Surveys in Differential Geometry

Volume 19 (2014)

Interior a priori estimates for the Monge-Ampère equation

Pages: 151 – 177

DOI: https://dx.doi.org/10.4310/SDG.2014.v19.n1.a7

Authors

Jiakun Liu (Institute for Mathematics and its Applications, School of Mathematics and Applied Statistics, University of Wollongong, New South Wales, Australia)

Xu-Jia Wang (Centre for Mathematics and its Applications, The Australian National University, Canberra, Australia)

Abstract

In this paper we prove the strict convexity, the interior $C^{1,\alpha}$, $C^{2,\alpha}$ and $W^{2,p}$ estimates for convex solutions to the Monge-Ampère type equation. For the strict convexity and $C^{1,\alpha}$ estimate, we assume that the inhomogeneous term $f$ satisfies a doubling condition. For the $C^{2,\alpha}$ and $W^{2,p}$ estimates, we assume that $f$ is Hölder continuous or continuous. These estimates are mainly due to Caffarelli. We also give a brief discussion on the regularity for more general Monge-Ampère type equations arising in optimal transportation.

Keywords

Monge-Ampère equations, $C^{2,\alpha}$ estimate, $W^{2,p}$ estimate

2010 Mathematics Subject Classification

35B45, 35J96

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Published 6 March 2015