Existence Theorems for a Crystal Surface Model Involving the $p$-Laplace Operator
Xiangsheng Xu · SIAM Journal on Mathematical Analysis · 2018
The manufacturing of crystal films lies at the heart of modern nanotechnology. How to accurately predict the motion of a crystal surface is of fundamental importance. Many continuum models have been developed for this purpose, including a number of PDE models, which are often obtained as the continuum limit of a family of kinetic Monte Carlo models of crystal surface relaxation that includes both the solid-on-solid and discrete Gaussian models. In this paper we offer an analytical perspective on some of these models. To be specific, we study the existence of a weak solution to the boundary value problem for the equation $- \Delta e^{-\mbox{div}(| abla u|^{p-2} abla u)}+au=f$, where $p>1, a>0$ are given numbers and $f$ is a given function. This problem is derived from a crystal surface model proposed by J. L. Marzuola and J. Weare [ Phys. Rev. E, 88 (2013), 032403]. The mathematical challenge is due to the fact that the principal term in our equation is an exponential function of a $p$-Laplacian. Existence of a suitably defined weak solution is established under the assumptions that $p\in(1,2], \, N\leq 4$, and $f\in W^{1,p}$. Our investigations reveal that the key to our existence assertion is how to control the set where $-\mbox{div}(| abla u|^{p-2} abla u)$ is $\pm\infty$.