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

Michele Caselli

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

Analysis of PDEs 2025-04-09 v3

Authors: Michele Caselli

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.

Keywords

stability theory

Cite

@article{arxiv.2412.06318,
  title  = {Stable $s$-minimal cones in $\mathbb{R}^2$ are flat for $s \sim 0$},
  author = {Michele Caselli},
  journal= {arXiv preprint arXiv:2412.06318},
  year   = {2025}
}

Comments

Introduction expanded adding Subsection 1.1 (Min-max curves and model singularities) and Subsection 1.2 (On the different notions of stability). Added references and fixed typos

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

Abstract

Keywords

Cite

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