Title: Mathematical modeling of crystal surfaces:
From microscopic schemes to continuum laws

Dr. Dionisios Margetis
Department of Mathematics, M.I.T.

Abstract:

In traditional settings such as fluids and classical elasticity the 
starting point (truth) is identified with continuum equations.
But in many cases of mathematical modeling this perspective is incomplete:
The truth is atomistic, or takes the form of discrete schemes, by 
which continuum laws must be determined at the macroscale. The issue
of connections between different scales is largely unresolved.

In this talk I focus on the evolution of crystal surfaces as a 
prototypical case of coupling between scales, with implications 
in the design of novel devices. The governing, discrete equations represent
the motion of interacting line defects, atomic ``steps''. In the continuum
limit a nonlinear PDE is derived for the surface height, and free-boundary 
problems are formulated for the surface motion. I show analytically how
microscopic details of the crystal enter the requisite boundary 
conditions, and thus affect evolution at the macroscale.