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

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 $$K(P)\le -\frac{1+\epsilon }{r(x{)}^2\log r(x)}$$ and a pointwise pinching condition $$|K(P)|\le C_K|K(P^{\prime })|$$ for some constants $\epsilon >0$ and $C_K\ge 1$, where $P$ and $P^{\prime }$ are any 2-dimensional subspaces of $T_{x}M$ 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.

arXiv admin note: substantial text overlap with arXiv:1504.05378

Cited by in corpus (1)

• Existence and non-existence of minimal graphic and $p$-harmonic functions