MiS Preprint Repository

In this paper we study Lipschitz solutions of partial differential relations of the form $\mathrm{\nabla }u(x)\in K$ a.e. in $\mathrm{\Omega }$, where u is a (Lipschitz) mapping of an open set $\mathrm{\Omega }\subset R^n$ 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 $I(u)={\int }_{\mathrm{\Omega }}F(\mathrm{\nabla }u)$, where $\mathrm{\Omega }$ is the unit disc in $R^2$, u is a mapping of $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$.