On Lipschitz approximations in second order Sobolev spaces and the change of variables formula

Paz Hashash; Alexander Ukhlov

doi:10.1016/j.jmaa.2021.125659

On Lipschitz approximations in second order Sobolev spaces and the change of variables formula

Paz Hashash, Alexander Ukhlov

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper we study approximations of functions of Sobolev spaces W_p,loc²(Ω), Ω⊂Rⁿ, by Lipschitz continuous functions. We prove that if f∈W_p,loc²(Ω), 1≤p<∞, then there exists a sequence of closed sets {A_k}_k=1^∞,A_k⊂A_k+1⊂Ω, such that the restrictions f|_{A_k} are Lipschitz continuous functions and cap_p(S)=0, S=Ω∖⋃_k=1^∞A_k. Using these approximations we prove the change of variables formula in the Lebesgue integral for mappings of Sobolev spaces W_p,loc²(Ω;Rⁿ) with the Luzin capacity-measure N-property.

Original language	English
Article number	125659
Journal	Journal of Mathematical Analysis and Applications
Volume	506
Issue number	2
DOIs	https://doi.org/10.1016/j.jmaa.2021.125659
State	Published - 15 Feb 2022

Keywords

Geometric measure theory
Potential theory
Sobolev spaces

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.jmaa.2021.125659

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abstract = "In this paper we study approximations of functions of Sobolev spaces Wp,loc2(Ω), Ω⊂Rn, by Lipschitz continuous functions. We prove that if f∈Wp,loc2(Ω), 1≤p<∞, then there exists a sequence of closed sets {Ak}k=1∞,Ak⊂Ak+1⊂Ω, such that the restrictions f|Ak are Lipschitz continuous functions and capp(S)=0, S=Ω∖⋃k=1∞Ak. Using these approximations we prove the change of variables formula in the Lebesgue integral for mappings of Sobolev spaces Wp,loc2(Ω;Rn) with the Luzin capacity-measure N-property.",

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AU - Ukhlov, Alexander

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N2 - In this paper we study approximations of functions of Sobolev spaces Wp,loc2(Ω), Ω⊂Rn, by Lipschitz continuous functions. We prove that if f∈Wp,loc2(Ω), 1≤p<∞, then there exists a sequence of closed sets {Ak}k=1∞,Ak⊂Ak+1⊂Ω, such that the restrictions f|Ak are Lipschitz continuous functions and capp(S)=0, S=Ω∖⋃k=1∞Ak. Using these approximations we prove the change of variables formula in the Lebesgue integral for mappings of Sobolev spaces Wp,loc2(Ω;Rn) with the Luzin capacity-measure N-property.

AB - In this paper we study approximations of functions of Sobolev spaces Wp,loc2(Ω), Ω⊂Rn, by Lipschitz continuous functions. We prove that if f∈Wp,loc2(Ω), 1≤p<∞, then there exists a sequence of closed sets {Ak}k=1∞,Ak⊂Ak+1⊂Ω, such that the restrictions f|Ak are Lipschitz continuous functions and capp(S)=0, S=Ω∖⋃k=1∞Ak. Using these approximations we prove the change of variables formula in the Lebesgue integral for mappings of Sobolev spaces Wp,loc2(Ω;Rn) with the Luzin capacity-measure N-property.

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KW - Potential theory

KW - Sobolev spaces

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On Lipschitz approximations in second order Sobolev spaces and the change of variables formula

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Keywords

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