Hydrodynamic limit for the Ginzburg-Landau $\mathrm{\nabla }\varphi$ interface model with non-convex potential

Hydrodynamic limit for the Ginzburg-Landau $\mathrm{\nabla }\varphi$ interface model was established in [Nishikawa, 2003] under the Dirichlet boundary conditions. This paper studies the similar problem, but with non-convex potentials. Because of the lack of strict convexity, a lot of difficulties arise, especially, on the identification of equilibrium states. We give a proof of the equivalence between the stationarity and the Gibbs property under quite general settings, and as its conclusion, we complete the identification of equilibrium states under the high temparature regime in [Deuschel and Cotar, 2008]. We also establish some uniform estimates for variances of extremal Gibbs measures under quite general settings.