Zhang, Kewei On some semiconvex envelopes. (English) Zbl 1012.49012 NoDEA, Nonlinear Differ. Equ. Appl. 9, No. 1, 37-44 (2002). 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. Reviewer: Riccardo De Arcangelis (Napoli) Cited in 2 Documents MSC: 49J45 Methods involving semicontinuity and convergence; relaxation 49J10 Existence theories for free problems in two or more independent variables 52A40 Inequalities and extremum problems involving convexity in convex geometry Keywords:vector-valued problems; quasiconvexity; relaxation; rank-one convexity PDFBibTeX XMLCite \textit{K. Zhang}, NoDEA, Nonlinear Differ. Equ. Appl. 9, No. 1, 37--44 (2002; Zbl 1012.49012) Full Text: DOI