Regularity at infinity of Hadamard manifolds with respect to some elliptic operators and applications to asymptotic Dirichlet problems

TL;DR

This paper shows that Hadamard manifolds satisfying a strict convexity condition are regular at infinity for certain elliptic operators, enabling solutions to asymptotic Dirichlet problems for minimal hypersurfaces and p-Laplacian equations.

Contribution

It establishes the link between the SC condition and regularity at infinity for a class of elliptic operators on Hadamard manifolds, and provides geometric conditions ensuring the SC condition.

Findings

SC condition implies regularity at infinity for the operator Q[u]

Solvability of Dirichlet problems for minimal hypersurface and p-Laplacian equations under SC

Geometric conditions like rotational symmetry or curvature bounds imply SC condition

Abstract

Let $M$ be Hadamard manifold with sectional curvature $K_{M}\leq-k^{2}$, $k>0$. Denote by $\partial_{\infty}M$ the asymptotic boundary of $M$. We say that $M$ satisfies the strict convexity condition (SC condition) if, given $x\in\partial_{\infty}M$ and a relatively open subset $W\subset\partial_{\infty}M$ containing $x$, there exists a $C^{2}$ open subset $\Omega\subset M$ such that $x\in\operatorname*{Int}(\partial_{\infty}\Omega) \subset W$ and $M\setminus\Omega$ is convex. We prove that the SC condition implies that $M$ is regular at infinity relative to the operator $$\mathcal{Q}[u] :=\mathrm{{div}}(\frac{a(|\nabla u|)}{|\nabla u|}\nabla u),$$ subject to some conditions. It follows that under the SC condition, the Dirichlet problem for the minimal hypersurface and the $p$-Laplacian ($p>1$) equations are solvable for any prescribed continuous asymptotic boundary data. It is also proved that if $M$ is rotationally symmetric or if $\inf_{B_{R+1}}K_{M}\geq-e^{2kR}/R^{2+2\epsilon}, R\geq R^{\ast},$ for some $R^{\ast}$ and $\epsilon>0,$ where $B_{R+1}$ is the geodesic ball with radius $R+1$ centered at a fixed point of $M,$ then $M$ satisfies the SC condition.