Stable $s$-minimal cones in $\mathbb{R}^2$ are flat for $s \sim 0$

Michele Caselli

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arXiv · 2412.06318

Stable $s$-minimal cones in $\mathbb{R}^2$ are flat for $s \sim 0$

Michele Caselli

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Abstract

For $s \in (0,1)$ small, we show that the only cones in $\mathbb{R}^2$ stationary for the $s$-perimeter and stable in $\mathbb{R}^2 \setminus \{0\}$ are half-planes. This is in direct contrast with the case of the classical perimeter or the regime $s$ close to $1$, where nontrivial cones as $\{xy>0\} \subset \mathbb{R}^2$ are stable for inner variations.

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Michele Caselli. 2024-12-09. Stable $s$-minimal cones in $\mathbb{R}^2$ are flat for $s \sim 0$. https://arxiv.org/abs/2412.06318

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