Eigenfunctions of the Laplace Operator and Harmonic Functions on Model Riemannian Manifolds

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.