On Pogorelov estimates for Monge-Ampère type equations

Jiakun Liu; Neil S. Trudinger

doi:10.3934/dcds.2010.28.1121

On Pogorelov estimates for Monge-Ampère type equations

Jiakun Liu^*, Neil S. Trudinger

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

21 Citations (Scopus)

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.

Original language	English
Pages (from-to)	1121-1135
Number of pages	15
Journal	Discrete and Continuous Dynamical Systems
Volume	28
Issue number	3
DOIs	https://doi.org/10.3934/dcds.2010.28.1121
Publication status	Published - Nov 2010

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10.3934/dcds.2010.28.1121

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title = "On Pogorelov estimates for Monge-Amp{\`e}re type equations",

abstract = "In this paper, we prove interior second derivative estimates of Pogorelov type for a general form of Monge-Amp{\`e}re 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 = "Optimal transportation, Pogorelov estimates",

author = "Jiakun Liu and Trudinger, \{Neil S.\}",

year = "2010",

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doi = "10.3934/dcds.2010.28.1121",

language = "English",

volume = "28",

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journal = "Discrete and Continuous Dynamical Systems",

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T1 - On Pogorelov estimates for Monge-Ampère type equations

AU - Liu, Jiakun

AU - Trudinger, Neil S.

PY - 2010/11

Y1 - 2010/11

N2 - 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.

AB - 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.

KW - Optimal transportation

KW - Pogorelov estimates

UR - https://www.scopus.com/pages/publications/77955326794

U2 - 10.3934/dcds.2010.28.1121

DO - 10.3934/dcds.2010.28.1121

M3 - Article

SN - 1078-0947

VL - 28

SP - 1121

EP - 1135

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

IS - 3

ER -

On Pogorelov estimates for Monge-Ampère type equations

Abstract

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