ASYMPTOTICS FOR THE FRACTIONAL ALLEN-CAHN EQUATION AND STATIONARY NONLOCAL MINIMAL SURFACES

Vincent Millot (1) , Yannick Sire (2) , Kelei Wang (3)

1 LJLL - Laboratoire Jacques-Louis Lions
2 JHU - Johns Hopkins University
3 Wuhan Institute of Physics and Mathematics

This article is mainly devoted to the asymptotic analysis of a fractional version of the (elliptic) Allen-Cahn equation in a bounded domain Ω ⊆ R n , with or without a source term in the right hand side of the equation (commonly called chemical potential). Compare to the usual Allen-Cahn equation, the Laplace operator is here replaced by the fractional Laplacian (−∆) s with s ∈ (0, 1/2), as defined in Fourier space. In the singular limit ε → 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 L.A. Caffarelli, J.M. Roquejoffre, and O. Savin [16]. 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 ′-perimeter in Ω for every s ′ ∈ (0, 1/2).

Domaines: Equations aux dérivées partielles [math.AP]

Vincent Millot, Yannick Sire, Kelei Wang. ASYMPTOTICS FOR THE FRACTIONAL ALLEN-CAHN EQUATION AND STATIONARY NONLOCAL MINIMAL SURFACES. Archive for Rational Mechanics and Analysis, 2019, 231. ⟨hal-01386330⟩