Stable s-minimal cones in R2 are flat for s∼0

Abstract

For $s\in (0,1)$ small, we show that the only cones in $R^2$ stationary for the $s$-perimeter and stable in $R^2\setminus \left\{0\right\}$ 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 $\{xy>0\}\subset R^2$ are stable for inner variations.