Optimal regularity for planar mappings of finite distortion

Astala, Kari  ; Gill, James T. ; Rohde, Steffen ; Saksman, Eero 

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 1, pp. 1-19

Résumé

Let $f:\Omega \to {ℝ}^2$ be a mapping of finite distortion, where $\Omega \subset {ℝ}^2$. Assume that the distortion function $K(x,f)$ satisfies $e^{K(·,f)}\in L_{\mathrm{𝑙𝑜𝑐}}^p\left(\Omega \right)$ for some $p>0$. We establish optimal regularity and area distortion estimates for f. In particular, we prove that ${\left|Df\right|}^2{\log }^{\beta -1}(e+|Df\left|\right)\in L_{\mathrm{𝑙𝑜𝑐}}^1\left(\Omega \right)$ 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].

Détail

DOI : 10.1016/j.anihpc.2009.01.012

Keywords: Mappings of finite distortion, Exponential distortion, Optimal regularity, Area distortion