On solvability of boundary value problems of the Poisson equation on non-compact Riemannian manifolds

This article is devoted to the investigation of the behavior of solutions of the Poisson equation in relation to the geometry of the manifold in question. Such problems originate in the classification theory of non-compact Riemannian surfaces and manifolds. For a noncompact Riemann surface, the well-known problem of conformal type identification can be stated as follows: Does a nontrivial positive superharmonic function exist on this surface?
Many questions of this kind fit into the pattern of a Liouville-type theorem saying that the space of bounded solutions of some elliptic equation is trivial. However, the class of manifolds admitting nontrivial solutions of some elliptic equations is wide. For example, conditions ensuring the solvability of the Dirichlet problem with continuous boundary conditions “at infinity” for several noncompact manifolds has been found in many papers (see, e.g., [12;18;21]). Notice that the very statement of the Dirichlet problem on such manifolds could turn out nontrivial, since it is unclear how we should interpret the boundary data.
In this article we study questions of existence and belonging to given functional class of bounded solutions of the Poisson equation
$$ \Delta u=g(x), \,(1) $$
where $g(x)\in C^{\gamma}(\Omega)$ for any subset $\Omega\subset\subset M$, $0<\gamma<1$ on a noncompact Riemannian manifold $M$ without boundary.
Of keen interest is the interrelation between problems of existence of solutions of equation (1) on $M$ and off some compact $B \subset M$ with the same growth “at infinity”. In our research we use a new approach wich is based on the consideration of equivalence classes of functions on $M$ (this approach for bounded solutions of the Schrödinger equation has been realized in [9]).
Let $M$ be an arbitrary smooth connected noncompact Riemannian manifold without boundary and let $\{B_k\}_{k=1}^{\infty}$ be an exhaustion of $M$. Throughout the sequel, we assume that boundaries $\partial B_k$ are $C^1$-smooth submanifolds.
Let $f_1$ and $f_2$ be arbitrary bounded continuous functions on $M$. Say that $f_1$ and $f_2$ are equivalent on $M$ and write $f_1\sim f_2$ if for some exhaustion $\{B_k\}_{k=1}^{\infty}$ of $M$ we have
$$ \lim_{k\to\infty}\sup_{M\setminus B_k}|f_1-f_2|=0. $$

It is easy to verify that the relation $``\sim``$ is an equivalence which does not depend on the choice of the exhaustion of the manifold and so partitions the set of all continuous functions on $M$ into equivalence classes. Denote the equivalence class of a function $f$ by $[f]$.
Let $B\subset M$ be an arbitrary connected compact subset and the boundary of $B$ is a $C^1$-smooth submanifold. Assume that the interior of $B$ is non-empty and $B\subset B_k$ for all $k$.
Observe that if the manifold $M$ has compact boundary or there is a natural geometric compactification of $M$ (for example, on manifolds of negative sectional curvature or spherically symmetric manifolds) which adds the boundary at infinity, then this approach leads naturally to the classical statement of the Dirichlet problem (see, for instance, [7;12;18;21]).
Denote by $v_k$ the harmonic function in $B_k\setminus B$ which satisfies to conditions
$$ \left.v_k\right|_{\partial{B}}=1,\qquad \left.v_k\right|_{\partial{B_k}}=0. $$
Using the maximum principle, we can easily verify that the sequence $v_k$ is uniformly bounded on $M\setminus B$ and so is compact in the class of twice continuously differentiable functions over every compact subset $G\subset M\setminus B$. Moreover, as $k\to\infty$ this sequence increases monotonically and converges on $M\setminus B$ to harmonic function
$$ v=\lim_{k\to\infty}v_k,\qquad 0<v\leq 1,\qquad \left.v\right|_{\partial{B}}=1. $$

Also, note that the function $v$ is independent of the choice of exhaustion $\{B_k\}_{k=1}^{\infty}$.
The function $v$ is nothing but the capacity potential of the compact set $B$ relative to the manifold $M$ (see [14]).
Call the manifold $M$ $\Delta$-strict if for some compact set $B\subset M$ there is a ecapacity potential $v$ of $B$ such that $v\in[0]$ (see [9]).
Say that a boundary value problem for (1) is solvable on $M$ with boundary conditions of class $[f]$ whenever there exists a solution $u(x)$ to (1) on $M$ with $u\in [f]$.
Say that for a continuous function $\Phi(x)$ on $\partial B$ the exterior boundary value problem for (1) is solvable on $M\setminus B$ with boundary conditions of class $(\Phi, [f])$ whenever on $M \setminus B$ there exists a solution $u(x)$ to (1) with $u\in[f]$ and $u|_{\partial B}=\Phi$.
Similarly we can state boundary value problems on arbitrary noncompact Riemannian manifolds for a series of other second order elliptic differential equations (see [5;8-11;16]).
We now formulate the main result.
Theorem 1. Suppose that for every positive constant $A$ the exterior boundary value problems for the equations (1) are solvable on $M \setminus B$ with boundary conditions $(A,[f])$. Then the boundary value problem for (1) is solvable on $M$ with boundary conditions of class $[f]$.
Theorem 2. Let $M$ be an $\Delta$-strict manifold. Suppose that the boundary value problem for (1) is solvable on $M$ with boundary conditions of class $[f]$. Then for every continuous function $\Phi(x)$ on $\partial B$ the exterior boundary value problem for (1) is solvable on $M \setminus B$ with boundary conditions $(\Phi, [f])$.