Asymptotic Dirichlet problem for $\mathcal{A}$-harmonic functions on manifolds with pinched curvature

TL;DR

This paper solves the asymptotic Dirichlet problem for a-harmonic functions on certain negatively curved manifolds, extending results to Laplacian and p-Laplacian cases under specific curvature conditions.

Contribution

It establishes existence results for the asymptotic Dirichlet problem on manifolds with pinched curvature bounds, including Laplacian and p-Laplacian operators.

Findings

Solves the asymptotic Dirichlet problem for a-harmonic functions.

Extends results to Laplacian and p-Laplacian operators.

Works under curvature bounds involving logarithmic decay.

Abstract

We study the asymptotic Dirichlet problem for $\mathcal{A}$-harmonic functions on a Cartan-Hadamard manifold whose radial sectional curvatures outside a compact set satisfy an upper bound $$ K(P)\le - \frac{1+\varepsilon}{r(x)^2 \log r(x)} $$ and a pointwise pinching condition $$ |K(P)|\le C_K |K(P')| $$ for some constants $\varepsilon>0$ and $C_K\ge 1$, where $P$ and $P'$ are any 2-dimensional subspaces of $T_xM$ containing the (radial) vector $\nabla r(x)$ and $r(x)=d(o,x)$ is the distance to a fixed point $o\in M$. We solve the asymptotic Dirichlet problem with any continuous boundary data $f\in C(\partial_{\infty} M)$. The results apply also to the Laplacian and $p$-Laplacian, $1<p<\infty,$ as special cases.