Boundary regularity for solutions of the equation of prescribed Gauss curvature

J. I.E. Urbas

doi:10.1016/S0294-1449(16)30259-1

Boundary regularity for solutions of the equation of prescribed Gauss curvature

J. I.E. Urbas

Research output: Contribution to journal › Article › peer-review

5 Citations (Scopus)

Abstract

We study the boundary regularity of convex solutions of the equation of prescribed Gauss curvature in a domain Ω ⊂ ℝⁿ in the case that the gradient of the solution is infinite on some relatively open, uniformly convex portion Γ of ∂Ω. Under suitable conditions on the data we show that near Γ × ℝ the graph of u is a smooth hypersurface (as a submanifold of ℝ^{n + 1}) and that u|_Γ is smooth. In particular, u is Hölder continuous with exponent 1/2 near Г.

Original language	English
Pages (from-to)	499-522
Number of pages	24
Journal	Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume	8
Issue number	5
DOIs	https://doi.org/10.1016/S0294-1449(16)30259-1
Publication status	Published - 1 Sept 1991

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title = "Boundary regularity for solutions of the equation of prescribed Gauss curvature",

abstract = "We study the boundary regularity of convex solutions of the equation of prescribed Gauss curvature in a domain Ω ⊂ ℝn in the case that the gradient of the solution is infinite on some relatively open, uniformly convex portion Γ of ∂Ω. Under suitable conditions on the data we show that near Γ × ℝ the graph of u is a smooth hypersurface (as a submanifold of ℝn + 1) and that u|Γ is smooth. In particular, u is H{\"o}lder continuous with exponent 1/2 near Г.",

keywords = "Gauss curvature, Legendre transform, boundary regularity, free boundary",

author = "Urbas, {J. I.E.}",

note = "Publisher Copyright: {\textcopyright} 2016 L'Association Publications de l'Institut Henri Poincar{\'e}",

year = "1991",

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AU - Urbas, J. I.E.

PY - 1991/9/1

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N2 - We study the boundary regularity of convex solutions of the equation of prescribed Gauss curvature in a domain Ω ⊂ ℝn in the case that the gradient of the solution is infinite on some relatively open, uniformly convex portion Γ of ∂Ω. Under suitable conditions on the data we show that near Γ × ℝ the graph of u is a smooth hypersurface (as a submanifold of ℝn + 1) and that u|Γ is smooth. In particular, u is Hölder continuous with exponent 1/2 near Г.

AB - We study the boundary regularity of convex solutions of the equation of prescribed Gauss curvature in a domain Ω ⊂ ℝn in the case that the gradient of the solution is infinite on some relatively open, uniformly convex portion Γ of ∂Ω. Under suitable conditions on the data we show that near Γ × ℝ the graph of u is a smooth hypersurface (as a submanifold of ℝn + 1) and that u|Γ is smooth. In particular, u is Hölder continuous with exponent 1/2 near Г.

KW - Gauss curvature

KW - Legendre transform

KW - boundary regularity

KW - free boundary

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U2 - 10.1016/S0294-1449(16)30259-1

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JO - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

JF - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

IS - 5

ER -

Boundary regularity for solutions of the equation of prescribed Gauss curvature

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