$\Gamma$-convergence of the non-local Massari functional and applications to inhomogeneous Allen-Cahn equations

TL;DR

This paper proves that the fractional Massari functional converges to the classical one, providing insights into the asymptotic behavior of solutions to inhomogeneous Allen-Cahn equations and introducing a new geometric concept.

Contribution

It establishes the $ ext{Gamma}$-convergence of the non-local Massari functional to the classical functional, linking non-local and local geometric analysis.

Findings

Fractional Massari functional $ ext{Gamma}$-converges to the classical functional.

Convergence preserves minimizers in the $L^1_{ ext{loc}}$ topology.

Introduces the concept of non-local hybrid mean curvature.

Abstract

We present several asymptotic results concerning the non-local Massari Problem for sets with prescribed mean curvature. In particular, we show that the fractional Massari functional $\Gamma$-converges to the classical one, and this convergence preserves minimizers in the $L^1_{\mbox{loc}}$-topology. This returns useful information about the asymptotic behavior of the solutions of the inhomogeneous Allen-Cahn equation in the forced and the mass-prescribed settings. In this context, a new geometric object, which we refer to as "non-local hybrid mean curvature", naturally appears.