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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## Abstract

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 |
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UniBasel Contributors: | Harbrecht, Helmut |

Item Type: | Preprint |

Publisher: | Universität Basel |

Language: | English |

Last Modified: | 08 May 2019 19:16 |

Deposited On: | 28 Mar 2019 09:51 |

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