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Interior a priori estimates for the Monge-Ampere equation

Journal Article


Abstract


  • In this paper we prove the strict convexity, the interior C1,α,

    C2,α and W2,p estimates for convex solutions to the Monge-Amp`ere

    type equation. For the strict convexity and C1,α estimate, we assume

    that the inhomogeneous term f satisfies a doubling condition. For the

    C2,α and W2,p estimates, we assume that f is H¨older continuous or

    continuous. These estimates are mainly due to Caffarelli. We also give a

    brief discussion on the regularity for more general Monge-Amp`ere type

    equations arising in optimal transportation.

Authors


Publication Date


  • 2014

Citation


  • Liu, J. & Wang, X. (2014). Interior a priori estimates for the Monge-Ampere equation. Surveys in Differential Geometry, 19 151-177.

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/4019

Number Of Pages


  • 26

Start Page


  • 151

End Page


  • 177

Volume


  • 19

Place Of Publication


  • http://www.intlpress.com/site/pub/pages/journals/items/sdg/content/vols/0019/0001/a007/index.html

Abstract


  • In this paper we prove the strict convexity, the interior C1,α,

    C2,α and W2,p estimates for convex solutions to the Monge-Amp`ere

    type equation. For the strict convexity and C1,α estimate, we assume

    that the inhomogeneous term f satisfies a doubling condition. For the

    C2,α and W2,p estimates, we assume that f is H¨older continuous or

    continuous. These estimates are mainly due to Caffarelli. We also give a

    brief discussion on the regularity for more general Monge-Amp`ere type

    equations arising in optimal transportation.

Authors


Publication Date


  • 2014

Citation


  • Liu, J. & Wang, X. (2014). Interior a priori estimates for the Monge-Ampere equation. Surveys in Differential Geometry, 19 151-177.

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/4019

Number Of Pages


  • 26

Start Page


  • 151

End Page


  • 177

Volume


  • 19

Place Of Publication


  • http://www.intlpress.com/site/pub/pages/journals/items/sdg/content/vols/0019/0001/a007/index.html