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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Mappings with integrable dilatation in higher dimensions

Authors: Juan J. Manfredi and Enrique Villamor
Journal: Bull. Amer. Math. Soc. 32 (1995), 235-240
MSC: Primary 30C65; Secondary 35J70
MathSciNet review: 1313107
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Abstract: Let ${F \in W_{{\text {loc}}}^{1,n}(\Omega ;{\mathbb {R}^n})}$ be a mapping with nonnegative Jacobian ${{J_F}(x) = \det DF(x) \geq 0}$ for a.e. x in a domain ${\Omega \subset {\mathbb {R}^n}}$. The dilatation of F is defined (almost everywhere in ${\Omega }$) by the formula \[ K(x) = \frac {{|DF(x){|^n}}}{{{J_F}(x)}}.\] Iwaniec and Šverák [IS] have conjectured that if ${p \geq n - 1}$ and ${K \in L_{loc}^p(\Omega )}$ then F must be continuous, discrete and open. Moreover, they have confirmed this conjecture in the two-dimensional case n = 2. In this article, we verify it in the higher-dimensional case ${n \geq 2}$ whenever ${p > n - 1}$.

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Keywords: Quasiregular mappings, degenerate elliptic equations, nonlinear elasticity
Article copyright: © Copyright 1995 American Mathematical Society