Minimal energy problems for strongly singular Riesz kernels

TL;DR

This paper investigates minimal energy problems involving strongly singular Riesz kernels on manifolds, introducing a regularization via Hadamard's finite part integral and establishing existence, uniqueness, and Sobolev space properties of solutions.

Contribution

It formulates a regularized approach using pseudodifferential operators for strongly singular kernels and connects continuous and discrete energy problems.

Findings

Unique solutions exist for the regularized energy problem.

Measures with finite energy belong to specific Sobolev spaces.

The continuous approach aligns with previously studied discrete problems.

Abstract

We study minimal energy problems for strongly singular Riesz kernels on a manifold. Based on the spatial energy of harmonic double layer potentials, we are motivated to formulate the natural regularization of such problems by switching to Hadamard's partie finie integral operator which defines a strongly elliptic pseudodifferential operator on the manifold. The measures with finite energy are shown to be elements from the corresponding Sobolev space, and the associated minimal energy problem admits a unique solution. We relate our continuous approach also to the discrete one, which has been worked out earlier by D.P. Hardin and E.B. Saff.