Asymptotic Analysis of Second Order Nonlocal Cahn-Hilliard-Type Functionals

TL;DR

This paper investigates the asymptotic behavior of a family of nonlocal second order Cahn-Hilliard functionals, characterizing their limit energy as anisotropic surface energy through Gamma convergence.

Contribution

It provides a novel Gamma convergence analysis of nonlocal Cahn-Hilliard functionals with fractional seminorm kernels, revealing their limit as anisotropic surface energies.

Findings

Limit energy characterized as anisotropic surface energy

Gamma convergence established for the family of functionals

Inclusion of fractional seminorm kernels in the analysis

Abstract

In this paper the study of a nonlocal second order Cahn-Hilliard-type singularly perturbed family of functions is undertaken. The kernels considered include those leading to Gagliardo fractional seminorms for gradients. Using Gamma convergence the integral representation of the limit energy is characterized leading to an anisotropic surface energy on interfaces separating different phases.