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 $\mathrm{C}(\mathcal{M})$, $c(x)\leqslant 0$, $c(x)\in \mathrm{C}(\mathcal{M})$), 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.
Keywords:non-compact Riemannian manifold, end of a manifold, Perron's method, equivalence class, spaces $\mathrm{CM}_b$, $\mathrm{CM}$.
Fulltext:
PDF file (895 kB)
First page:
PDF file