  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.
