Asymptotic Dirichlet problem for A-harmonic and minimal graph equations in Cartan-Hadamard manifolds

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Casteras , J-B , Holopainen , I & Ripoll , J B 2019 , ' Asymptotic Dirichlet problem for A-harmonic and minimal graph equations in Cartan-Hadamard manifolds ' , Communications in Analysis and Geometry , vol. 27 , no. 4 , pp. 809-855 . https://doi.org/10.4310/cag.2019.v27.n4.a3

Title: Asymptotic Dirichlet problem for A-harmonic and minimal graph equations in Cartan-Hadamard manifolds
Author: Casteras, Jean-Baptiste; Holopainen, Ilkka; Ripoll, Jaime B.
Contributor organization: Department of Mathematics and Statistics
Geometric Analysis and Partial Differential Equations
Date: 2019-10-08
Language: eng
Number of pages: 47
Belongs to series: Communications in Analysis and Geometry
ISSN: 1019-8385
DOI: https://doi.org/10.4310/cag.2019.v27.n4.a3
URI: http://hdl.handle.net/10138/306970
Abstract: We study the asymptotic Dirichlet problem for A-harmonic equations and for the minimal graph equation on a Cartan-Hadamard manifold M whose sectional curvatures are bounded from below and above by certain functions depending on the distance r = d(., o) to a fixed point o is an element of M. We are, in particular, interested in finding optimal (or close to optimal) curvature upper bounds. In the special case of the Laplace-Beltrami equation we are able to solve the asymptotic Dirichlet problem in dimensions n >= 3 if radial sectional curvatures satisfy -(logr(x))(2 (epsilon) over bar)/r(x)(2 ) outside a compact set for some epsilon > (epsilon) over bar > 0. The upper bound is close to optimal since the nonsolvability is known if K >= -1/(2r(x)(2)log r(x)). Our results (in the non-rotationally symmetric case) improve on the previously known case of the quadratically decaying upper bound.
Subject: 111 Mathematics
ISOPERIMETRIC-INEQUALITIES
ELLIPTIC-OPERATORS
BROWNIAN-MOTION
KILLING GRAPHS
INFINITY
SURFACES
DIFFEOMORPHISMS
NONSOLVABILITY
THEOREMS
Peer reviewed: Yes
Rights: other
Usage restriction: openAccess
Self-archived version: acceptedVersion


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