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

Abstract This paper continues the study of the asymptotic development of order 2 by $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> -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. We show that, in the absence of internal interfaces, the energy concentrates in a transition layer near the domain boundary.