Fine properties of symmetric and positive matrix fields with bounded divergence

This paper is concerned with various fine properties of the functional $$\mathbb{D}(A)={\int }_{{\mathbb{T}}^n}{\text{det}}^{\frac{1}{n-1}}(A(x))dx$$ introduced in [33]. This functional is defined on $X_p$, which is the cone of matrix fields $A\in L^p({\mathbb{T}}^n;{\text{Sym}}^{+}(n))$ with $\text{div }(A)$ a bounded measure. We start by correcting a mistake we noted in our [13, Corollary 7], which concerns the upper semicontinuity of $\mathbb{D}(A)$ in $X_p$. We give a proof of a refined correct statement, and we will use it to study the behaviour of $\mathbb{D}(A)$ when $A\in X_{\frac{n}{n-1}}$, which is the critical integrability for $\mathbb{D}(A)$. One of our main results gives an explicit bound of the measure generated by $\mathbb{D}(A_k)$ for a sequence of such matrix fields $\{A_k{\}}_k$. In particular it allows us to characterize the upper semicontinuity of $\mathbb{D}(A)$ in the case $A\in X_{\frac{n}{n-1}}$ in terms of the measure generated by the variation of $\{\text{div }{A}_k{\}}_k$. We show by explicit example that this characterization fails in $X_p$ if $p<\frac{n}{n-1}$. As a by-product of our characterization we also recover and generalize a result of P.-L. Lions [25,26] on the lack of compactness in the study of Sobolev embeddings. Furthermore, in analogy with Monge-Ampère theory, we give sufficient conditions under which ${\text{det}}^{\frac{1}{n-1}}(A)$ is Hardy when $A\in X_{\frac{n}{n-1}}$, generalising the celebrated result of S. Müller [29] when $A=\text{cof }{D}^2\phi$, for a convex function $\phi$.