Quasiconvexity, null Lagrangians, and Hardy space integrability under constant rank constraints

TL;DR

This paper advances the theory of compensated compactness under constant rank conditions, characterizes null Lagrangians as weakly continuous functions with Hardy space integrability, and offers methods to compute these null Lagrangians.

Contribution

It provides a systematic treatment of compensated compactness under constant rank assumptions, characterizes null Lagrangians, and introduces a way to compute them explicitly.

Findings

Null Lagrangians are exactly the compensated compactness quantities with Hardy space integrability.

A short proof of a sharp weak lower semicontinuity result for signed integrands.

An effective method for computing null Lagrangians associated with a given operator.

Abstract

We present a systematic treatment of the theory of Compensated Compactness under Murat's constant rank assumption. We give a short proof of a sharp weak lower semicontinuity result for signed integrands, extending the results of Fonseca--M\"uller. The null Lagrangians are an important class of signed integrands, since they are the weakly continuous functions. We show that they are precisely the compensated compactness quantities with Hardy space integrability, thus proposing an answer to a question raised by Coifman-Lions-Meyer-Semmes. Finally we provide an effective way of computing the null Lagrangians associated with a given operator.