Boundary Classes of Non-Compact Riemannian Manifolds and Perron’s Method

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.