Minimal energy problems for strongly singular Riesz kernels. Issue 1 (31st July 2017)

• Record Type: Journal Article
• Authors: Harbrecht, Helmut; Wendland, Wolfgang L.; Zorii, Natalia
• Is Part Of: Mathematische Nachrichten. Volume 291:Issue 1(2018)
• Page Start: 55
• Page End: 85
• Publication Date: 2017-07-31
• Subjects: Minimal energy problem -- strongly singular Riesz kernel -- pseudodifferential operators -- 31B10 -- 31C15 -- 45L10 -- 49J35; Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201600024
• Languages: English
• ISSNs: 0025-584X

Abstract

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 R n . 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 H β / 2 ( Γ ), 0 < β < 2, and the corresponding minimal energy problem admits a unique solution. We relate our continuous approach also to the discrete one, which has been worked out earlier by D. P. Hardin and E. B. Saff.