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

Serena Dipierro; Enrico Valdinoci; Riccardo Villa

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arXiv · 2506.15946

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

Serena Dipierro·Enrico Valdinoci·Riccardo Villa

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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.

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Serena Dipierro, Enrico Valdinoci, Riccardo Villa. 2025-06-19. $\Gamma$-convergence of the non-local Massari functional and applications to inhomogeneous Allen-Cahn equations. https://arxiv.org/abs/2506.15946

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