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Minimal energy problems for strongly singular Riesz kernels

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  • Helmut Harbrecht
  • Wolfgang L. Wendland
  • Natalia Zorii

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∞†manifolds Γ immersed into Rn. 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 β=1−α on Γ. The measures with finite energy are shown to be elements from the Sobolev space Hβ/2(Γ), 0

Suggested Citation

  • Helmut Harbrecht & Wolfgang L. Wendland & Natalia Zorii, 2018. "Minimal energy problems for strongly singular Riesz kernels," Mathematische Nachrichten, Wiley Blackwell, vol. 291(1), pages 55-85, January.
    • Handle: RePEc:bla:mathna:v:291:y:2018:i:1:p:55-85
    DOI: 10.1002/mana.201600024
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