Astala, Kari; Gill, James T.; Rohde, Steffen; Saksman, Eero
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
Zbl 1191.30007
doi : 10.1016/j.anihpc.2009.01.012
URL stable : http://www.numdam.org/item?id=AIHPC_2010__27_1_1_0

Let f:Ω 2 be a mapping of finite distortion, where Ω 2 . Assume that the distortion function K(x,f) satisfies e K(·,f) L 𝑙𝑜𝑐 p (Ω) for some p>0. We establish optimal regularity and area distortion estimates for f. In particular, we prove that |Df| 2 log β-1 (e+|Df|)L 𝑙𝑜𝑐 1 (Ω) for every β<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].

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