## Mappings with integrable dilatation in higher dimensions

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- by Juan J. Manfredi and Enrique Villamor PDF
- Bull. Amer. Math. Soc.
**32**(1995), 235-240 Request permission

## 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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## Additional Information

- © Copyright 1995 American Mathematical Society
- Journal: Bull. Amer. Math. Soc.
**32**(1995), 235-240 - MSC: Primary 30C65; Secondary 35J70
- DOI: https://doi.org/10.1090/S0273-0979-1995-00583-5
- MathSciNet review: 1313107