Fine properties of symmetric and positive matrix fields with bounded divergence

TL;DR

This paper investigates the detailed properties of a determinant-based functional on symmetric positive matrix fields with bounded divergence, correcting previous errors and characterizing its behavior at critical integrability levels.

Contribution

It corrects a key mistake regarding upper semicontinuity, provides explicit bounds, and characterizes the functional's behavior at the critical integrability case, extending previous results.

Findings

Corrected upper semicontinuity result for the functional

Explicit bounds for measures generated by sequences of matrix fields

Characterization of behavior at critical integrability and generalization of Monge-Ampère related results

Abstract

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 $\varphi$.