ON SOLVABILITY OF THE BOUNDARY VALUE PROBLEMS FOR HARMONIC FUNCTION ON NONCOMPACT RIEMANNIAN MANIFOLDS

Abstract

We study questions of existence and belonging to the given functional class of solutions of the Laplace-Beltrami equations on a noncompact Riemannian manifold M with no boundary. In the present work we suggest the concept of φ-equivalency in the class of continuous functions and establish some interrelation between problems of existence of solutions of the Laplace-Beltrami equations on M and off some compact B ⊂ M with the same growth "at infinity". A new conception of φ-equivalence classes of functions on M develops and generalizes the concept of equivalence of function on M and allows us to more accurately estimate the rate of convergence of the solution to boundary conditions.

Keywords

• riemannian manifold
• harmonic function
• boundaryvalue problems
• φ-equivalency