Interior curvature bounds for a class of curvature equations

Weimin Sheng; John Urbas; Xu Jia Wang

doi:10.1215/S0012-7094-04-12321-8

Interior curvature bounds for a class of curvature equations

Weimin Sheng^*, John Urbas, Xu Jia Wang

^*Corresponding author for this work

Research output: Contribution to journal › Review article › peer-review

44 Citations (Scopus)

Abstract

We derive interior curvature bounds for admissible solutions of a class of curvature equations subject to affine Dirichlet data, generalizing a well-known estimate of Pogorelov for equations of Mange-Ampère type. For equations for which convexity of the solution is the natural ellipticity assumption, the curvature bound is proved for solutions with C^1,1 Dirichlet data. We also use the curvature bounds to improve and extend various existence results for the Dirichlet and Plateau problems.

Original language	English
Pages (from-to)	235-264
Number of pages	30
Journal	Duke Mathematical Journal
Volume	123
Issue number	2
DOIs	https://doi.org/10.1215/S0012-7094-04-12321-8
Publication status	Published - 1 Jun 2004

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title = "Interior curvature bounds for a class of curvature equations",

abstract = "We derive interior curvature bounds for admissible solutions of a class of curvature equations subject to affine Dirichlet data, generalizing a well-known estimate of Pogorelov for equations of Mange-Amp{\`e}re type. For equations for which convexity of the solution is the natural ellipticity assumption, the curvature bound is proved for solutions with C1,1 Dirichlet data. We also use the curvature bounds to improve and extend various existence results for the Dirichlet and Plateau problems.",

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T1 - Interior curvature bounds for a class of curvature equations

AU - Sheng, Weimin

AU - Urbas, John

AU - Wang, Xu Jia

PY - 2004/6/1

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N2 - We derive interior curvature bounds for admissible solutions of a class of curvature equations subject to affine Dirichlet data, generalizing a well-known estimate of Pogorelov for equations of Mange-Ampère type. For equations for which convexity of the solution is the natural ellipticity assumption, the curvature bound is proved for solutions with C1,1 Dirichlet data. We also use the curvature bounds to improve and extend various existence results for the Dirichlet and Plateau problems.

AB - We derive interior curvature bounds for admissible solutions of a class of curvature equations subject to affine Dirichlet data, generalizing a well-known estimate of Pogorelov for equations of Mange-Ampère type. For equations for which convexity of the solution is the natural ellipticity assumption, the curvature bound is proved for solutions with C1,1 Dirichlet data. We also use the curvature bounds to improve and extend various existence results for the Dirichlet and Plateau problems.

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U2 - 10.1215/S0012-7094-04-12321-8

DO - 10.1215/S0012-7094-04-12321-8

M3 - Review article

SN - 0012-7094

VL - 123

SP - 235

EP - 264

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

IS - 2

ER -

Interior curvature bounds for a class of curvature equations

Abstract

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