Fine properties of symmetric and positive matrix fields with bounded divergence

Abstract

This paper is concerned with various fine properties of the functionalD(A)≐∫Tndet1n−1⁡(A(x))dx introduced in [34]. This functional is defined on Xp, which is the cone of matrix fields A∈Lp(Tn;Sym+(n)) with div(A) a bounded measure. We start by correcting a mistake we noted in our [13, Corollary 7], which concerns the upper semicontinuity of D(A) in Xp. We give a proof of a refined correct statement, and we will use it to study the behaviour of D(A) when A∈Xnn−1, which is the critical integrability for D(A). One of our main results gives an explicit bound of the measure generated by D(Ak) for a sequence of such matrix fields {Ak}k. In particular it allows us to characterize the upper semicontinuity of D(A) in the case A∈Xnn−1 in terms of the measure generated by the variation of {divAk}k. We show by explicit example that this characterization fails in Xp if p<nn−1. As a by-product of our characterization we also recover and generalize a result of P.-L. Lions [26,27] on the lack of compactness in the study of Sobolev embeddings. Furthermore, in analogy with Monge-Ampère theory, we give sufficient conditions under which det1n−1⁡(A)∈H1(Tn) when A∈Xnn−1, generalising the celebrated result of S. Müller [30] when A=cofD2φ, for a convex function φ.