Sobolev and SBV Representation Theorems for large volume limit Gibbs measures

TL;DR

This paper investigates the limiting behavior of large volume equilibrium Gibbs measures for complex Hamiltonians, establishing integral representation theorems and a homogenization result relevant to nonlinear elasticity and fracture mechanics.

Contribution

It extends existing theories by providing new integral representation theorems and homogenization results for Gibbs measures with Hamiltonians from elasticity and fracture mechanics.

Findings

Integral representation for limit measures established

Homogenization result demonstrated for nonlinear elasticity

Applicable to Hamiltonians with surface terms in fracture mechanics

Abstract

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}.