Second order variational heuristics for the Monge problem on compact manifolds

Abstract.

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.