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In this paper, we study second-order and fourth-order elliptic problems whichinclude not only a Poisson equation in the bulk but also an inhomogeneousLaplace--Beltrami equation on the boundary of the domain. The bulk and thesurface PDE are coupled by a boundary condition that is either of Dirichlet orRobin type. We point out that both the Dirichlet and the Robin type boundarycondition can be handled simultaneously through our formalism without having tochange the framework. Moreover, we investigate the eigenvalue problemsassociated with these second-order and fourth-order elliptic systems. Wefurther discuss the relation between these elliptic problems and certainparabolic problems, especially the Allen--Cahn equation and the Cahn--Hilliardequation with dynamic boundary conditions.

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