Bessel capacities on compact manifolds and their relation to Poisson capacities
A motivation for this paper comes from the role the Choquet capacities play in the study of semilinear elliptic partial differential equations. In particular, recent progress in the classification of all positive solutions of the equation \(L u = u^ {\alpha }\) in a bounded smooth domain from \(\mathbb{R}^d\) was achieved by using, as a tool, capacities on the boundary of the manifold. The authors introduce two versions of Bessel capacities, on compact Lipschitz, respectively \(C^2\), \(N\)-dimensional manifold. They establish that these classes coincide when both are defined. Moreover, they prove that both versions of the Bessel capacities are equivalent to the so-called Poisson capacities associated with the Poisson kernel. If the manifold is the boundary of a bounded \(C^2\)-domain from \(\mathbb{R}^d\), then both versions of the Bessel capacities are equivalent to the Poisson capacities.
scientific article; zbMATH DE number 5116363
scientific article