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arXiv · 2605.07543

Stability of the ball in isoperimetric inequalities between two fractional perimeters

G. Alberti·G. Cozzi·A. Massaccesi·J. Mirmina
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Abstract

We consider the isoperimetric inequality involving the $s$-perimeter and the $t$-perimeter with $0<s<t<1$, and show that the ball is a local minimizer of the (scale-invariant) isoperimetric ratio $\mathcal{F}(E):=P_t(E)^{\frac{1}{n-t}}/ P_s(E)^{\frac{1}{n-s}}$ among sets $E$ that are nearly spherical. To this end, we rewrite $\mathcal{F}$ as a functional of $u$, where $u$ is a scalar function on the unit sphere in $\mathbb{R}^n$ that parametrizes the boundary of $E$, and prove a quantitative stability result for $\mathcal{F}$ around $u=0$ with respect to a suitable Sobolev norm. This parallels known results where the $s$-perimeter is replaced by the volume.

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G. Alberti, G. Cozzi, A. Massaccesi, J. Mirmina. 2026-05-08. Stability of the ball in isoperimetric inequalities between two fractional perimeters. https://arxiv.org/abs/2605.07543

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