On second-order and fourth-order elliptic systems consisting of bulk and surface PDEs: Well-posedness, regularity theory and eigenvalue problems

TL;DR

This paper analyzes coupled second- and fourth-order elliptic PDE systems involving bulk and surface equations, establishing well-posedness, regularity, and eigenvalue properties, and relating them to parabolic models like Allen--Cahn and Cahn--Hilliard.

Contribution

It introduces a unified framework for handling bulk-surface elliptic systems with Dirichlet or Robin boundary conditions, including eigenvalue analysis and connections to parabolic problems.

Findings

Well-posedness and regularity results for coupled bulk-surface elliptic systems.

Eigenvalue problems for second- and fourth-order systems are characterized.

Connections established between elliptic systems and dynamic boundary condition parabolic equations.

Abstract

In this paper, we study second-order and fourth-order elliptic problems which include not only a Poisson equation in the bulk but also an inhomogeneous Laplace--Beltrami equation on the boundary of the domain. The bulk and the surface PDE are coupled by a boundary condition that is either of Dirichlet or Robin type. We point out that both the Dirichlet and the Robin type boundary condition can be handled simultaneously through our formalism without having to change the framework. Moreover, we investigate the eigenvalue problems associated with these second-order and fourth-order elliptic systems. We further discuss the relation between these elliptic problems and certain parabolic problems, especially the Allen--Cahn equation and the Cahn--Hilliard equation with dynamic boundary conditions.