The aim of this note is to give a Beckmann-type problem as well as the corresponding optimal mass transportation problem associated with a degenerate Hamilton-Jacobi equation
H
(
x
,
∇
u
)
=
0
,
H(x,\nabla u)=0,
coupled with non-zero Dirichlet condition
u
=
g
u=g
on
∂
Ω
\partial \Omega
. In this article, the Hamiltonian
H
H
is continuous in both arguments, coercive and convex in the second, but not enjoying any property of existence of a smooth strict sub-solution. We also provide numerical examples to validate the correctness of theoretical formulations.