$\Gamma$-Convergence of Free-discontinuity Problems

TL;DR

This paper establishes general $ ext{Gamma}$-convergence results for free-discontinuity problems involving vector-valued functions, extending classical homogenisation theories to non-periodic and stochastic settings with minimal assumptions.

Contribution

It provides the first comprehensive $ ext{Gamma}$-convergence analysis for free-discontinuity functionals under very general conditions, including non-periodic and unbounded surface integrands.

Findings

Proved compactness and $ ext{Gamma}$-limit representation for general free-discontinuity functionals.

Derived homogenisation formulas without periodicity assumptions.

Extended classical homogenisation results to stochastic settings.

Abstract

We study the $\Gamma$-convergence of sequences of free-discontinuity functionals depending on vector-valued functions $u$ which can be discontinuous across hypersurfaces whose shape and location are not known a priori. The main novelty of our result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Further, we consider the case of surface integrands which are not bounded from below by the amplitude of the jump of $u$. We obtain three main results: compactness with respect to $\Gamma$-convergence, representation of the $\Gamma$-limit in an integral form and identification of its integrands, and homogenisation formulas without periodicity assumptions. In particular, the classical case of periodic homogenisation follows as a by-product of our analysis. Moreover, our result covers also the case of stochastic homogenisation, as we will show in a forthcoming paper.