首页/论文/2011.11562

arXiv 2011.115622020-11-30math.FAmath.APmath.MG

Fractional perimeters on the sphere

Andreas Kreuml, Olaf Mordhorst

备注：: Page 2 and 7 refernce for Blaschke-Petkantchin formula added

英文摘要：

This note treats several problems for the fractional perimeter or $s$-perimeter on the sphere. The spherical fractional isoperimetric inequality is established. It turns out that the equality cases are exactly the spherical caps. Furthermore, the convergence of fractional perimeters to the surface area as $s \nearrow 1$ is proven. It is shown that their limit as $s \searrow -\infty$ can be expressed in terms of the volume.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Anisotropic fractional perimeters

The anisotropic fractional $s$-perimeter with respect to a convex body $K$ in $\rn$ is shown to converge to the anisotropic perimeter with respect to the moment body of $K$ as $s\to 1^-$. A corresponding result is established for anisotropic fractional $s$-seminorms on $BV(\rn)$ (generalizing results of Bourgain, Brezis & Mironescu and Dávila). The minimizers of the anisotropic fractional $s$-isoperimetric inequality with respect to $K$ are shown to converge to the moment body of $K$. Anisotropic fractional Sobolev inequalities are established.

2013-06-06math.APmath.FAmath.MG

A quantitative isoperimetric inequality for fractional perimeters

Recently Frank and Seiringer have shown an isoperimetric inequality for nonlocal perimeter functionals arising from Sobolev seminorms of fractional order. This isoperimetric inequality is improved here in a quantitative form.

2011-02-11math.APmath.FA

关于具有有限分布分数周长的集合

On sets with finite distributional fractional perimeter

本文研究局部有限分布分数周长集合的精细性质，改进其爆破刻画，证明其与有限分数周长集合相交的Leibniz法则，并描述相关非局部边界。

2024-01-18math.FA

A note on the fractional perimeter and interpolation

We present the fractional perimeter as a set-function interpolation between the Lebesgue measure and the perimeter in the sense of De Giorgi. Our motivation comes from a new fractional Boxing inequality that relates the fractional perimeter and the Hausdorff content and implies several known inequalities involving the Gagliardo seminorm of the Sobolev spaces $W^{α, 1}$ of order $0 < α< 1$.

2017-07-19math.FA

The $0$-fractional perimeter between fractional perimeters and Riesz potentials

This paper provides a unified point of view on fractional perimeters and Riesz potentials. Denoting by $H^σ$ - for $σ\in (0,1)$ - the $σ$-fractional perimeter and by $J^σ$ - for $σ\in (-d,0)$ - the $σ$-Riesz energies acting on characteristic functions, we prove that both functionals can be seen as limits of renormalized self-attractive energies as well as limits of repulsive interactions between a set and its complement. We also show that the functionals $H^σ$ and $J^σ$\,, up to a suitable additive renormalization diverging when $σ\to 0$, belong to a continuous one-parameter family of function…

2019-06-14math.FAmath.OC

A distributional approach to fractional Sobolev spaces and fractional variation: existence of blow-up

We introduce the new space $BV^α(\mathbb{R}^n)$ of functions with bounded fractional variation in $\mathbb{R}^n$ of order $α\in (0, 1)$ via a new distributional approach exploiting suitable notions of fractional gradient and fractional divergence already existing in the literature. In analogy with the classical $BV$ theory, we give a new notion of set $E$ of (locally) finite fractional Caccioppoli $α$-perimeter and we define its fractional reduced boundary $\mathscr{F}^α E$. We are able to show that $W^{α,1}(\mathbb{R}^n)\subset BV^α(\mathbb{R}^n)$ continuously and, similarly, that sets with (…

2019-10-29math.FA

更多相关论文收起更多论文

与分数周长相关的几何不等式：度量测度空间中的分数Poincaré、等周与boxing不等式

Geometric inequalities related to fractional perimeter: fractional Poincaré, isoperimetric, and boxing inequalities in metric measure spaces

本文在支撑$(1,1)$-Poincaré不等式的度量测度空间中建立了带缩放常数$(1-\theta)$的分数Poincaré不等式，并由此导出分数等周与boxing不等式，进而刻画了Poincaré不等式和下Ahlfors正则性。

