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

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 on boundary of domain. The surface PDE are coupled by condition that is either Dirichlet or Robin type. We point out both type can be handled simultaneously through our formalism without having to change framework. Moreover, investigate eigenvalue associated with these systems. further discuss relation between certain parabolic problems, especially Allen--Cahn Cahn--Hilliard dynamic conditions.