Hydrodynamic limit for the Ginzburg-Landau ∇ϕ interface model with a conservation law and Dirichlet boundary conditions

Hydrodynamic limit for the Ginzburg-Landau ∇ϕ interface model with a conservation law was established in Nishikawa (2002) under periodic boundary conditions. This paper studies the same problem on a bounded domain imposing the Dirichlet boundary condition. A nonlinear partial differential equation of fourth order satisfying the boundary conditions is derived as the macroscopic equation. Its solution converges to the Wulff shape derived by Deuschel et al. (2000) as the time t→∞.