Communications in Analysis and Geometry

Volume 27 (2019)

Number 4

Asymptotic Dirichlet problem for ${\mathcal A}$-harmonic and minimal graph equations in Cartan–Hadamard manifolds

Pages: 809 – 855

DOI: https://dx.doi.org/10.4310/CAG.2019.v27.n4.a3

Authors

Jean-Baptiste Casteras (Departement de Mathematique, Universite libre de Bruxelles, Belgium)

Ilkka Holopainen (Department of Mathematics and Statistics, University of Helsinki, Finland)

Jaime B. Ripoll (Instituto de Matemática, Universidad Federal do Rio Grande do Sul Porto Alegre, RS, Brazil)

Abstract

We study the asymptotic Dirichlet problem for $\mathcal{A}$-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 $r =d(\cdot , o)$ to a fixed point $o \in M$. 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 $n \geq 3$ if radial sectional curvatures satisfy\[\dfrac {(\log (r(x))^{2 \overline{\varepsilon}}}{r(x)^2} \leq K \leq - \dfrac {1 + \varepsilon}{r(x)^2 \log r(x)}\]outside a compact set for some $\varepsilon \gt \overline{\varepsilon} \gt 0$. The upper bound is closeto optimal since the nonsolvability is known if\[K \geq -1 / (2r(x)^2 \log r(x)) \quad \textrm{.}\]Our results (in the non-rotationally symmetric case) improve on the previously known case of the quadratically decaying upper bound.

Received 6 March 2016

Accepted 24 April 2017

Published 8 October 2019