Quasiconformal, Lipschitz, and BV mappings in metric spaces

IF 1.3 3区数学 Q1 MATHEMATICS

Advances in Calculus of Variations Pub Date : 2024-01-12 DOI:10.1515/acv-2022-0071

Panu Lahti

{"title":"Quasiconformal, Lipschitz, and BV mappings in metric spaces","authors":"Panu Lahti","doi":"10.1515/acv-2022-0071","DOIUrl":null,"url":null,"abstract":"Consider a mapping <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>f</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>→</m:mo> <m:mi>Y</m:mi> </m:mrow> </m:mrow> </m:math> {f\\colon X\\to Y} between two metric measure spaces. We study generalized versions of the local Lipschitz number <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>Lip</m:mi> <m:mo>⁡</m:mo> <m:mi>f</m:mi> </m:mrow> </m:math> {\\operatorname{Lip}f} , as well as of the distortion number <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>H</m:mi> <m:mi>f</m:mi> </m:msub> </m:math> {H_{f}} that is used to define quasiconformal mappings. Using these numbers, we give sufficient conditions for f being a BV mapping <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msub> <m:mi>BV</m:mi> <m:mi>loc</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>X</m:mi> <m:mo>;</m:mo> <m:mi>Y</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {f\\in\\mathrm{BV}_{\\mathrm{loc}}(X;Y)} or a Newton–Sobolev mapping <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msubsup> <m:mi>N</m:mi> <m:mi>loc</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>X</m:mi> <m:mo>;</m:mo> <m:mi>Y</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {f\\in N_{\\mathrm{loc}}^{1,p}(X;Y)} , with <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mi>p</m:mi> <m:mo><</m:mo> <m:mi mathvariant=\"normal\">∞</m:mi> </m:mrow> </m:math> {1\\leq p<\\infty} .","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"45 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Calculus of Variations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/acv-2022-0071","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

Consider a mapping f : X → Y

{f\colon X\to Y}

between two metric measure spaces. We study generalized versions of the local Lipschitz number Lip ⁡ f

{\operatorname{Lip}f}

, as well as of the distortion number H f

{H_{f}}

that is used to define quasiconformal mappings. Using these numbers, we give sufficient conditions for f being a BV mapping f ∈ BV loc ⁢ ( X ; Y )

{f\in\mathrm{BV}_{\mathrm{loc}}(X;Y)}

or a Newton–Sobolev mapping f ∈ N loc 1 , p ⁢ ( X ; Y )

{f\in N_{\mathrm{loc}}^{1,p}(X;Y)}

, with 1 ≤ p < ∞

{1\leq p<\infty}

查看原文本刊更多论文

度量空间中的准共形、Lipschitz 和 BV 映射

考虑两个度量空间之间的映射 f : X → Y {f\colon X\to Y} 。我们研究局部 Lipschitz 数 Lip f {\operatorname{Lip}f} 的广义版本，以及用于定义准共形映射的变形数 H f {H_{f}} 的广义版本。利用这些数字，我们给出了 f 是 BV 映射 f∈ BV loc ( X ; Y ) {f\in\mathrm{BV}_{\mathrm{loc}}(X. Y)} 或牛顿映射 f∈ BV loc ( X ; Y ) {f\in\mathrm{BV}_{\mathrm{loc}}(X. Y)} 的充分条件；Y)} 或者牛顿-索博列夫映射 f∈ N loc 1 , p ( X ; Y ) {f\in N_{\mathrm{loc}}^{1,p}(X;Y)} , 其中 1 ≤ p < ∞ {1\leq p<\infty} 。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Advances in Calculus of Variations MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

3.90

自引率

5.90%

发文量

审稿时长

>12 weeks

期刊介绍： Advances in Calculus of Variations publishes high quality original research focusing on that part of calculus of variation and related applications which combines tools and methods from partial differential equations with geometrical techniques.