Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals
type: Item
instance of: scholarly article
title: Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals
zbMATH Open document ID: 0573.49010
DOI: 10.1007/BF01168345
author: Paolo Marcellini
published in: Manuscripta Mathematica
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155063
review text: The author studies the semicontinuity of the following functional in the calculus of variations: \(F(u)=\int_{G}f(x,u,Du)dx\) where u is a vector- valued function. The function f(x,s,\(\xi)\) is assumed to verify the Carathéodory property and to be quasi-convex in Morrey's sense. Growth conditions for f are given to ensure that F is sequentially lower semicontinuous in the weak topology of \(H^{1,p}(G,R^ N)\). The proof is based on some interesting approximation results for f. In particular, it is possible to approximate f with a non decreasing sequence of quasi- convex functions \(f_ K\), which are convex and independent of (x,s) for \(| \xi | \geq K\). Finally, polyconvex functionals in Ball's sense are considered and some counterexamples are given.
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 26B25
Mathematics Subject Classification ID: 54C08
Mathematics Subject Classification ID: 41A30
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 3915071
zbMATH Keywords: semicontinuity
zbMATH Keywords: quasi-convex functions
zbMATH Keywords: polyconvex functionals
reviewed by: Elvira Mascolo
MaRDI profile type: Publication
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OpenAlex ID: W2155476332
description: scientific article; zbMATH DE number 3915071