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

Paz Hashash; Alexander Ukhlov

doi:https://doi.org/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

Ben-Gurion University of the Negev

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	American 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

All Science Journal Classification (ASJC) codes

Analysis
Applied Mathematics

Access to Document

https://doi.org/10.1016/j.jmaa.2021.125659

Cite this

@article{f5976da97b5346b4be66d1bb3160c4ba,

title = "On Lipschitz approximations in second order Sobolev spaces and the change of variables formula",

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.",

keywords = "Geometric measure theory, Potential theory, Sobolev spaces",

author = "Paz Hashash and Alexander Ukhlov",

note = "Publisher Copyright: {\textcopyright} 2021 Elsevier Inc.",

year = "2022",

month = feb,

day = "15",

doi = "https://doi.org/10.1016/j.jmaa.2021.125659",

language = "American English",

volume = "506",

journal = "Journal of Mathematical Analysis and Applications",

issn = "0022-247X",

publisher = "Academic Press Inc.",

number = "2",

}

TY - JOUR

T1 - On Lipschitz approximations in second order Sobolev spaces and the change of variables formula

AU - Hashash, Paz

AU - Ukhlov, Alexander

PY - 2022/2/15

Y1 - 2022/2/15

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.

KW - Geometric measure theory

KW - Potential theory

KW - Sobolev spaces

UR - http://www.scopus.com/inward/record.url?scp=85115245106&partnerID=8YFLogxK

U2 - https://doi.org/10.1016/j.jmaa.2021.125659

DO - https://doi.org/10.1016/j.jmaa.2021.125659

M3 - Article

SN - 0022-247X

VL - 506

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

M1 - 125659

ER -

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

Abstract

Keywords

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Fingerprint

Cite this