Minimal energy problems for strongly singular Riesz kernels

We study minimal energy problems for strongly singular Riesz kernels ${|\mathbf{x}-\mathbf{y}|}^{\alpha -n}$, where $n\ge 2$ and $\alpha \in (-1,1)$, considered for compact $(n-1)$‐dimensional $C^{\infty }$‐manifolds Γ immersed into ${\mathbb{R}}^n$. Based on the spatial energy of harmonic double layer potentials, we are motivated to formulate the natural regularization of such minimization problems by switching to Hadamard's partie finie integral operator which defines a strongly elliptic pseudodifferential operator of order $\beta =1-\alpha$ on Γ. The measures with finite energy are shown to be elements from the Sobolev space $H^{\beta /2}(\mathrm{\Gamma })$, $0<\beta <2$, and the corresponding 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.