A Framework with Differential Forms

Abstract

During the year 1974–1975 which I spent at UW, Madison, WI, Joel ROBBIN showed me a different proof of the div–curl lemma, using differential forms and the Hodge theorem, but it was only a few years later, after obtaining general results of compensated compactness with FrançoisMURAT, that I fully understood the example of differential forms, and what Joel ROBBIN said.

In the fall of 1975, I heard about the sequential weak continuity of Jacobian determinants proven by Yuri RESHETNYAK,1 and I saw that it is just the divcurl lemma for N = 2, and I deduced the case N = 3 fromthe div–curl lemma by noticing that grad(u) × grad(v) is divergence free, but I did not see that the corresponding algebraic manipulations to perform for N >3 are natural in the framework of differential forms, and that Yuri RESHETNYAK’s result almost follows from Lemma 9.1, a natural extension of Joel ROBBIN’s proof.²