# On a question of D. Serre

## TL;DR

This paper investigates the integrability properties of determinants of divergence-free, non-negative definite matrix-fields and provides a negative answer to a previously posed open question, also analyzing the Hessian of convex functions.

## Contribution

It offers a negative resolution to an open question about integrability gains and examines the case of Hessians of convex functions.

## Key findings

- Negative answer to the open question on integrability gains
- Analysis of Hessians of convex functions
- Insights into divergence-free matrix-fields

## Abstract

In this paper we give a negative answer to the question posed in [15, Open Question 2.1] about possible gains of integrability of determinants of divergence-free, non-negative definite matrix-fields. We also analyze the case in which the matrix-field is given by the Hessian of a convex function.