Hydrodynamic limit for the Ginzburg-Landau $\nabla \varphi$ interface model with a conservation law and Dirichlet boundary conditions

Abstract

Hydrodynamic limit for the Ginzburg-Landau $\nabla \varphi$ interface model with a conservation law was established in [Nishikawa 2002] under the periodic boundary conditions. This paper studies the same problem on the bounded domain imposing Dirichlet boundary conditions. A nonlinear partial equation of fourth order with boundary conditions is derived as the macroscopic equation, which is related to the Wulff shape derived by [Deuschel et.al. 2000].