Optimal regularity for planar mappings of finite distortion

  • Kari Astala

    Department of Mathematics and Statistics, University of Helsinki, PO Box 68, Gustaf Hällströmin katu 2b, FI-00014 University of Helsinki, Finland
  • James T. Gill

    Department of Mathematics, University of Washington, C-337 Padelford Hall, Box 354350, Seattle, WA 98195-4350, USA
  • Steffen Rohde

    Department of Mathematics, University of Washington, C-337 Padelford Hall, Box 354350, Seattle, WA 98195-4350, USA
  • Eero Saksman

    Department of Mathematics and Statistics, University of Helsinki, PO Box 68, Gustaf Hällströmin katu 2b, FI-00014 University of Helsinki, Finland

Abstract

Let be a mapping of finite distortion, where . Assume that the distortion function satisfies for some . We establish optimal regularity and area distortion estimates for . In particular, we prove that for every . This answers positively, in dimension , 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].

Cite this article

Kari Astala, James T. Gill, Steffen Rohde, Eero Saksman, Optimal regularity for planar mappings of finite distortion. Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), no. 1, pp. 1–19

DOI 10.1016/J.ANIHPC.2009.01.012