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Pogorelov estimates for the Monge-Ampère equations


Authors: JuHua Shi and Feida Jiang
Journal: Proc. Amer. Math. Soc. 147 (2019), 2561-2571
MSC (2010): Primary 35J96, 35J70, 35J60
DOI: https://doi.org/10.1090/proc/14400
Published electronically: February 20, 2019
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Abstract: In this paper, we study the Pogorelov estimate for the Monge-Ampère equation $ \det D^{2}u=f(x)$ under the assumption $ f^{\frac {1}{n-1}}\in C^{1,1}(\bar \Omega )$. When $ n\ge 3$, we improve the Pogorelov estimate $ (w-u)^\alpha \vert D^2u\vert\le C$ by Błocki [Bull. Austral. Math. Soc. 68 (2003), pp. 81-92] from $ \alpha =n-1$ to all $ \alpha >1$. Some applications of the Pogorelov estimate are discussed.


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Additional Information

JuHua Shi
Affiliation: School of Science, Nanjing University of Science and Technology, Nanjing 210094, People’s Republic of China
Email: ashijuhua@163.com

Feida Jiang
Affiliation: College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, People’s Republic of China
Email: jfd2001@163.com

DOI: https://doi.org/10.1090/proc/14400
Keywords: Pogorelov estimate, degenerate equations, Monge--Amp\`ere equations
Received by editor(s): November 16, 2017
Received by editor(s) in revised form: September 12, 2018, and September 15, 2018
Published electronically: February 20, 2019
Additional Notes: The second author served as corresponding author for this paper. The second author was supported by the National Natural Science Foundation of China (No. 11771214).
Communicated by: Joachim Krieger
Article copyright: © Copyright 2019 American Mathematical Society