Stochastic Homogenisation of Free-Discontinuity Problems

TL;DR

This paper investigates the stochastic homogenisation of free-discontinuity functionals, establishing the existence of a deterministic limit in ergodic cases and characterising the limit integrands via asymptotic cell formulas.

Contribution

It proves the existence of a homogenised functional for stochastic free-discontinuity problems and links convergence to the Subadditive Ergodic Theorem.

Findings

Existence of a homogenised deterministic functional in ergodic cases

Characterisation of limit integrands via asymptotic cell formulas

Connection established between functional convergence and ergodic theorem

Abstract

In this paper we study the stochastic homogenisation of free-discontinuity functionals. Assuming stationarity for the random volume and surface integrands, we prove the existence of a homogenised random free-discontinuity functional, which is deterministic in the ergodic case. Moreover, by establishing a connection between the deterministic convergence of the functionals at any fixed realisation and the pointwise Subadditive Ergodic Theorem by Akcoglou and Krengel, we characterise the limit volume and surface integrands in terms of asymptotic cell formulas.