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

2019/10/23 by Xiangsheng Xu, Xu, Xiangsheng
Computer Science · Engineering · Materials Science · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Thin Films #Solidification and crystal growth phenomena

paper · pdf · doi:10.48550/arxiv.1910.11153

openalex publication_date 2019/10/23 · openalex created_date 2019/11/01 · openalex updated_date 2026/07/28

Abstract

In this paper we study the boundary value problem for the equation div(D(∇ u)∇(div(|∇ u|^p-2∇ u+β(∇ u)/(|∇ u|))))+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(∇ u) is a 2× 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, |∇ u| is shown to be bounded when p>(4)/(3).

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