Calculus of Variations and Geometric Measure Theory

G. De Philippis - A. Guerra - R. Tione

Unique continuation for differential inclusions

created by tione on 12 Dec 2023
modified on 04 Nov 2024

[BibTeX]

Published Paper

Inserted: 12 dec 2023
Last Updated: 4 nov 2024

Journal: Annales de l'Institut Henri Poincaré C
Year: 2023
Doi: 10.4171/AIHPC/146

ArXiv: 2312.05022 PDF

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.