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

TL;DR

This paper advances the understanding of the second-order $ ext{Gamma}$-limit for the Cahn-Hilliard functional with Dirichlet boundary conditions, especially when boundary data are not well separated from the wells, revealing boundary transition layers.

Contribution

It extends previous work by analyzing the case where boundary data are not well separated, identifying boundary transition layers in the second-order $ ext{Gamma}$-limit.

Findings

Existence of boundary transition layers without interfaces.

Extension of second-order $ ext{Gamma}$-convergence analysis.

Characterization of boundary effects in the limit.

Abstract

This paper continues the study of the asymptotic development of order 2 by $\Gamma$ -convergence of the Cahn-Hilliard functional with Dirichlet boundary conditions initiated in [8]. While in the first paper, the Dirichlet data are assumed to be well separated from one of the two wells, here this is no longer the case. In the case where there are no interfaces, it is shown that there is a transition layer near the boundary of the domain.