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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

$\displaystyle K(x) = \frac{{\vert DF(x){\vert^n}}}{{{J_F}(x)}}.$

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

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