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

This paper continues the study of the asymptotic development of order 2 by $\mathrm{\Gamma }$ -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.