On Pogorelov estimates for Monge-Ampère type equations

Jiakun Liu; Neil S. Trudinger

doi:10.3934/dcds.2010.28.1121

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2010, Volume 28, Issue 3: 1121-1135. Doi: 10.3934/dcds.2010.28.1121

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On Pogorelov estimates for Monge-Ampère type equations

Jiakun Liu^1, and
Neil S. Trudinger^2,

1.
Centre for Mathematics and Its Applications, The Australian National University, Canberra, ACT 0200
2.
Centre for Mathematics and Its Applications, the Australian National University, Canberra, ACT 0200

Received: February 28, 2010

Revised: March 31, 2010

Published: August 2010

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Abstract

In this paper, we prove interior second derivative estimates of Pogorelov type for a general form of Monge-Ampère equation which includes the optimal transportation equation. The estimate extends that in a previous work with Xu-Jia Wang and assumes only that the matrix function in the equation is regular with respect to the gradient variables, that is it satisfies a weak form of the condition introduced previously by Ma,Trudinger and Wang for regularity of optimal transport mappings. We also indicate briefly an application to optimal transportation.

Keywords:

Pogorelov estimates,
optimal transportation.

Mathematics Subject Classification: Primary: 35J60, 35B45; Secondary: 49Q20.

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