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Unique continuation for differential inclusions
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2026-01-01
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Journal Article
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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 of R(2 & times;2 i)s an elliptic, smooth, closed curve. More precisely, we prove that the distance of Du 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-Amp & egrave;re equation: concerning the latter, we obtain a new proof of the W-2,W-1+epsilon-regularity for two-dimensional solutions.
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ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE
Volume
43 (1)
Pages / Article No.
127 - 154
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unique continuation; differential inclusions; quasiregular maps; Monge-Ampere equation