On some semiconvex envelopes
type: Item
instance of: scholarly article
title: On some semiconvex envelopes
zbMATH Open document ID: 1012.49012
author: Kewei Zhang
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 2002-05-30T00:00:00Z
review text: Let \(M^{N\times n}\) denote the set of the (\(N\times n\))-matrices with real entries. For every \(f\colon M^{N\times n}\to{\mathbb R}\) let \(Cf\), \(Qf\), and \(Rf\) denote, respectively, the convex, quasiconvex, rank-one convex envelopes of \(f\). Then \[ Cf\leq Qf\leq Rf\leq f. \] In the paper it is proved that, if \(f\) is superlinear in the sense that \[ \lim_{|A|\to\infty}{f(A)\over|A|}=+\infty, \] then \(Cf=Qf\) if and only if \(Cf=Rf\). I particular, if a rank-one convex superlinear function is not convex, then its quasiconvex envelope is not convex. This remark provides a way for testing whether the quasiconvex envelope of a superlinear function is trivial (i.e., convex) by just calculating its rank-one convex envelope.
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 49J10
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 1747957
zbMATH Keywords: vector-valued problems
zbMATH Keywords: quasiconvexity
zbMATH Keywords: relaxation
zbMATH Keywords: rank-one convexity
MaRDI profile type: Publication
full work available at URL: s00030-002-8117-x
OpenAlex ID: W2043742452
DOI: 10.1007/S00030-002-8117-X
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description: scientific article; zbMATH DE number 1747957