Regularity of quasiconvex envelopes
type: Item
instance of: scholarly article
title: Regularity of quasiconvex envelopes
zbMATH Open document ID: 0972.49024
author: J. M. Ball
author: Bernd Kirchheim
author: Jan Kristensen
published in: Calculus of Variations and Partial Differential Equations
publication date: 2001-11-06T00:00:00Z
review text: In this very interesting paper the following question is addressed: consider the variational integral (\(\Omega\) denoting a domain in \(\mathbb{R}^n\)) \[ I(u)= \int_\Omega F(x,u,\nabla u) dx\tag{2} \] such that for some \(p\in (1,\infty)\) the integrand \(F:\Omega\times \mathbb{R}^N\times \mathbb{R}^{nN}\to \mathbb{R}\) satisfies the growth condition \[ c_1|\xi|^p- c_2\leq F(x,v,\xi)\leq c_3(|\xi|^p+ 1).\tag{2} \] Then \(I(u)\) makes sense for functions \(u: \Omega\to\mathbb{R}^N\) of Sobolev class \(W^{1,p}\) but without further assumptions, \(I\) in general fails to be lower-semicontinuous w.r.t. the weak topology of \(W^{1,p}\). So one passes to the l.s.c. envelope \(\overline I\) of (1) which has the representation \[ \overline I(u)= \int_\Omega\overline F(x,u,\nabla u) dx,\tag{3} \] \(\overline F\) denoting the quasiconvex envelope of \(F\) w.r.t. the last variable. Now, in order to justify the Euler-Lagrange equation associated to the functional (3), it is necessary to know what differentiability properties can be expected to hold for \(\overline F\). Up to now this question has only been settled by deriving explicit formulas for \(\overline F\) in certain examples which is extremely difficult. For integrands \(F\) which depend also on \(u\) and satisfy (2) smoothness of \(\overline F\) in general fails to hold even if \(F\) is of class \(C^\infty\), examples are presented in Section 5 of the paper. The main results of the paper concern the case ``\(F(\nabla u)\)'' (see Theorem A and B) for which it is shown that differentiability or Lipschitz continuity of \(F\) together with some weaker growth conditions than stated in (2) implies \(C^1\)-regularity of the quasiconvex envelope \(\overline F\). Also an important class of non-differentiable integrands \(F(\nabla u)\) is given (see Theorem C and 5.5) such that \(\overline F\) is of type \(C^{1,1}_{\text{loc}}\). In Section 4 of the paper the reader will find further comments on the behaviour of the polyconvex…
Mathematics Subject Classification ID: 49N60
Mathematics Subject Classification ID: 49J45
zbMATH DE Number: 1563305
zbMATH Keywords: integral functional
zbMATH Keywords: quasiconvex envelope
zbMATH Keywords: \(C^1\)-regularity
zbMATH Keywords: polyconvex envelope
reviewed by: M. Fuchs
MaRDI profile type: Publication
full work available at URL: s005260000041
OpenAlex ID: W2045970486
Wikidata QID: Q116038301
DOI: 10.1007/S005260000041
description: scientific article; zbMATH DE number 1563305