G-convergence of linear differential equations

TL;DR

This paper explores the G-convergence of linear integro-differential-algebraic equations in Hilbert spaces, revealing conditions under which limit equations develop memory effects and analyzing classes of equations preserved under G-convergence, with applications to homogenization and Maxwell's equations.

Contribution

It provides new insights into the behavior of G-convergence for complex equations and identifies conditions leading to memory effects in the limit.

Findings

Memory effects emerge in limit equations under certain G-convergence conditions

Classes of equations are shown to be closed under G-convergence

Applications demonstrated for Maxwell's equations with specific constitutive relations

Abstract

We discuss $G$-convergence of linear integro-differential-algebaric equations in Hilbert spaces. We show under which assumptions it is generic for the limit equation to exhibit memory effects. Moreover, we investigate which classes of equations are closed under the process of $G$-convergence. The results have applications to the theory of homogenization. As an example we treat Maxwell's equation with the Drude-Born-Fedorov constitutive relation.