Mappings of finite distortion: the degree of regularity |
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Authors: | Daniel Faraco Pekka Koskela Xiao Zhong |
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Institution: | a Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, Leipzig D-04103, Germany;b Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, FIN-40014, Finland |
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Abstract: | This paper investigates the self-improving integrability properties of the so-called mappings of finite distortion. Let K(x)1 be a measurable function defined on a domain ΩRn, n2, and such that exp(βK(x))Lloc1(Ω), β>0. We show that there exist two universal constants c1(n),c2(n) with the following property: Let f be a mapping in Wloc1,1(Ω,Rn) with |Df(x)|nK(x)J(x,f) for a.e. xΩ and such that the Jacobian determinant J(x,f) is locally in L1 log−c1(n)βL. Then automatically J(x,f) is locally in L1 logc2(n)βL(Ω). This result constitutes the appropriate analog for the self-improving regularity of quasiregular mappings and clarifies many other interesting properties of mappings of finite distortion. Namely, we obtain novel results on the size of removable singularities for bounded mappings of finite distortion, and on the area distortion under this class of mappings. |
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Keywords: | 30C65 26B10 73C50 |
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