Second-Order $\Gamma$-Limit for the Cahn-Hilliard Functional with Dirichlet Boundary Conditions, I

TL;DR

This paper investigates the second-order Gamma-convergence of the Cahn-Hilliard functional with Dirichlet boundary conditions, revealing boundary transition layers when no interfaces are present.

Contribution

It provides a detailed asymptotic analysis of the second-order Gamma-limit for the Cahn-Hilliard functional under Dirichlet conditions, including boundary layer characterization.

Findings

Existence of boundary transition layers in the absence of interfaces.

Second-order Gamma-limit development for the functional.

Insights into boundary effects under Dirichlet conditions.

Abstract

This paper addresses the asymptotic development of order 2 by the $\Gamma$ -convergence of the Cahn-Hilliard functional with Dirichlet boundary conditions. The Dirichlet data are assumed to be well separated from one of the two wells. In the case where there are no interfaces, it is shown that there is a transition layer near the boundary of the domain.