  In this paper, we propose a variational formulation to study the singular
evolution equations that govern the dynamics of surface modulations on crystals
below the roughening temperature. The basic idea of the formulation is to
expand the surface shape in terms of a complete set of basis functions and to
use a variational principle equivalent to the continuum evolution equations to
obtain coupled nonlinear ordinary differential equations for the expansion
coefficients. Unlike several earlier approaches that rely on ad hoc
regularization procedures to handle the singularities in the evolution
equations, the only inputs required in the present approach are the orientation
dependent surface energies and the diffusion constants. The method is applied
to study the morphological equilibration of patterned unidirectional and
bidirectional sinusoidal modulations on semiconductor surfaces through surface
diffusion.
