The two-well problem in three dimensions

Georg Dolzmann, Bernd Kirchheim, Stefan Müller, and Vladimír Šverák

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Submission date: 18. Mar. 1999
Pages: 17
published in: Calculus of variations and partial differential equations, 10 (2000) 1, p. 21-40 
DOI number (of the published article): 10.1007/PL00013455
Bibtex
MSC-Numbers: 73C50, 49J40, 52A30
Keywords and phrases: young measure, minors relations, incompatible wells, generalized convex hulls
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Abstract:
We study properties of generalized convex hulls of the set with . If K contains no rank-1 one connection we show that the quasiconvex hull of K is trivial if H belongs to a certain (large) neighbourhood of the identity. We also show that the polyconvex hull of K can be nontrivial if H is sufficiently far from the identity, while the (functional) rank-1 convex hull is always trivial. If the second well is replaced by a point then the polyconvex hull is trivial provided that there are no rank-1 connections.