Optimal regularity for planar mappings of finite distortion

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

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

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010), p. 1-19 / Harvested from Numdam

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Résumé

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

Publié le : 2010-01-01

Zbl 1191.30007

DOI : https://doi.org/10.1016/j.anihpc.2009.01.012

@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},
     volume = {27},
     year = {2010},
     pages = {1-19},
     doi = {10.1016/j.anihpc.2009.01.012},
     zbl = {1191.30007},
     language = {en},
     url = {http://dml.mathdoc.fr/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, Tome 27 (2010) pp. 1-19. doi : 10.1016/j.anihpc.2009.01.012. http://gdmltest.u-ga.fr/item/AIHPC_2010__27_1_1_0/

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