This paper is concerned with time periodic solutions of
Hamilton-Jacobi equations in which the hamiltonian is increasing
wrt to the unknown variable. When the uniqueness of the periodic
solution is not guaranteed, we define a notion of extremal
solution and propose two different ways to attain it, together
with the corresponding numerical simulations. In the course of the
analysis, the ode case, where we show that things are rather
explicit, is also visited.