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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    to diffeomorphisms. If a diffeomorphism is stationary for   , we know that it admits a potential function. If it realizes a local minimum of   , 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    to that of an elliptic solution of the Monge–Ampère 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.

Second order variational heuristics for the Monge problem on compact manifolds