Γ-convergence of the non-local Massari functional and applications to inhomogeneous Allen-Cahn equations

2025/06/19 by Dipierro, Serena, Valdinoci, Enrico, Valdioci, Enrico +1
Computer Science · Materials Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Solidification and crystal growth phenomena

paper · pdf · doi:10.48550/arxiv.2506.15946

openalex publication_date 2025/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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 Γ-converges to the classical one, and this convergence preserves minimizers in the L¹_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.

Citations

• On non-local almost minimal sets and an application to the non-local Massari's Problem
• Asymptotics for the fractional Allen-Cahn equation and stationary nonlocal minimal surfaces
• Nonlocal problems with Neumann boundary conditions
• Hitchhiker's guide to the fractional Sobolev spaces
• Hitchhikerʼs guide to the fractional Sobolev spaces
• Local and global minimizers for a variational energy involving a\n fractional norm
• Density estimates for a variational model driven by the Gagliardo norm
• Nonlocal minimal surfaces
• Regularity of the obstacle problem for a fractional power of the laplace operator

Related

• Asymptotics for the fractional Allen-Cahn equation and stationary nonlocal minimal surfaces
• Convergence of a mass conserving Allen-Cahn equation whose Lagrange\n multiplier is nonlocal and local
• On non-local almost minimal sets and an application to the non-local Massari's Problem
• Convergence of the Nonlocal Allen-Cahn Equation to Mean Curvature Flow
• Time-fractional Allen-Cahn equations versus powers of the mean curvature