Analytical validation of a continuum model for the evolution of a crystal surface in multiple space dimensions

Abstract

In this paper we are concerned with the existence of a weak solution to the initial boundary value problem for the equation \(\partial_t u= \Delta(\Delta u)^{-3}\). This problem arises in the mathematical modeling of the evolution of a crystal surface. Existence of a weak solution \(u\) with \(\Delta u\geq 0\) is obtained via a suitable substitution. Our investigations reveal the close connection between this problem and the equation \(\partial_t\rho+\rho^2\Delta^2\rho^3=0\), another crystal surface model first proposed by H. Al Hajj Shehadeh, R. V. Kohn and J. Weare in \cite{AKW}.