  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.
