Pogorelov estimates for the Monge–Ampère equations

Shi, JuHua; Jiang, Feida

doi:10.1090/proc/14400

Pogorelov estimates for the Monge–Ampère equations
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by JuHua Shi and Feida Jiang PDF

Proc. Amer. Math. Soc. 147 (2019), 2561-2571 Request permission

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 $(w-u)^\alpha |D^2u|\le C$ 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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