H^n中无界区域上Laplacian特征函数的存在性和不存在性
Existence and nonexistence results for eigenfunctions of the Laplacian in unbounded domains of H^n
摘要：
本文研究了双曲空间H^n的Laplacian算子在无界区域上的零到第一特征值之间的特征值的特征函数的存在性和不存在性.如果一个区域包含在一个horoball，我们证明了没有正的有界特征函数，在边界上消失。然而，如果区域的渐近边界包含H^n的渐近边界的开集，则存在在无穷远处收敛到0的解，并且可以连续扩展到渐近边界。特别是，这一结果适用于超球。
英文摘要：
We investigate, for the Laplacian operator, the existence and nonexistence of eigenfunctions of eigenvalue between zero and the first eigenvalue of the hyperbolic space H^n, for unbounded domains of H^n. If a domain is contained in a horoball, we prove that there is no positive bounded eigenfunction that vanishes on the boundary. However, if the asymptotic boundary of a domain contains an open set of the asymptotic boundary of H^n, there is a solution that converges to 0 at infinity and can be extended continuously to the asymptotic boundary. In particular, this result holds for hyperballs.
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