Cahn-Hilliard equation with dynamic boundary conditions and mass constraint on the boundary

TL;DR

This paper studies a modified Cahn-Hilliard equation with dynamic boundary conditions and a boundary mass constraint, introducing Lagrange multipliers to ensure mass conservation and establishing well-posedness of the system.

Contribution

It introduces a novel boundary mass constraint into the Cahn-Hilliard equation with dynamic boundary conditions, analyzing the resulting system with Lagrange multipliers.

Findings

Existence and uniqueness of solutions are proved.

The system's well-posedness is established using subdifferential evolution theory.

Two Lagrange multipliers are identified for bulk and boundary mass constraints.

Abstract

The well-known Cahn-Hilliard equation entails mass conservation if a suitable boundary condition is prescribed. In the case when the equation is also coupled with a dynamic boundary condition, including the Laplace-Beltrami operator on the boundary, the total mass on the inside of the domain and its trace on the boundary should be conserved. The new issue of this paper is the setting of a mass constraint on the boundary. The effect of this additional constraint is the appearance of a Lagrange multiplier; in fact, two Lagrange multipliers arise, one for the bulk, the other for the boundary. The well-posedness of the resulting Cahn-Hilliard system with dynamic boundary condition and mass constraint on the boundary is obtained. The theory of evolution equations governed by subdifferentials is exploited and a complete characterization of the solution is given.