Lower semicontinuity of quasi-convex integrals in BV
type: Item
instance of: scholarly article
title: Lower semicontinuity of quasi-convex integrals in BV
zbMATH Open document ID: 0915.49007
author: Jan Kristensen
published in: Calculus of Variations and Partial Differential Equations
publication date: 1999-04-20T00:00:00Z
review text: The paper deals with the lower semicontinuity of integral functionals \(F(u)=\int_\Omega f(\nabla u) dx\) (with \(\Omega\subset{\mathbb R}^m\) open, \(u:\Omega\to{\mathbb R}^n\)) under the assumption that \(f\) is quasi convex and satisfies a linear growth condition from below and a \(q\)-growth condition from above, namely \[ c(| p| -1)\leq f(p)\leq C(| p| ^q+1)\qquad\qquad\forall p\in M^{n\times m}. \] The main result of the paper, obtained under the assumption that \(q<m/(m-1)\), is that \[ \liminf_{h\to\infty}\int_\Omega f(\nabla u_h) dx\geq\int_\Omega f(\nabla u) dx\tag{*} \] whenever \((u_h)\subset W^{1,1}(\Omega,{\mathbb R}^n)\) converges to \(u\) in \(L^1_{\text{loc}}\). In (*) \(\nabla u\) is understood as the approximate differential of the BV function \(u\). This result implies in particular an estimate from below of the relaxed functional of \(F\) in the space \(\text{BV}(\Omega,{\mathbb R}^n)\). The proof relies on the blow-up technique of I. Fonseca and S. Müller and on sharp estimates for suitable extension operators.
Mathematics Subject Classification ID: 49J45
zbMATH DE Number: 1221537
zbMATH Keywords: lower semicontinuity
zbMATH Keywords: quasi convexity
zbMATH Keywords: functions of bounded variation
reviewed by: Luigi Ambrosio
MaRDI profile type: Publication
full work available at URL: s005260050108
OpenAlex ID: W2089903613
DOI: 10.1007/S005260050108
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description: scientific article; zbMATH DE number 1221537