The interior \(C^2\) estimate for the Monge-Ampère equation in dimension \(n = 2\)
type: Item
instance of: scholarly article
title: The interior \(C^2\) estimate for the Monge-Ampère equation in dimension \(n = 2\)
zbMATH Open document ID: 1353.35175
published in: Analysis \& PDE
publication date: 2016-10-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/1512.01146
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 35J96
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 6637201
zbMATH Keywords: interior \(C^2\) a priori estimate
zbMATH Keywords: Monge-Ampère equation
zbMATH Keywords: \(\sigma^2\)-Hessian equation
zbMATH Keywords: optimal concavity
MaRDI profile type: Publication
OpenAlex ID: W3102073546
arXiv ID: 1512.01146
DOI: 10.2140/APDE.2016.9.1419
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review text: The paper deals with convex solutions of the Monge-Ampère equation NEWLINE\[NEWLINE\det D^2u=f(x)\quad \text{in }B_R(0)\subset\mathbb R^2.NEWLINE\]NEWLINE The interior \(C^2\) estimates for the convex solutions have been proved by \textit{E. Heinz} [J. Anal. Math. 7, 1--52 (1959; Zbl 0152.30901)] for planar planar domains, and by \textit{A. V. Pogorelov} [The Minkowski multidimensional problem. New York etc.: John Wiley \& Sons (1978; Zbl 0387.53023)] for dimensions \(n\geq3\).NEWLINENEWLINEThe authors rediscover the Heinz result by considering suitable new auxiliary functions.
description: scientific article; zbMATH DE number 6637201