TY - JOUR
T1 - On Asymptotic Behaviour and W 2, p Regularity of Potentials in Optimal Transportation
AU - Liu, Jiakun
AU - Trudinger, Neil S.
AU - Wang, Xu Jia
N1 - Publisher Copyright:
© 2014, Springer-Verlag Berlin Heidelberg.
PY - 2014/3
Y1 - 2014/3
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=84921354939&partnerID=8YFLogxK
U2 - 10.1007/s00205-014-0797-y
DO - 10.1007/s00205-014-0797-y
M3 - Article
SN - 0003-9527
VL - 215
SP - 867
EP - 905
JO - Archive for Rational Mechanics and Analysis
JF - Archive for Rational Mechanics and Analysis
IS - 3
ER -