Lack of regularity of the transport density in the Monge problem

In this paper, we provide a family of counter-examples to the regularity of the transport density in the classical Monge-Kantorovich problem. We prove that the W1,p regularity of the source and target measures f± does not imply that the transport density σ is W1,p, that the BV regularity of f± does not imply that σ is BV and that f±∈C∞ does not imply that σ is W1,p, for large p.