Regularity and monotonicity for solutions to a continuum model of epitaxial growth with nonlocal elastic effects

We study a nonlocal 4th order degenerate equation deriving from the epitaxial growth on crystalline materials. We first prove the global existence of evolution variational inequality solution with a general initial data using the gradient flow structure. Then with a monotone initial data, we prove the subdifferential of the associated convex functional is indeed single-valued, which gives higher regularities of the global solution. Particularly, higher regularites imply that the strict monotonicity maintains for all time, which provides rigorous justification for global-in time monotone solution to epitaxial growth model with nonlocal elastic effects on vicinal surface.