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Asymptotics for the Fractional Allen–Cahn Equation and Stationary Nonlocal Minimal Surfaces

2018; Springer Science+Business Media; Volume: 231; Issue: 2 Linguagem: Inglês

10.1007/s00205-018-1296-3

1432-0673

Vincent Millot, Yannick Sire, Kelei Wang,

Differential Equations and Numerical Methods

This article is mainly devoted to the asymptotic analysis of a fractional version of the (elliptic) Allen–Cahn equation in a bounded domain $${\Omega \subseteq \mathbb{R}^{n}}$$ , with or without a source term in the right hand side of the equation (commonly called chemical potential). In contrast to the usual Allen–Cahn equation, the Laplace operator is here replaced by the fractional Laplacian $${(-\Delta)^s}$$ with $${s \in (0,1/2)}$$ , as defined in Fourier space. In the singular limit $${\varepsilon \to 0}$$ , we show that arbitrary solutions with uniformly bounded energy converge both in the energetic and geometric sense to surfaces of prescribed nonlocal mean curvature in Ω whenever the chemical potential remains bounded in suitable Sobolev spaces. With no chemical potential, the notion of surface of prescribed nonlocal mean curvature reduces to the stationary version of the nonlocal minimal surfaces introduced by Caffarelli et al. (Commun Pure Appl Math 63:1111–1144, 2010). Under the same Sobolev regularity assumption on the chemical potential, we also prove that surfaces of prescribed nonlocal mean curvature have a Minkowski codimension equal to one, and that the associated sets have a locally finite fractional $${2s^\prime}$$ -perimeter in Ω for every $${s^\prime \in (0,1/2)}$$ .