We consider Monge's optimal transport problem posed on compact manifolds (possibly with boundary) for a lower semi-continuous cost function $c$. When all data are smooth and the given measures, positive, we restrict the total cost ${\cal C}$ to diffeomorphisms. If a diffeomorphism is stationary for ${\cal C}$, we know that it admits a potential function. If it realizes a local minimum of ${\cal C}$, we prove that the $c$-Hessian of its potential function must be non-negative, positive if the cost function $c$ is non degenerate. If $c$ is generating non-degenerate, we reduce the existence of a local minimizer of ${\cal C}$ to that of an elliptic solution of the Monge--Ampére equation expressing the measure transport; moreover, the local minimizer is unique. It is global, thus solving Monge's problem, provided $c$ is superdifferentiable with respect to one of its arguments.