Mathematical validation of a continuum model for relaxation of interacting steps in crystal surfaces in $2$ space dimensions
Xiangsheng Xu · arXiv (Cornell University) · 2019
In this paper we study the boundary value problem for the equation $\mbox{div}\left(D( abla u) abla\left(\mbox{div}\left(| abla u|^{p-2} abla u+β\frac{ abla u}{| abla u|}\right)\right)\right)+au=f$ in the $z=(x,y)$ plane. This problem is derived from a continuum model for the relaxation of a crystal surface below the roughing temperature. The mathematical challenge is of two folds. First, the mobility $D( abla u)$ is a $2\times 2$ matrix whose smallest eigenvalue is not bounded away from $0$ below. Second, the equation contains the $1$-Laplace operator, whose mathematical properties are still not well-understood. Existence of a weak solution is obtained. In particular, $| abla u|$ is shown to be bounded when $p>\frac{4}{3}$.