$\Gamma$-convergence for nonlocal phase transitions

TL;DR

This paper studies the asymptotic behavior of a family of nonlocal energy functionals involving fractional Sobolev norms and double-well potentials, showing their convergence to classical or nonlocal minimal surface functionals depending on the fractional parameter.

Contribution

It establishes the $ ext{Gamma}$-convergence of nonlocal energy functionals with fractional Sobolev norms to minimal surface functionals, revealing a phase transition at $s=1/2$.

Findings

For $s o 1$, the functional converges to the classical minimal surface energy.

For $s o 0$, it converges to a nonlocal minimal surface functional.

Identifies a phase transition at $s=1/2$ between local and nonlocal limits.

Abstract

We discuss the $\Gamma$-convergence, under the appropriate scaling, of the energy functional $$ \|u\|_{H^s(\Omega)}^2+\int_\Omega W(u)dx,$$ with $s \in (0,1)$, 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. When $s\in [1/2,\,1)$, we show that the energy $\Gamma$-converges to the classical minimal surface functional -- while, when $s\in(0,\,1/2)$, it is easy to see that the functional $\Gamma$-converges to the nonlocal minimal surface functional.