Unique continuation for differential inclusions

Abstract

We consider the following question arising in the theory of differential inclusions: given an elliptic set $\Gamma$ and a Sobolev map $u$ whose gradient lies in the quasiconformal envelope of $\Gamma$ and touches $\Gamma$ on a set of positive measure, must $u$ be affine? We answer this question positively for a suitable notion of ellipticity, which for instance encompasses the case where $\Gamma \subset \mathbb R^{2\times 2}$ is an elliptic, smooth, closed curve. More precisely, we prove that the distance of $D u$ to $\Gamma$ satisfies the strong unique continuation property. As a by-product, we obtain new results for nonlinear Beltrami equations and recover known results for the reduced Beltrami equation and the Monge--Amp\`ere equation: concerning the latter, we obtain a new proof of the $W^{2,1+\varepsilon}$-regularity for two-dimensional solutions.