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

Lack of regularity of the transport density in the monge problem