# Lack of regularity of the transport density in the monge problem

## TL;DR

This paper presents counterexamples showing that regularity of source and target measures does not guarantee regularity of the transport density in the Monge problem, challenging assumptions about smoothness transfer.

## Contribution

It provides the first explicit counterexamples demonstrating the failure of regularity transfer from measures to transport density in the Monge-Kantorovich problem.

## Key findings

- W^{1,p} regularity of measures does not imply W^{1,p} regularity of transport density
- BV regularity of measures does not imply BV regularity of transport density
- Smooth measures do not necessarily lead to regular transport densities for large p

## Abstract

In this paper, we provide a family of counterexamples to the regularity of the transport density in the classical Monge-Kantorovich problem. We prove that the W^{1,p} regularity of the source and target measures f ^\pm does not imply that the transport density $\sigma$ is W^{1,p} , that the BV regularity of f ^\pm does not imply that $\sigma$ is BV and that f^\pm $\in$ C^\infty does not imply that $\sigma$ is W^{1,p} , for large p.