# On the upper semicontinuity of a quasiconcave functional

## TL;DR

This paper investigates the conditions under which a divergence-quasiconcave functional related to determinants of positive definite matrices exhibits weak upper semicontinuity, establishing a precise threshold for the integrability exponent.

## Contribution

It proves that the divergence-quasiconcavity inequality implies weak upper semicontinuity of the functional if and only if the integrability exponent exceeds a specific critical value.

## Key findings

- Weak upper semicontinuity holds if and only if p > n/(n-1).
- The divergence-quasiconcavity inequality is key to the semicontinuity result.
- The paper clarifies the relationship between divergence-quasiconcavity and semicontinuity for this class of functionals.

## Abstract

In the recent paper \cite{SER}, the second author proved a divergence-quasiconcavity inequality for the following functional $ \mathbb{D}(A)=\int_{\mathbb{T}^n} det(A(x))^{\frac{1}{n-1}}\,dx$ defined on the space of $p$-summable positive definite matrices with zero divergence. We prove that this implies the weak upper semicontinuity of the functional $\mathbb{D}(\cdot)$ if and only if $p>\frac{n}{n-1}$.