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

Journal Article


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Abstract


  • © 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.

Authors


  •   Liu, Jiakun
  •   Trudinger, Neil (external author)
  •   Wang, Xu-Jia (external author)

Publication Date


  • 2015

Citation


  • 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

Scopus Eid


  • 2-s2.0-84921354939

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=4437&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/3420

Number Of Pages


  • 38

Start Page


  • 867

End Page


  • 905

Volume


  • 215

Issue


  • 3

Abstract


  • © 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.

Authors


  •   Liu, Jiakun
  •   Trudinger, Neil (external author)
  •   Wang, Xu-Jia (external author)

Publication Date


  • 2015

Citation


  • 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

Scopus Eid


  • 2-s2.0-84921354939

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=4437&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/3420

Number Of Pages


  • 38

Start Page


  • 867

End Page


  • 905

Volume


  • 215

Issue


  • 3