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

Abstract

The well-posedness for a system of partial differential equations and dynamic boundary conditions is discussed. This system is a sort of transmission problem between the dynamics in the bulk $\Omega$ and on the boundary $\Gamma$. The Poisson equation for the chemical potential, the Allen-Cahn equation for the order parameter in the bulk $\Omega$ are considered as auxiliary conditions for solving the Cahn-Hilliard equation on the boundary $\Gamma$. Recently the well-posedness for the equation and 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.

Key words: Cahn-Hilliard equation, bulk condition, dynamic boundary condition, well-posedness. The interested reader can also see the related preprint arXiv:1502.05159 whose results are recalled and used for the analysis carried out in this paper