Sobolev regularity of quasiconformal mappings on domains

  • Martí PratsEmail author


Consider a Lipschitz domain Ω and a measurable function μ supported in \(\overline{\Omega}\)with ‖μL < 1. Then the derivatives of a quasiconformal solution of the Beltrami equation \(\overline{\partial}f=\mu\;\partial{f}\) inherit the Sobolev regularity Wn,p(Ω) of the Beltrami coefficient μ as long as Ω is regular enough. The condition obtained is that the outward unit normal vector N of the boundary of the domain is in the trace space, that is, \(N\in{B}_{p,p}^{n-1/p}(\partial\Omega)\).


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The author was funded by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement 320501. Also, he was partially supported by grants 2014-SGR-75 (Generalitat de Catalunya), MTM-2010-16232 and MTM-2013-44304-P (Spanish government) and by a FI-DGR grant from the Generalitat de Catalunya, (2014FI-B2 00107).

The author would like to thank Xavier Tolsa for advice on his Ph.D. thesis, which gave rise to this work; Cruz,Mateu, Orobitg andVerdera for their advice and interest; and the editor and referee for their patient work and valuable comments.


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© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaCataloniaSpain

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