Permeability parameter asymptotics in a Cahn--Hilliard system with third type transmission conditions

Abstract

In this paper, we study a system in which the Cahn--Hilliard system is imposed in the bulk domain and an Allen--Cahn type equation is prescribed on the boundary, connected through a third type transmission condition characterized by a permeability parameter. This setting is closely related to the theory of transmission problems, and the third type transmission condition can be viewed as a remnant of a thin-boundary description, with the permeability parameter controlling the degree of coupling between the bulk and the boundary. The main objective is to perform a rigorous asymptotic analysis with respect to the permeability parameter. We investigate two limits: the parameter tending to zero, corresponding to a completely impermeable boundary, and the parameter tending to infinity, corresponding to a perfectly permeable boundary where the bulk and boundary phases are fully coupled. For each limiting regime, we establish the convergence of solutions to the respective limit problems and characterize the resulting equations.

29 pages