Density estimates for a nonlocal variational model via the Sobolev inequality

TL;DR

This paper proves that minimizers of a nonlocal energy functional with a double-well potential have sublevel sets occupying a volume proportional to the domain, using a fractional Sobolev inequality for a new proof.

Contribution

The paper introduces a novel proof technique employing fractional Sobolev inequalities to analyze volume estimates of minimizers in nonlocal variational models.

Findings

Sublevel sets of minimizers occupy a volume comparable to the domain.

The proof utilizes fractional Sobolev inequalities.

Results apply to minimizers in large balls.

Abstract

We consider the minimizers of the energy $$ \|u\|_{H^s(\Omega)}^2+\int_\Omega W(u)\,dx,$$ with $s \in (0,1/2)$, where $\|u\|_{H^s(\Omega)}$ 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 $B_R$.