# American Institute of Mathematical Sciences

July  2010, 28(3): 1121-1135. doi: 10.3934/dcds.2010.28.1121

## On Pogorelov estimates for Monge-Ampère type equations

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

Received  March 2010 Revised  April 2010 Published  April 2010

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.
Citation: Jiakun Liu, Neil S. Trudinger. On Pogorelov estimates for Monge-Ampère type equations. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1121-1135. doi: 10.3934/dcds.2010.28.1121
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