nonlinear elasticity. Moreover, one of the recent important contributions to nonlinear analysis has been the study of the behavior of nonlinear functionals under various types of convergence, particularly weak convergence. Two well studied theories have now been developed, namely \(\Gamma\)-convergence and compensated compactness. They both include as a particular case the direct methods of the calculus of variations, but they are also, both, inspired and have as main examples these direct methods.'' A brief description of the contents of the monograph is given in the first introductory chapter. The second chapter is devoted to the necessary background for the succeeding four parts: weak convergence in \(L^ p\), Sobolev spaces and convex analysis. The necessary facts are listed and some of them are proved. The general and fundamental results concerning with direct methods are described in the next three chapters. The…
Mathematics Subject Classification ID: 49-02
Mathematics Subject Classification ID: 49J10
Mathematics Subject Classification ID: 49K10
zbMATH DE Number: 194305
zbMATH Keywords: direct methods
zbMATH Keywords: vector valued functions
zbMATH Keywords: relaxation of non convex problems
zbMATH Keywords: nonlinear elasticity
zbMATH Keywords: optimal design
zbMATH Keywords: \(\Gamma \) -convergence
zbMATH Keywords: compensated compactness
zbMATH Keywords: relaxed problem
reviewed by: Ernst G. Al'brekht
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 194305