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

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.