The paper provides some partial differential equation methods that improve upon the \(L^p\) estimates of the ‘transport densities’ in certain Monge-Kantorovich mass transfer problems, as derived in an earlier paper by L. De Pascale and A. Pratelli [Calc. Var. Partial Differ. Equ. 14, No. 3, 249–274 (2002; Zbl 1032.49043)].