Optimal regularity for planar mappings of finite distortion

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

doi:10.1016/j.anihpc.2009.01.012

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 $f : Ω \to ℝ^{2}$ 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].

Zbl

DOI : 10.1016/j.anihpc.2009.01.012

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

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     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},
     pages = {1--19},
     year = {2010},
     publisher = {Elsevier},
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     doi = {10.1016/j.anihpc.2009.01.012},
     zbl = {1191.30007},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2009.01.012/}
}

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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, pp. 1-19. doi: 10.1016/j.anihpc.2009.01.012

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