Optimal regularity for planar mappings of finite distortion
Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 1, p. 1-19

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].

DOI : https://doi.org/10.1016/j.anihpc.2009.01.012
Keywords: Mappings of finite distortion, Exponential distortion, Optimal regularity, Area distortion
@article{AIHPC_2010__27_1_1_0,
author = {Astala, Kari and Gill, James T. and Rohde, Steffen and Saksman, Eero},
title = {Optimal regularity for planar mappings of finite distortion},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {27},
number = {1},
year = {2010},
pages = {1-19},
doi = {10.1016/j.anihpc.2009.01.012},
zbl = {1191.30007},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2010__27_1_1_0}
}

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, Volume 27 (2010) no. 1, pp. 1-19. doi : 10.1016/j.anihpc.2009.01.012. http://www.numdam.org/item/AIHPC_2010__27_1_1_0/

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