Harmonic potentials on non-compact Riemannian manifolds

The work is carried out within the framework of the topic devoted to the asymptotic behavior of solutions of partial differential equations on non-compact Riemannian manifolds. The most popular sections of such studies are theorems of the Liouville type on the triviality of spaces of solutions of elliptic equations on non-compact Riemannian manifolds, as well as questions of solvability of boundary value problems. The currently considered classical formulation of the Liouville theorem states that any bounded harmonic function in is identically constant. Recently, a tendency has emerged towards a more general approach to theorems of the Liouville type, namely, the dimensions of various spaces of solutions of linear equations of elliptic type are estimated. In particular, in the work by A.A. Grigory’an (1990), an exact estimate of the dimensions of spaces of bounded harmonic functions on non-compact Riemannian manifolds in terms of massive sets was proved. In general, studies of the last decades have shown the extremely high efficiency of using capacitive techniques in solving the above problems. This work is devoted to the development of capacitive techniques related to the concepts of a massive set in the study of the asymptotic behavior of harmonic functions on non-compact Riemannian manifolds.