黎曼流形上的分数索波列夫空间

IF 1.3 2区 数学 Q1 MATHEMATICS Mathematische Annalen Pub Date : 2024-06-03 DOI:10.1007/s00208-024-02894-w
Michele Caselli, Enric Florit-Simon, Joaquim Serra
{"title":"黎曼流形上的分数索波列夫空间","authors":"Michele Caselli, Enric Florit-Simon, Joaquim Serra","doi":"10.1007/s00208-024-02894-w","DOIUrl":null,"url":null,"abstract":"<p>This article studies the canonical Hilbert energy <span>\\(H^{s/2}(M)\\)</span> on a Riemannian manifold for <span>\\(s\\in (0,2)\\)</span>, with particular focus on the case of closed manifolds. Several equivalent definitions for this energy and the fractional Laplacian on a manifold are given, and they are shown to be identical up to explicit multiplicative constants. Moreover, the precise behavior of the kernel associated with the singular integral definition of the fractional Laplacian is obtained through an in-depth study of the heat kernel on a Riemannian manifold. Furthermore, a monotonicity formula for stationary points of functionals of the type <span>\\({\\mathcal {E}}(v)=[v]^2_{H^{s/2}(M)}+\\int _M F(v) \\, dV\\)</span>, with <span>\\(F\\ge 0\\)</span>, is given, which includes in particular the case of nonlocal <i>s</i>-minimal surfaces. Finally, we prove some estimates for the Caffarelli–Silvestre extension problem, which are of general interest. This work is motivated by Caselli et al. (Yau’s conjecture for nonlocal minimal surfaces, arxiv preprint, 2023), which defines nonlocal minimal surfaces on closed Riemannian manifolds and shows the existence of infinitely many of them for any metric on the manifold, ultimately proving the nonlocal version of a conjecture of Yau (Ann Math Stud 102:669–706, 1982). Indeed, the definitions and results in the present work serve as an essential technical toolbox for the results in Caselli et al. (Yau’s conjecture for nonlocal minimal surfaces, arxiv preprint, 2023).</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Fractional Sobolev spaces on Riemannian manifolds\",\"authors\":\"Michele Caselli, Enric Florit-Simon, Joaquim Serra\",\"doi\":\"10.1007/s00208-024-02894-w\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>This article studies the canonical Hilbert energy <span>\\\\(H^{s/2}(M)\\\\)</span> on a Riemannian manifold for <span>\\\\(s\\\\in (0,2)\\\\)</span>, with particular focus on the case of closed manifolds. Several equivalent definitions for this energy and the fractional Laplacian on a manifold are given, and they are shown to be identical up to explicit multiplicative constants. Moreover, the precise behavior of the kernel associated with the singular integral definition of the fractional Laplacian is obtained through an in-depth study of the heat kernel on a Riemannian manifold. Furthermore, a monotonicity formula for stationary points of functionals of the type <span>\\\\({\\\\mathcal {E}}(v)=[v]^2_{H^{s/2}(M)}+\\\\int _M F(v) \\\\, dV\\\\)</span>, with <span>\\\\(F\\\\ge 0\\\\)</span>, is given, which includes in particular the case of nonlocal <i>s</i>-minimal surfaces. Finally, we prove some estimates for the Caffarelli–Silvestre extension problem, which are of general interest. This work is motivated by Caselli et al. (Yau’s conjecture for nonlocal minimal surfaces, arxiv preprint, 2023), which defines nonlocal minimal surfaces on closed Riemannian manifolds and shows the existence of infinitely many of them for any metric on the manifold, ultimately proving the nonlocal version of a conjecture of Yau (Ann Math Stud 102:669–706, 1982). Indeed, the definitions and results in the present work serve as an essential technical toolbox for the results in Caselli et al. (Yau’s conjecture for nonlocal minimal surfaces, arxiv preprint, 2023).</p>\",\"PeriodicalId\":18304,\"journal\":{\"name\":\"Mathematische Annalen\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-06-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematische Annalen\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00208-024-02894-w\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematische Annalen","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00208-024-02894-w","RegionNum":2,"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

