Unique continuation for differential inclusions

  • Guido De Philippis

    New York University, New York, USA
  • André Guerra

    ETH Zürich, Zürich, Switzerland
  • Riccardo Tione

    Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany; Università di Torino, Torino, Italy
Unique continuation for differential inclusions cover

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Abstract

We consider the following question arising in the theory of differential inclusions: Given an elliptic set and a Sobolev map whose gradient lies in the quasiconformal envelope of and touches on a set of positive measure, must be affine? We answer this question positively for a suitable notion of ellipticity, which for instance encompasses the case where is an elliptic, smooth, closed curve. More precisely, we prove that the distance of to 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 -regularity for two-dimensional solutions.

Cite this article

Guido De Philippis, André Guerra, Riccardo Tione, Unique continuation for differential inclusions. Ann. Inst. H. Poincaré C Anal. Non Linéaire (2024), published online first

DOI 10.4171/AIHPC/146