Cahn-Hilliard equation on the boundary with bulk condition of Allen-Cahn type

Abstract

The well-posedness of a system of partial differential equations with dynamic boundary conditions is discussed. This system is a sort of transmission problem between the dynamics in the bulk Ω and on the boundary Γ. The Poisson equation for the chemical potential and the Allen-Cahn equation for the order parameter in the bulk Ω are considered as auxiliary conditions for solving the Cahn-Hilliard equation on the boundary Γ. Recently, the well-posedness of this equation with a dynamic boundary condition, both of Cahn-Hilliard type, was discussed. Based on this result, the existence of the solution and its continuous dependence on the data are proved.