Density estimates for a nonlocal variational model via the Sobolev inequality

Abstract

We consider the minimizers of the energy $$\Vert u{\Vert }_{H^s(\Omega )}^2+{\int }_{\Omega }W(u)dx,$$ with $s\in (0,1/2)$, where $\Vert u{\Vert }_{H^s(\Omega )}$ denotes the total contribution from $\Omega$ in the $H^s$ norm of $u$, and $W$ is a double-well potential. By using a fractional Sobolev inequality, we give a new proof of the fact that the sublevel sets of a minimizer $u$ in a large ball $B_R$ occupy a volume comparable with the volume of $B_R$.