$\Gamma$-convergence for nonlocal phase transitions involving the $H^{1/2}$ norm

TL;DR

This paper establishes the $ ext{Gamma}$-convergence of a family of nonlocal functionals involving the $H^{1/2}$ norm to a classical surface tension functional, revealing the limiting behavior of phase transition models with nonlocal interactions.

Contribution

It proves the $ ext{Gamma}$-convergence of nonlocal phase transition functionals involving the $H^{1/2}$ norm to the classical surface tension functional under specific scaling conditions.

Findings

Compactness in $BV$ space for the functionals.

$ ext{Gamma}$-convergence to the classical surface tension functional.

Results applicable to phase transition models with nonlocal interactions.

Abstract

We study functionals \begin{equation*} F_\varepsilon (u) := \lambda_\varepsilon \int_\Omega W(u) \, dx + \varepsilon \|u\|_{H^{1/2}}^2 \end{equation*} for a double well potential $W$ and the Gagliardo seminorm $\|\cdot\|_{H^{1/2}}$ when $\varepsilon \ln(\lambda_\varepsilon) \rightarrow k$ as $\varepsilon \rightarrow 0^+$ and show compactness in the space of $BV$ functions on $\Omega$ and the $\Gamma$-convergence to the classical surface tension functional.