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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.07543 |
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| _version_ | 1866918489354141696 |
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| author | Alberti, G. Cozzi, G. Massaccesi, A. Mirmina, J. |
| author_facet | Alberti, G. Cozzi, G. Massaccesi, A. Mirmina, J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_07543 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stability of the ball in isoperimetric inequalities between two fractional perimeters Alberti, G. Cozzi, G. Massaccesi, A. Mirmina, J. Analysis of PDEs 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. |
| title | Stability of the ball in isoperimetric inequalities between two fractional perimeters |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.07543 |