  We study the continuum limit in 2+1 dimensions of nanoscale anisotropic
diffusion processes on crystal surfaces relaxing to become flat below
roughening. Our main result is a continuum law for the surface flux in terms of
a new continuum-scale tensor mobility. The starting point is the Burton,
Cabrera and Frank (BCF) theory, which offers a discrete scheme for atomic steps
whose motion drives surface evolution. Our derivation is based on the
separation of local space variables into fast and slow. The model includes: (i)
anisotropic diffusion of adsorbed atoms (adatoms) on terraces separating steps;
(ii) diffusion of atoms along step edges; and (iii) attachment-detachment of
atoms at step edges. We derive a parabolic fourth-order, fully nonlinear
partial differential equation (PDE) for the continuum surface height profile.
An ingredient of this PDE is the surface mobility for the adatom flux, which is
a nontrivial extension of the tensor mobility for isotropic terrace diffusion
derived previously by Margetis and Kohn. Approximate, separable solutions of
the PDE are discussed.
