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

Abstract

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