Partial regularity for an exponential PDE in crystal surface models

Abstract

We study the regularity properties of a weak solution to the boundary value problem for the equation $-\Delta \rho +au=f$ in a bounded domain $\Omega \subset {\mathbb{R}}^N$, where $ρ=e^{-\mbox{div}\left(|\nabla u|^{p-2}\nabla u+β_0|\nabla u|^{-1}\nabla u\right)}$. This problem is derived from the mathematical modeling of crystal surfaces. It is known that the exponent term can exhibit singularity. In this paper we obtain a partial regularity result for the weak solution. It asserts that there exists an open subset ${\Omega }_0\subset \Omega$ such that $|\Omega \setminus {\Omega }_0|=0$ and the exponent term is locally bounded in ${\Omega }_0$. Furthermore, if $x_0\in \Omega \setminus {\Omega }_0$, then $\rho$ vanishes of $N+2-\epsilon$ order at $x_0$ for each $\epsilon \in (0,2)$. Our results reveal that the exponent term behaves well if it stays away from negative infinity.