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MiS Preprint
26/1999
Convex integration for Lipschitz mappings and counterexamples to regularity
Stefan Müller and Vladimír Šverák
Abstract
In this paper we study Lipschitz solutions of partial differential relations of the form $\nabla u (x) \in K$ a.e. in $\Omega$, where u is a (Lipschitz) mapping of an open set $\Omega \subset R^n$ into $R^m$ and K is a subset of the set $M^{m \times n}$ of all real $m \times n$ 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 $I(u) = \int_\Omega F(\nabla u)$, where $\Omega$ is the unit disc in $R^2$, u is a mapping of $\Omega$ into $R^2$, 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 $C^1$ in any open subset of $\Omega$.
