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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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.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Computational Mathematics (Harbrecht)
12 Special Collections > Preprints Fachbereich Mathematik
UniBasel Contributors:Harbrecht, Helmut
Item Type:Preprint
Publisher:Universität Basel
edoc DOI:
Last Modified:08 May 2019 19:16
Deposited On:28 Mar 2019 09:51

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