Boundary and outer boundary-value problems for the Poisson equation on noncompact Riemannian manifolds

In this paper, we examine the existence of solutions of the Poisson equations on a noncompact Riemannian manifold $M$ without boundary. To describe the asymptotic behavior of a solution, we is introduce the notion of $\varphi$-equivalence on the set of continuous functions on a Riemannian manifold and establish a relationship between the solvability of boundary-value problems for the Poisson equations on the manifold $M$ and outside some compact subset $B\subset M$ with the same growth “at infinity.” Moreover, the notion of $\varphi$-equivalence of continuous functions on $M$ allows one to estimate the rate of asymptotic convergence of solutions of boundary-value and outer boundary-value problems to boundary data.