On asymptotic behaviour and W 2, p regularity of potentials in optimal transportation

Abstract

In this paper we study local properties of cost and potential functions in optimal transportation. We prove that in a proper normalization process, the cost function is uniformly smooth and converges locally smoothly to a quadratic cost x · y, while the potential function converges to a quadratic function. As applications we obtain the interior W2,p estimates and sharp C1,α estimates for the potentials, which satisfy a Monge–Ampère type equation. The W2,p estimate was previously proved by Caffarelli for the quadratic transport cost and the associated standard Monge–Ampère equation. Show more