The interior \(C^2\) estimate for the Monge-Ampère equation in dimension \(n = 2\) (Q325826)
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scientific article; zbMATH DE number 6637201
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | The interior \(C^2\) estimate for the Monge-Ampère equation in dimension \(n = 2\) |
scientific article; zbMATH DE number 6637201 |
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The interior \(C^2\) estimate for the Monge-Ampère equation in dimension \(n = 2\) (English)
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11 October 2016
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interior \(C^2\) a priori estimate
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Monge-Ampère equation
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\(\sigma^2\)-Hessian equation
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optimal concavity
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0.99239457
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0.98237884
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0.9670688
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0.9557302
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0.9353684
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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.
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