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Rank-one convexity implies quasi-convexity on certain hypersurfaces. (English) Zbl 1054.49018
Let \({\mathbb M}^{2\times 2}\) be the set of the \(2\times 2\) matrices with real entries, and let \(S^{2\times 2}\) be the set of the symmetric elements of \({\mathbb M}^{2\times 2}\). For \(D\geq0\) and \(c>0\), let \(H^-_D=\{X=(X_{ij})_{1\leq i,j\leq2}\in S^{2\times 2} : \det X=-D,\;X_{11}\geq c\}\).
In the paper it is proved that every \(f: {\mathbb M}^{2\times 2}\to{\mathbb R}\) that is rank-one convex on \(H^-_D\) can be approximated by quasiconvex functions on \({\mathbb M}^{2\times 2}\) uniformly on the compact subsets of \(H^-_D\).
The result is deduced from a structure theorem for Young measures stating that every gradient Young measure supported on a compact subset of \(H^-_D\) is a laminate.

49J45 Methods involving semicontinuity and convergence; relaxation
49J10 Existence theories for free problems in two or more independent variables
52A01 Axiomatic and generalized convexity
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