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 $K_{M}\leq -k^{2}$, $k>0$. 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 $x\in \operatorname *{Int}\left ( \partial _{\infty }\Omega \right ) \subset W$ and $M\setminus \Omega$ is convex. We prove that the SC condition implies that $M$ is regular at infinity relative to the operator \[ \mathcal {Q}\left [ u\right ] :=\mathrm {{ div }}\left ( \frac {a(|\nabla u|)}{|\nabla u|}\nabla u\right ) , \] 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.