The Allen-Cahn equation with dynamic boundary conditions and mass constraints

TL;DR

This paper studies a generalized Allen-Cahn equation with dynamic boundary conditions and mass constraints, formulating it as a variational inequality and proving well-posedness of the initial value problem.

Contribution

It introduces a novel mass constraint framework involving the solution inside the domain or on the boundary, with a rigorous well-posedness proof for the resulting nonlinear PDE system.

Findings

Formulation as a variational inequality with constraints

Existence and uniqueness of solutions established

Inclusion of Lagrange multipliers in the model

Abstract

The Allen-Cahn equation, coupled with dynamic boundary conditions, has recently received a good deal of attention. The new issue of this paper is the setting of a rather general mass constraint which may involve either the solution inside the domain or its trace on the boundary. The system of nonlinear partial differential equations can be formulated as variational inequality. The presence of the constraint in the evolution process leads to additional terms in the equation and the boundary condition containing a suitable Lagrange multiplier. A well-posedness result is proved for the related initial value problem.