The Div-Curl Lemma Revisited

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 $L^2(\mathrm{\Omega })$ and $L^p(\mathrm{\Omega })$, $p\mathrm{\ne }2$, 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.