$G$-convergence of elliptic and parabolic operators depending on vector fields

TL;DR

This paper investigates the G-convergence of sequences of elliptic and parabolic divergence-form operators influenced by vector fields, employing compensated compactness to establish new compactness results without relying on affine functions.

Contribution

It introduces novel compactness results for G- and H-convergence of operators depending on vector fields, extending previous theories to settings lacking affine functions.

Findings

Established compactness results for G- and H-convergence

Applied compensated compactness in a new setting

Extended convergence theory to operators without affine functions

Abstract

We consider sequences of elliptic and parabolic operators in divergence form and depending on a family of vector fields. We show compactness results with respect to G-convergence, or H-convergence, by means of the compensated compactness theory, in a setting in which the existence of affine functions is not always guaranteed, due to the nature of the family of vector fields.