  Mathematical model that allows for direct tracking of the homoepitaxial
crystal growth out of the window etched in the solid, pre-deposited layer on
the substrate is described. The growth is governed by the normal (to the
crystal-vapor interface) flux from the vapor phase and by the interface
diffusion. The model accounts for possibly inhomogeneous energy of the mask
surface and for strong anisotropies of crystal-vapor interfacial energy and
kinetic mobility. Results demonstrate that the motion of the crystal-mask
contact line slows down abruptly as radius of curvature of the mask edge
approaches zero. Numerical procedure is suggested to overcome difficulties
associated with ill-posedness of the evolution problem for the interface with
strong energy anisotropy.
  Keywords: Thin films, epitaxy, MOCVD, surface diffusion, interface dynamics,
contact lines, rough surfaces, wetting, regularization of ill-posed evolution
problems.