2025-11-06math.FAmath.MG

分数阶周长的 Boxing 不等式

A Boxing Inequality for the Fractional Perimeter

本文证明了分数阶周长的 Boxing 不等式，并据此推导出分数阶 Sobolev 空间中的迹不等式，统一包含 Sobolev 嵌入、Lorentz 空间改进及 Hardy 不等式，且在 $\alpha$ 趋于 $0$ 和 $1$ 时保持适当渐近性并恢复经典一阶不等式。

2018-02-28math.APmath.FA

带BBM因子的John域上相对分数阶周长的Boxing不等式与分数阶Poincaré型不等式

Boxing inequalities for relative fractional perimeter and fractional Poincaré-type inequalities on John domains with the BBM factor

Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University（湖南师范大学数学与统计学院计算与随机数学教育部重点实验室）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）

本文针对s-John域证明了带BBM因子的相对分数阶周长Boxing不等式及其泛函形式，推导了对应分数阶Poincaré型系列不等式，验证了John域条件的最优性，相关结果在Lipschitz域上也属全新。

2026-04-23math.APmath.CVmath.FA

A derivation of the sharp Moser-Trudinger-Onofri inequalities from the fractional Sobolev inequalities

We derive the sharp Moser-Trudinger-Onofri inequalities on the standard $n$-sphere and CR $(2n+1)$- sphere as the limit of the sharp fractional Sobolev inequalities for all $n\ge 1$. On the $2$-sphere and $4$-sphere, this was established recently by S.-Y. Chang and F. Wang. Our proof uses an alternative and elementary argument.

2018-09-14math.APmath.DGmath.FA

A spherical rearrangement proof of the stability of a Riesz-type inequality and an application to an isoperimetric type problem

We prove the stability of the ball as global minimizer of an attractive shape functional under volume constraint, by means of mass transportation arguments. The stability exponent is $1/2$ and it is sharp. Moreover, we use such stability result together with the quantitative (possibly fractional) isoperimetric inequality to prove that the ball is a global minimizer of a shape functional involving both an attractive and a repulsive term with a sufficiently large fixed volume and with a suitable (possibly fractional) perimeter penalization.

2021-10-19math.APmath.FA

Fractional Laplacians on the sphere, the Minakshisundaram zeta function and semigroups

In this paper we show novel underlying connections between fractional powers of the Laplacian on the unit sphere and functions from analytic number theory and differential geometry, like the Hurwitz zeta function and the Minakshisundaram zeta function. Inspired by Minakshisundaram's ideas, we find a precise pointwise description of $(-Δ_{\mathbb{S}^{n-1}})^s u(x)$ in terms of fractional powers of the Dirichlet-to-Neumann map on the sphere. The Poisson kernel for the unit ball will be essential for this part of the analysis. On the other hand, by using the heat semigroup on the sphere, addition…

2018-06-20math.APmath.CAmath.DG

Fractional div-curl quantities and applications to nonlocal geometric equations

We investigate a fractional notion of gradient and divergence operator. We generalize the div-curl estimate by Coifman-Lions-Meyer-Semmes to fractional div-curl quantities, obtaining, in particular, a nonlocal version of Wente's lemma. We demonstrate how these quantities appear naturally in nonlocal geometric equations, which can be used to obtain a theory for fractional harmonic maps analogous to the local theory. Firstly, regarding fractional harmonic maps into spheres, we obtain a conservation law analogous to Shatah's conservation law and give a new regularity proof analogous to Hélein's f…

2017-03-01math.APmath.FA

Hypercontractivity of the semigroup of the fractional laplacian on the n-sphere

For $1<p\leq q$ we show that the Poisson semigroup $e^{-t\sqrt{-Δ}}$ on the $n$-sphere is hypercontractive from $L^{p}$ to $L^{q}$ in dimensions $n \leq 3$ if and only if $e^{-t\sqrt{n}} \leq \sqrt{\frac{p-1}{q-1}}$. We also show that the equivalence fails in large dimensions.

2021-01-15math.CAmath.FA