Large Deviations for $\nabla_{\varphi}$ Interface Model and Derivation of Free Boundary Problems

Abstract

We consider the $\nabla_{\varphi}$ interface model with weak self potential (one-body potential) under general Dirichlet boundary conditions on a large bounded domain and establish the large deviation principle for the macroscopically scaled interface height variables. As its application the law of large numbers is proved and the limit profile is characterized by a variational problem which was studied by Alt-Caffarelli [1], Alt-Caffarelli-Friedman [2] and others. The minimizers generate free boundaries inside the domain. We also discuss the $\nabla_{\varphi}$ interface model with $\delta$-pinning potential in one dimension.