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MiS Preprint
68/2002
Rank-one convexity implies quasiconvexity on certain hypersurfaces
Nirmalendu Chaudhuri and Stefan Müller
Abstract
We show that, if $f : \Bbb M^{2\times 2}\longrightarrow \Bbb R$ is rank-1 convex on the hyperboloid $$H^{-}_{D}:=\left\{X\in S^{2\times 2}\, : \, \text{det}\,X=-D, X_{11}\geq c>0\right\}, D\geq 0, S^{2\times 2}$$ is the set of $2\times2$ real symmetric matrices, then $f$ can be approximated by quasiconvex functions on $\Bbb M^{2\times 2}$ uniformly on compact subsets of $H^{-}_{D}$. Equivalently, every gradient Young measure supported on a compact subset of $H^{-}_{D}$ is a laminate.
