Abstract

This article explores and develops opportunities Fourier method of separation of variables for the study of the asymptotic behavior of harmonic functions on noncompact Riemannian manifolds of a special form. These manifolds generalize spherically symmetric manifold and are called model ones in a series of works. In the first part of the paper, an estimate of the eigenfunctions of the Laplace operator is obtained on compact Riemannian manifolds \(S\) in the norm \(C^{m}(S)\). In the second part of the paper, the conditions for the unique solvability of the Dirichlet problem for harmonic functions on model manifolds with smooth boundary data at ‘‘infinity’’ are found. It was shown that the solution of this boundary value problem converges to the boundary data in the \(C^{1}\)-norm.

中文翻译：

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黎曼流形上的拉普拉斯算子的本征函数和调和函数

摘要

本文探讨并开发了机会分离的傅立叶方法，用于研究特殊形式的非紧黎曼流形上的调和函数的渐近行为。这些流形泛化了球对称流形，并在一系列工作中被称为模型流形。在本文的第一部分中，在范数\（C ^ {m}（S）\）中的紧致黎曼流形\（S \）上获得了Laplace算子本征函数的估计。在本文的第二部分中，找到了具有在“无穷大”处具有光滑边界数据的模型流形上的调和函数的Dirichlet问题的唯一可解性的条件。结果表明，该边值问题的解收敛于边界数据的边界。\（C ^ {1} \）-规范。