epitaxial growth is an important physical process for forming solid films or other nano-structures. it occurs as atoms, deposited from above, adsorb and diffuse on a crystal surface. modeling the rates that atoms hop and break bonds leads in the continuum limit to degenerate 4th-order pde that involve exponential nonlinearity and the p-laplacian with p=1, for example. we discuss a number of analytical results for such models, some of which involve subgradient dynamics for radon measure solutions and a new notion of weak solutions.