## Advanced Studies in Pure Mathematics

### 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.

#### Article information

Dates
Received: 24 January 2003
Revised: 28 May 2003
First available in Project Euclid: 1 January 2019

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1546369038

Digital Object Identifier
doi:10.2969/aspm/03910173

Mathematical Reviews number (MathSciNet)
MR2073334

Zentralblatt MATH identifier
1221.60138

#### Citation

Funaki, Tadahisa; Sakagawa, Hironobu. Large Deviations for $\nabla_{\varphi}$ Interface Model and Derivation of Free Boundary Problems. Stochastic Analysis on Large Scale Interacting Systems, 173--211, Mathematical Society of Japan, Tokyo, Japan, 2004. doi:10.2969/aspm/03910173. https://projecteuclid.org/euclid.aspm/1546369038