The Div-Curl Lemma Revisited

TL;DR

This paper offers a new proof of the Div-Curl Lemma, a fundamental result in compensated compactness theory, using classical integral formulas and compactness properties, simplifying previous proofs.

Contribution

It provides a streamlined proof of the Div-Curl Lemma relying solely on Green-Gauss and Rellich-Kondrachov compactness, avoiding more complex arguments.

Findings

Simplified proof of the Div-Curl Lemma

Utilizes classical integral and compactness tools

Enhances understanding of compensated compactness

Abstract

The Div-Curl Lemma, which is the basic result of the compensated compactness theory in Sobolev spaces, was introduced by F. Murat (1978) with distinct proofs for the $L^2(\Omega)$ and $L^p(\Omega)$, $p \neq 2$, cases. In this note we present a slightly different proof, relying only on a Green-Gauss integral formula and on the usual Rellich-Kondrachov compactness properties.