Preprint 26/1999

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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In this paper we study Lipschitz solutions of partial differential relations of the form tex2html_wrap_inline28 a.e. in tex2html_wrap_inline30, where u is a (Lipschitz) mapping of an open set tex2html_wrap_inline34 into tex2html_wrap_inline36 and K is a subset of the set tex2html_wrap_inline40 of all real tex2html_wrap_inline42 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 tex2html_wrap_inline46, where tex2html_wrap_inline30 is the unit disc in tex2html_wrap_inline50, u is a mapping of tex2html_wrap_inline30 into tex2html_wrap_inline50, 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 tex2html_wrap_inline62 in any open subset of tex2html_wrap_inline30.

04.01.2023, 02:10