Bessel capacities on compact manifolds and their relation to Poisson capacities

Abstract

A motivation for this paper comes from the role of Choquet capacities in the study of semilinear elliptic partial differential equations. In particular, the recent progress in the classification of all positive solutions of L u = u α in a bounded smooth domain E ⊂ R d was achieved by using, as a tool, capacities on a smooth manifold ∂ E. Either the Poisson capacities (associated with the Poisson kernel in E) or the Bessel capacities (related to the Bessel kernel) have been used. In this and many other applications there is no advantage in choosing any special member in a class of equivalent capacities. (Two capacities are called equivalent if their ratio is bounded away from 0 and ∞.) In the literature Bessel capacities are considered mostly in the space R d . We introduce two versions of Bessel capacities on a compact N-dimensional manifold. A class C ap ℓ , p of equivalent capacities is defined, for ℓ p ⩽ N , on every compact Lipschitz manifold. Another class C B ℓ , p is defined (for all ℓ > 0 , p > 1 ) in terms of a diffusion process on a C 2 -manifold. These classes coincide when both are defined. If the manifold is the boundary of a bounded C 2 -domain E ⊂ R d , then both versions of the Bessel capacities are equivalent to the Poisson capacities.