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

TL;DR

This paper investigates the asymptotic Dirichlet problem for A-harmonic and minimal graph equations on Cartan-Hadamard manifolds, focusing on optimal curvature bounds to understand boundary value problems at infinity.

Contribution

It provides new insights into curvature bounds that ensure solvability of the asymptotic Dirichlet problem for these equations on negatively curved manifolds.

Findings

Established conditions on curvature bounds for solvability

Derived optimal or near-optimal curvature estimates

Extended understanding of boundary behavior for harmonic and minimal graphs

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 to a fixed point in M. We are, in particular, interested in finding optimal (or close to optimal) curvature upper bounds.