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

TL;DR

This paper extends the hydrodynamic limit analysis of the Ginzburg-Landau $ abla heta$ interface model to bounded domains with Dirichlet boundary conditions, deriving a fourth-order nonlinear PDE linked to the Wulff shape.

Contribution

It introduces the hydrodynamic limit for the model on bounded domains with Dirichlet boundary conditions, deriving a new macroscopic PDE related to the Wulff shape.

Findings

Derived a fourth-order nonlinear PDE as the macroscopic limit.

Extended previous periodic boundary results to bounded domains.

Connected the PDE to the Wulff shape concept.

Abstract

Hydrodynamic limit for the Ginzburg-Landau $\nabla\phi$ 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].