Published Paper

Inserted: 20 dec 2022
Last Updated: 31 aug 2023

Journal: Adv. in Math.
Year: 2022
Doi: https://doi.org/10.1016/j.aim.2023.109130

Abstract:

This paper is concerned with various fine properties of the functional \[ D(A) \doteq \int_{\mathbb{T}^n}{\det}^\frac{1}{n-1}(A(x))\,dx \] introduced in $[33]$. This functional is defined on $X_p$, which is the cone of matrix fields $A \in L^p(\mathbb{T}^n;Sym^+(n))$ 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 $X_p$. We give a proof of a refined correct statement, and we will use it to study the behaviour of $D(A)$ when $A \in X_\frac{n}{n-1}$, which is the critical integrability for $D(A)$. One of our main results gives an explicit bound of the measure generated by $D(A_k)$ for a sequence of such matrix fields $\{A_k\}_k$. In particular it allows us to characterize the upper semicontinuity of $D(A)$ in the case $A \in X_\frac{n}{n - 1}$ in terms of the measure generated by the variation of $\{div A_k\}_k$. We show by explicit example that this characterization fails in $X_p$ if $p<\frac{n}{n-1}$. As a bi-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 $\det^\frac{1}{n-1}(A) \in \mathcal{H}^1(\mathbb{T}^n)$ when $A \in X_\frac{n}{n - 1}$, generalising the celebrated result of S. Müller $[29]$ when $A=$ cof $D^2\varphi$, for a convex function $\varphi$.