Homogenisation and the Weak Operator Topology

TL;DR

This paper surveys the relationship between homogenisation problems in PDEs and convergence in the weak operator topology, covering static and dynamic equations with an operator-theoretic perspective.

Contribution

It introduces and characterizes various homogenisation notions, including recent nonlocal H-convergence, through weak operator topology convergence, extending to dynamic PDEs.

Findings

Homogenisation concepts are characterized by weak operator topology convergence.

The survey extends static homogenisation notions to dynamic equations.

Operator-theoretic aspects clarify homogenisation in autonomous PDEs.

Abstract

This article surveys results that relate homogenisation problems for partial differential equations and convergence in the weak operator topology of a suitable choice of linear operators. More precisely, well-known notions like $G$-convergence, $H$-convergence as well as the recent notion of nonlocal $H$-convergence are discussed and characterised by certain convergence statements under the weak operator topology. Having introduced and described these notions predominantly made for static or variational type problems, we further study these convergences in the context of dynamic equations like the heat equation, the wave equation or Maxwell's equations. The survey is intended to clarify the ideas and highlight the operator theoretic aspects of homogenisation theory in the autonomous case.