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In this paper we study the boundary value problem for the equation$\mbox{div}\left(D(\nabla u)\nabla\left(\mbox{div}\left(|\nabla u|^{p-2}\nablau+\beta\frac{\nabla u}{|\nabla u|}\right)\right)\right)+au=f$ in the $z=(x,y)$plane. This problem is derived from a continuum model for the relaxation of acrystal surface below the roughing temperature. The mathematical challenge isof two folds. First, the mobility $D(\nabla u)$ is a $2\times 2$ matrix whosesmallest eigenvalue is not bounded away from $0$ below. Second, the equationcontains the $1$-Laplace operator, whose mathematical properties are still notwell-understood. Existence of a weak solution is obtained. In particular,$|\nabla u|$ is shown to be bounded when $p>\frac{4}{3}$.

Mathematical validation of a continuum model for relaxation of interacting steps in crystal surfaces in $2$ space dimensions