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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.

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation 49J10 Existence theories for free problems in two or more independent variables 52A01 Axiomatic and generalized convexity
##### Keywords:
quasiconvexity; rank-one convexity; Young measures
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