Sobolev and SBV Representation Theorems for large volume limit Gibbs measures

We study the limit of large volume equilibrium Gibbs measures for a rather general Hamiltonians. In particular we study Hamiltonians which arise in naturally in Nonlinear Elasticity and Hamiltonians (containing surface terms) which arises naturally in Fracture Mechanics. In both of these settings we show that an integral representation holds for the limit. Moreover, we also show a homogenization result for the Nonlinear Elasticity setting. This extends a recent result of R. Koteck\'y and S. Luckhaus \cite{LK-comm-math-phys}.