Second Order Asymptotic Development for the Anisotropic Cahn-Hilliard Functional

TL;DR

This paper investigates the second order asymptotic behavior of an anisotropic Cahn-Hilliard functional, revealing that the second order term vanishes under certain conditions, thus refining the understanding of the convergence rate of the functional's minima.

Contribution

It demonstrates that the second order term in the asymptotic expansion is zero for the anisotropic Cahn-Hilliard functional with specific potential assumptions, providing new insights into convergence rates.

Findings

Second order term in asymptotic expansion is zero.

Provides estimate on convergence rate of minima.

Refines understanding of anisotropic Cahn-Hilliard behavior.

Abstract

The asymptotic behavior of an anisotropic Cahn-Hilliard functional with prescribed mass and Dirichlet boundary condition is studied when the parameter epsilon that determines the width of the transition layers tends to zero. The double-well potential is assumed to be even and equal to |s-1|^b near s=1, with 1<b<2. The first order term in the asymptotic development by Gamma-convergence is well-known, and is related to a suitable anisotropic perimeter of the interface. Here it is shown that, under these assumptions, the second order term is zero, which gives an estimate on the rate of convergence of the minimum values.