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Mappings of Lp-integrable distortion: regularity of the inverse

Part of: Functions of several variables Classical measure theory Geometric function theory

Published online by Cambridge University Press:  16 May 2016

Jani Onninen  and
Ville Tengvall
Jani Onninen
Affiliation:
Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35 (MaD), 40014University of Jyväskylä, Finland (jani.onninen@jyu.fi; ville.tengvall@jyu.fi)
Ville Tengvall
Affiliation:
Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35 (MaD), 40014University of Jyväskylä, Finland (jani.onninen@jyu.fi; ville.tengvall@jyu.fi)
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Extract

Let be an open set in ℝn and suppose that is a Sobolev homeomorphism. We study the regularity of f–1 under the Lp-integrability assumption on the distortion function Kf. First, if is the unit ball and p > n – 1, then the optimal local modulus of continuity of f–1 is attained by a radially symmetric mapping. We show that this is not the case when p ⩽ n – 1 and n ⩾ 3, and answer a question raised by S. Hencl and P. Koskela. Second, we obtain the optimal integrability results for ∣Df–1∣ in terms of the Lp-integrability assumptions of Kf.

Keywords

mappings of finite distortionregularity of the inversemodulus of continuityhigher integrabilitySobolev homeomorphism

MSC classification

Secondary: 30C65: Quasiconformal mappings in $^n$, other generalizations 28A78: Hausdorff and packing measures 26B10: Implicit function theorems, Jacobians, transformations with several variables
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 146 , Issue 3 , June 2016 , pp. 647 - 663
Copyright
Copyright © Royal Society of Edinburgh 2016 

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