© 2014, Springer-Verlag Berlin Heidelberg. 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.
Liu, J., Trudinger, N. & Wang, X. (2015). On asymptotic behaviour and W 2, p regularity of potentials in optimal transportation. Archive for Rational Mechanics and Analysis, 215 (3), 867-905. Archive for Rational Mechanics and Analysis