Mappings with Integrable Dilatation in Higher Dimensions

Abstract

Let $F\in W_{\text{loc}}^{1,n}(\Omega ;{\mathbb{R}}^n)$ be a mapping with nonnegative Jacobian $J_F(x)=\det DF(x)\ge 0$ for a.e. $x$ in a domain $\Omega \subset {\mathbb{R}}^n$. The {\it dilatation} of $F$ is defined (almost everywhere in $\Omega$) by the formula $$K(x)=\frac{|DF(x){|}^n}{J_F(x)}\cdot$$ Iwaniec and \v Sver\' ak \ncite{IS} have conjectured that if $p\ge n-1$ and $K\in L_{\text{loc}}^p(\Omega )$ 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 $n\ge 2$ whenever $p>n-1$.

6 pages