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-Amp\`ere 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. M\"uller [29] when $A=\text{cof } D^2\varphi$, for a convex function $\varphi$.