摘要

这篇文章研究了黎曼流形上的\(s\in (0,2)\)的典型希尔伯特能(H^{s/2}(M)\),尤其关注封闭流形的情况。给出了该能量与流形上分数拉普拉奇的几个等价定义,并证明它们在明确的乘法常数范围内是相同的。此外,通过深入研究黎曼流形上的热核,还获得了与分数拉普拉奇奇异积分定义相关的核的精确行为。此外,我们还给出了具有 \(F\ge 0\) 的函数类型 \({\mathcal {E}}(v)=[v]^2_{H^{s/2}(M)}+\int _M F(v) \, dV\) 的静止点的单调性公式,其中特别包括非局部 s 最小曲面的情况。最后,我们证明了对 Caffarelli-Silvestre 扩展问题的一些估计,这些估计具有普遍意义。这项工作受 Caselli 等人(Yau's conjecture for nonlocal minimal surfaces, arxiv preprint, 2023)的启发,他们定义了封闭黎曼流形上的非局部极小曲面,并证明了对于流形上的任意度量,存在无限多的非局部极小曲面,最终证明了 Yau 猜想的非局部版本(Ann Math Stud 102:669-706, 1982)。事实上,本研究中的定义和结果是卡塞利等人(Yau's conjecture for nonlocal minimal surfaces, arxiv preprint, 2023)成果的重要技术工具箱。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
Fractional Sobolev spaces on Riemannian manifolds

This article studies the canonical Hilbert energy \(H^{s/2}(M)\) on a Riemannian manifold for \(s\in (0,2)\), with particular focus on the case of closed manifolds. Several equivalent definitions for this energy and the fractional Laplacian on a manifold are given, and they are shown to be identical up to explicit multiplicative constants. Moreover, the precise behavior of the kernel associated with the singular integral definition of the fractional Laplacian is obtained through an in-depth study of the heat kernel on a Riemannian manifold. Furthermore, a monotonicity formula for stationary points of functionals of the type \({\mathcal {E}}(v)=[v]^2_{H^{s/2}(M)}+\int _M F(v) \, dV\), with \(F\ge 0\), is given, which includes in particular the case of nonlocal s-minimal surfaces. Finally, we prove some estimates for the Caffarelli–Silvestre extension problem, which are of general interest. This work is motivated by Caselli et al. (Yau’s conjecture for nonlocal minimal surfaces, arxiv preprint, 2023), which defines nonlocal minimal surfaces on closed Riemannian manifolds and shows the existence of infinitely many of them for any metric on the manifold, ultimately proving the nonlocal version of a conjecture of Yau (Ann Math Stud 102:669–706, 1982). Indeed, the definitions and results in the present work serve as an essential technical toolbox for the results in Caselli et al. (Yau’s conjecture for nonlocal minimal surfaces, arxiv preprint, 2023).

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
Mathematische Annalen
Mathematische Annalen 数学-数学
CiteScore
2.90
自引率
7.10%
发文量
181
审稿时长
4-8 weeks
期刊介绍: Begründet 1868 durch Alfred Clebsch und Carl Neumann. Fortgeführt durch Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück und Nigel Hitchin. The journal Mathematische Annalen was founded in 1868 by Alfred Clebsch and Carl Neumann. It was continued by Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguigon, Wolfgang Lück and Nigel Hitchin. Since 1868 the name Mathematische Annalen stands for a long tradition and high quality in the publication of mathematical research articles. Mathematische Annalen is designed not as a specialized journal but covers a wide spectrum of modern mathematics.
期刊最新文献
Coarsely holomorphic curves and symplectic topology On the uniqueness of periodic solutions for a Rayleigh–Liénard system with impulses Multifractality and intermittency in the limit evolution of polygonal vortex filaments Uniformly super McDuff $$\hbox {II}_1$$ factors Normalized solutions for Kirchhoff equations with Sobolev critical exponent and mixed nonlinearities
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1