On solvability of the boundary value problems for the inhomogeneous elliptic equations on noncompact Riemannian manifolds

Abstract

We study questions of existence and belonging to a given functional class of solutions of the inhomogeneous elliptic equations ∆u − c(x)u = g(x), where c(x) 0, g(x) are Hölder fuctions on a noncompact Riemannian manifold M without boundary. In this work we develop an approach to evaluation of solutions to boundary-value problems for linear and quasilinear equations of the elliptic type on arbitrary noncompact Riemannian manifolds. Our technique is essentially based on an approach from the papers by E. A. Mazepa and S. A. Korol'kov connected with an introduction of equivalency classes of functions and representations. We investigate the relationship between the existence of solutions of this equation on M and outside some compact set B ⊂ M with the same growth "at infinity".

On solvability of the boundary value problems for harmonic function on noncompact Riemannian manifolds

“…Using this approach has been established the interrelation between the solvability of boundary value problems and solvability of exterior boundary problems for the stationary Schrödinger equation on noncompact Rieman-nian manifold (see e. g. [12]). A similar result for inhomogeneous elliptic equations was obtained in [11].…”

“…We formulate and prove some auxiliary assertions. Analogy statements for class of equivalence functions were proved in [11,12]. The proof of all results is based on classical propositions of the theory of equations with partial derivatives: the Maximum Principle, the Comparison and Uniqueness Theorems for solutions to linear elliptic differential equations.…”

“…In the paper [4] it is shown that parabolicity of the type of a complete Riemannian manifold is equivalent to the fact that the capacity of any compact set is zero. Moreover, the capacitive technique has shown high efficiency in studying the behavior of solutions of elliptic equations and inequalities on noncompact Riemannian manifolds (see [4][5][6][7]11]).…”

“…Remark 1. The connections between solvability of boundary-value and exterior boundary-value problems for linear and quasilinear elliptic equations in terms of equivalent functions is investigated in detail, for example, in [11,12].…”