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 \mathrm{D}u to \Gamma satisfies the strong unique continuation property. As a by-product, we obtain new results for non-linear Beltrami equations and recover known results for the reduced Beltrami equation and the Monge–Ampère equation: concerning the latter, we obtain a new proof of the W^{2,1+\varepsilon} -regularity for two-dimensional solutions.