Asymptotic Dirichlet problems in warped products

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Casteras , J-B , Heinonen , E , Holopainen , I & Lira , J 2020 , ' Asymptotic Dirichlet problems in warped products ' , Mathematische Zeitschrift , vol. 295 , no. 1-2 , pp. 211-248 .

Title: Asymptotic Dirichlet problems in warped products
Author: Casteras, Jean-Babtiste; Heinonen, Esko; Holopainen, Ilkka; Lira, Jorge
Contributor: University of Helsinki, Department of Mathematics and Statistics
University of Helsinki, Geometric Analysis and Partial Differential Equations
Date: 2020-06
Language: eng
Number of pages: 38
Belongs to series: Mathematische Zeitschrift
ISSN: 0025-5874
Abstract: We study the asymptotic Dirichlet problem for Killing graphs with prescribed mean curvature H in warped product manifolds Mx rho R. In the first part of the paper, we prove the existence of Killing graphs with prescribed boundary on geodesic balls under suitable assumptions on H and the mean curvature of the Killing cylinders over geodesic spheres. In the process we obtain a uniform interior gradient estimate improving previous results by Dajczer and de Lira. In the second part we solve the asymptotic Dirichlet problem in a large class of manifolds whose sectional curvatures are allowed to go to 0 or to -infinity provided that H satisfies certain bounds with respect to the sectional curvatures of M and the norm of the Killing vector field. Finally we obtain non-existence results if the prescribed mean curvature function H grows too fast.
Subject: 111 Mathematics
Mean curvature equation
Killing graph
Dirichlet problem
Hadamard manifold
warped product

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