Optimal regularity for planar mappings of finite distortion
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 1 , p. 1-19
doi : 10.1016/j.anihpc.2009.01.012
URL stable : http://www.numdam.org/item?id=AIHPC_2010__27_1_1_0

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

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