Analytical validation of a 2+1 dimensional continuum model for epitaxial growth with elastic substrate

TL;DR

This paper provides an analytical validation of a 2+1 dimensional continuum model for epitaxial growth on elastic substrates, establishing mathematical properties like existence and uniqueness of solutions.

Contribution

It proves existence, uniqueness, and regularity of weak solutions for the proposed evolution equation modeling heteroepitaxial growth.

Findings

Proved existence and uniqueness of weak solutions.

Established Lipschitz regularity in time.

Validated the mathematical soundness of the model.

Abstract

An analytical validation is obtained for the evolution equation $$h_t=\Delta[ \mathcal{F}^{-1}(-aE \mathcal{F}(h)) - r/h^2 -\Delta h ],$$ introduced in {\cite{TS}} by W.T. Tekalign and B.J. Spencer to describe the heteroepitaxial growth of a two-dimensional thin film on an elastic substrate. In the expression above, $h$ denotes the surface height of the film, $\mathcal{F}$ is the Fourier transform, and $a$, $E$, $r$ are positive material constants. Existence, uniqueness, and Lipschitz regularity in time for weak solutions are proved, under suitable assumptions on the initial datum.