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arXiv · 2312.05022

Unique continuation for differential inclusions

Guido De Philippis·André Guerra·Riccardo Tione
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Abstract

We consider the following question arising in the theory of differential inclusions: given an elliptic set $Γ$ and a Sobolev map $u$ whose gradient lies in the quasiconformal envelope of $Γ$ and touches $Γ$ 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 $Γ\subset \mathbb R^{2\times 2}$ is an elliptic, smooth, closed curve. More precisely, we prove that the distance of $D u$ to $Γ$ 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--Ampère equation: concerning the latter, we obtain a new proof of the $W^{2,1+\varepsilon}$-regularity for two-dimensional solutions.

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Guido De Philippis, André Guerra, Riccardo Tione. 2023-12-08. Unique continuation for differential inclusions. https://arxiv.org/abs/2312.05022

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