Abstract
In the present work, we consider solvability of the generalized Dirichlet problem for the linear elliptic differential equation \(Lu=f\) , where \(L=\Delta +\langle B(x),\nabla\rangle+c(x)\) is a linear operator, ( \(B(x)\) is a vector field of class \(\mathrm{C}(\mathcal{M})\) , \(c(x)\leq0\) , \(c(x)\in \mathrm{C}(\mathcal{M})\) ), considered on a non-compact Riemannian manifold \((\mathcal{M},g)\) . We develop the approach to this problem, based on equivalence classes, introduced by E. A. Mazepa, which allows to state the problem on non-compact manifolds in the absence of a natural geometric compactification. We introduce and study linear spaces \(\mathrm{CM}_b\) and \(\mathrm{CM}\) of such classes. We give a version of the well-known Perron’s method with boundary data in these classes, and establish signs of \(L\) -parabolicity and \(L\) -hyperbolicity of the ends of the manifold \(\mathcal{M}\) depending on their geometric structure. The signs of hyperbolicity of a manifold play a key role in justifying solvability of the Dirichlet problem, while signs of parabolicity are important for establishing theorems of Liouville type for the manifold.
