# Minimal energy problems for strongly singular Riesz kernels

Harbrecht, Helmut and Wendland, Wolfgang L. and Zorii, Natalia. (2015) Minimal energy problems for strongly singular Riesz kernels. Preprints Fachbereich Mathematik, 2015 (41).

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Official URL: https://edoc.unibas.ch/69994/

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.