Convex integration for Lipschitz mappings and counterexamples to regularity

Stefan Müller and Vladimír Šverák

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Submission date: 03. May. 1999
Pages: 25
published in: Annals of mathematics, 157 (2003) 3, p. 715-742 
DOI number (of the published article): 10.4007/annals.2003.157.715
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Abstract:
In this paper we study Lipschitz solutions of partial differential relations of the form a.e. in , where u is a (Lipschitz) mapping of an open set into and K is a subset of the set of all real matrices. We extend Gromov's method of convex integration by replacing his P-convex hull by the larger rank-one convex hull, defined by duality with rank-one convex functions. There are a number of interesting examples for which the latter is nontrivial while the former is trivial.

As an application we give a solution of a long-standing problem regarding regularity of weak solutions of elliptic systems. We construct an example of a variational integral , where is the unit disc in , u is a mapping of into , and F is a smooth, strongly quasi-convex function with bounded second derivatives, such that the Euler-Lagrange equation of I has a large class of weak solutions which are Lipschitz but not in any open subset of .