Existence and regularity of weak solutions to a model for coarsening in molecular beam epitaxy

TL;DR

This paper proves the existence and uniqueness of weak solutions for a mathematical model describing surface growth in molecular beam epitaxy, focusing on regularity and slope selection effects.

Contribution

It establishes the first rigorous mathematical results on weak solutions for a model incorporating Ehrlich-Schwoebel effects in epitaxial growth.

Findings

Existence and uniqueness of weak solutions proved.

Regularity properties of solutions analyzed.

Model behavior with and without slope selection examined.

Abstract

Taking into account the occurrence of a zero of the surface diffusion current and the requirement of the Ehrlich-Schwoebel effect, Siegert et al \cite{Siegert94} formulate a model of Langevin type that describes the growth of pyramidlike structures on a surface under conditions of molecular beam epitaxy, and that the slope of these pyramids is selected by the crystalline symmetries of the growing film. In this article, the existence and uniqueness of weak solution to an initial boundary value problem for this model is proved, in the case that the noise is neglected. The regularity of the weak solution to models, with/without slope selection, is also investigated.