  In the absence of external material deposition, crystal surfaces usually
relax to become flat by decreasing their free energy. We study an asymmetry in
the relaxation of macroscopic plateaus, facets, of a periodic surface
corrugation in 1+1 dimensions via a continuum model below the roughening
transition temperature. The model invokes a highly degenerate parabolic partial
differential equation (PDE) for surface diffusion, which is related to the
weighted-$H^{-1}$ (nonlinear) gradient flow of a convex, singular surface free
energy in homoepitaxy. The PDE is motivated both by an atomistic broken-bond
model and a mesoscale model for steps. By constructing an explicit solution to
the PDE, we demonstrate the lack of symmetry in the evolution of top and bottom
facets in periodic surface profiles. Our explicit, analytical solution is
compared to numerical simulations of the PDE via a regularized surface free
energy.
