Density estimates for a nonlocal variational model with a degenerate double-well potential via the Sobolev inequality

TL;DR

This paper establishes density estimates for level sets of minimizers in a nonlocal variational model with degenerate double-well potentials, using Sobolev inequalities and iteration techniques, applicable even for the fractional Laplacian.

Contribution

It introduces new barrier methods and analytical techniques to handle degenerate potentials in nonlocal models, extending results to the fractional p-Laplacian case.

Findings

Density estimates for level sets are proven.

Results apply to degenerate double-well potentials with polynomial growth.

Method is robust for quasilinear nonlocal equations.

Abstract

We provide density estimates for level sets of minimizers of the energy $\frac{1}{2} \int_{\Omega}\int_{\Omega} \frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}dxdy+\int_{\Omega}\int_{\mathbb{R}^n\setminus\Omega} \frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}dxdy+\int_{\Omega}W(u(x))dx$ where $p\in(1,+\infty)$ and $s\in\left(0,\frac{1}{p}\right)$ and $W$ is a double-well potential with polynomial growth $m\in [p,+\infty)$ from the minima. These kinds of potentials are ''degenerate'', since they detach ''slowly'' from the minima, therefore they provide additional difficulties if one wishes to determine the relative sizes of the ''layers'' and the ''pure phases''. To overcome these challenges, we introduce new barriers allowing us to rely on the fractional Sobolev inequality and on a suitable iteration method. The proofs presented here are robust enough to consider the case of quasilinear nonlocal equations driven by the fractional $p$-Laplacian, but our results are new even for the case $p=2$.