Communications in Mathematical Sciences

Volume 16 (2018)

Number 5

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

Pages: 1379 – 1394



Xin Yang Lu (Department of Mathematical Sciences, Lakehead University, Ontario, Canada; and Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada)


We consider the evolution equation\begin{array}{lr}h_t = \Delta [ F^{-1} (-a E \mathcal{F} (h))-r / h^2 - \Delta h] \; \textrm{,} & \textrm{(0.1)}\end{array}introduced in [Wondimu Taye Tekalign and B.J. Spencer, J. Appl. Phys., 96(10):5505–5512, 2004] by Tekalign and 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. For simplicity, we set $aE=r=1$. As this equation does not have any particular structure, its analysis is quite challenging. Therefore, we introduce the auxiliary equation (with $c$ being a given constant)\begin{array}{lr}u_t = \nabla [ - \nabla \cdot u - (\nabla \cdot u + c)^{-2} - \Delta \nabla \cdot u ] \; \textrm{,} & \textrm{(0.2)}\end{array}which has a variational structure. Equivalency between (0.1) and (0.2) will hold under sufficient regularity on the solution. The main aim of this paper is to provide an analytical validation to (0.2), by proving existence and regularity properties for weak solutions, under suitable assumptions on the initial datum.


epitaxial growth, wetting, maximal monotone operators

2010 Mathematics Subject Classification

35K55, 35K67, 44A15, 74K35

Received 19 April 2017

Received revised 8 June 2018

Accepted 8 June 2018

Published 19 December 2018