The surface of an epitaxially growing thin film often exhibits a mound-like structure with its characteristic lateral size increasing in time. In this talk, we consider two competing mechanisms for such a coarsening process: (1) surface relaxation described by high-order gradients of the surface profile; and (2) the Ehrlich-Schwoebel (ES) effect which is the upper-lower terrace asymmetry in the adatom attachment and detachment to and from atomic steps. We present a theory based on a class of continuum models that are mathematically gradient-flows of some effective free-energy functionals describing these mechanisms. This theory consists of two parts: (1) variational properties of the energies, such as ``ground states'' and their large-system-size asymptotics, showing the unboundedness of surface slope and revealing the relation between some of the models; (2) rigorous bounds for the scaling law of the roughness, the rate of increase of surface slope, and the rate of energy dissipation, all of which characterize the coarsening process. Predictions on scaling laws made by our theory agree well with experiments.