{"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。
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).
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.