Existence theorems for a crystal surface model involving the p-Laplace operator

TL;DR

This paper proves the existence of weak solutions for a complex PDE model of crystal surface evolution involving the p-Laplace operator and exponential functions, advancing the mathematical understanding of nanotechnology surface models.

Contribution

It provides the first analytical proof of weak solutions for a nonlinear PDE with exponential p-Laplacian terms in the context of crystal surface modeling.

Findings

Existence of weak solutions established for specific parameter ranges.

Key control of infinite divergence sets is crucial for the proof.

Results apply to models with p in (1,2] and dimension up to 4.

Abstract

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 into 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}\left(|\nabla u|^{p-2}\nabla u\right)}+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 (2013 Physical Review, E 88, 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 $-\mbox{div}\left(|\nabla u|^{p-2}\nabla u\right)$ is $\pm\infty$.