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

Abstract

Let $M$ be Hadamard manifold with sectional curvature $K_M\le -k^2$, $k>0$. 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 $x\in \mathrm{Int}({\partial }_{\infty }\Omega )\subset W$ 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 ${\inf }_{B_{R+1}}{K}_M\ge -e^{2kR}/R^{2+2\epsilon },R\ge R^{\ast },$ 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.