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Gradient estimates for the constant mean curvature equation in hyperbolic space

Part of: Partial differential equations Elliptic equations and systems Classical differential geometry

Published online by Cambridge University Press:  09 December 2019

Rafael López
Rafael López*
Affiliation:
Departamento de Geometría y Topología, Instituto de Matemáticas (IEMath-GR), Universidad de Granada, 18071Granada, Spain (rcamino@ugr.es)
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Abstract

We establish gradient estimates for solutions to the Dirichlet problem for the constant mean curvature equation in hyperbolic space. We obtain these estimates on bounded strictly convex domains by using the maximum principles theory of Φ-functions of Payne and Philippin. These estimates are then employed to solve the Dirichlet problem when the mean curvature H satisfies H < 1 under suitable boundary conditions.

Keywords

hyperbolic spaceDirichlet problemmaximum principlecritical point

MSC classification

Primary: 35J62: Quasilinear elliptic equations
Secondary: 35J25: Boundary value problems for second-order elliptic equations 35J93: Quasilinear elliptic equations with mean curvature operator 35B38: Critical points 53A10: Minimal surfaces, surfaces with prescribed mean curvature
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 3216 - 3230
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Copyright
Copyright © 2019 The Royal Society of Edinburgh

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