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

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--Müller. 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.

Corrected typos and added an example in Section 5

References in corpus (5)

• Bounded solutions of ideal MHD with compact support in space-time
• On the Hardy space theory of compensated compactness quantities
• On the necessity of the constant rank condition for $L^p$ estimates
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Cited by in corpus (7)

• Oscillation and concentration in sequences of PDE constrained measures
• Endpoint $L^1$ estimates for Hodge systems
• Compensated compactness: continuity in optimal weak topologies
• The Dirichlet problem for the Jacobian equation in critical and supercritical Sobolev spaces
• On Scaling Properties for a Class of Two-Well Problems for Higher Order Homogeneous Linear Differential Operators
• A variational view on constitutive laws in parabolic problems
• Syzygies, constant rank, and beyond