Minimal energy problems for strongly singular Riesz kernels

We study minimal energy problems for strongly singular Riesz kernels $|x−y|^{α−n}$, where n ≥ 2 and α ∈ (−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 a minimization problem by switching to Hadamard's partie finie integral operator which defines a strongly elliptic pseudodifferential operator of order β = 1 − α on Γ. The measures with finite energy are thus elements from the Sobolev space $H^{β/2}(Γ)$, 0 < β < 2, and the corresponding minimal energy problem admits a unique solution. We relate this approach also to the common approach where for δ>0 the set |x−y|≤δ of Γ×Γ is cut out.