A global method for deterministic and stochastic homogenisation in $BV$

TL;DR

This paper develops a comprehensive approach to homogenisation of free-discontinuity functionals in BV spaces, covering both deterministic and stochastic cases without requiring periodicity, and characterizes the limit integrands via asymptotic formulas.

Contribution

It introduces a general homogenisation framework for BV functionals that does not rely on periodicity, extending to stochastic settings using ergodic theorems.

Findings

Established deterministic homogenisation under broad assumptions.

Proved stochastic homogenisation for stationary random integrands.

Characterized limit integrands with asymptotic cell formulas.

Abstract

In this paper we study the deterministic and stochastic homogenisation of free-discontinuity functionals under \emph{linear} growth and coercivity conditions. The main novelty of our deterministic 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. Combining this result with the pointwise Subadditive Ergodic Theorem by Akcoglu and Krengel, we prove a stochastic homogenisation result, in the case of stationary random integrands. In particular, we characterise the limit integrands in terms of asymptotic cell formulas, as in the classical case of periodic homogenisation.