Optimal regularity for planar mappings of finite distortion

Abstract

Let $f:\Omega \to {\mathbb{R}}^2$ be a mapping of finite distortion, where $\Omega \subset {\mathbb{R}}^2$. Assume that the distortion function $K(x,f)$ satisfies $e^{K(\cdot ,f)}\in L_{\mathrm{loc}}^p(\Omega )$ for some $p>0$. We establish optimal regularity and area distortion estimates for $f$. In particular, we prove that $|Df{|}^2{\log }^{\beta -1}(e+|Df|)\in L_{\mathrm{loc}}^1(\Omega )$ for every $\beta <p$. This answers positively, in dimension $n=2$, the well-known conjectures of Iwaniec and Sbordone [T. Iwaniec, C. Sbordone, Quasiharmonic fields, Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001) 519–572, Conjecture 1.1] and of Iwaniec, Koskela and Martin [T. Iwaniec, P. Koskela, G. Martin, Mappings of BMO-distortion and Beltrami-type operators, J. Anal. Math. 88 (2002) 337–381, Conjecture 7.1].