Mappings with integrable dilatation in higher dimensions

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

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

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 ${p > n - 1}$.