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Minimal energy problems for strongly singular Riesz kernels
Author(s) -
Harbrecht Helmut,
Wendland Wolfgang L.,
Zorii Natalia
Publication year - 2018
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201600024
Subject(s) - mathematics , riesz transform , sobolev space , hadamard transform , riesz potential , regularization (linguistics) , operator (biology) , energy (signal processing) , pure mathematics , order (exchange) , mathematical analysis , biochemistry , chemistry , statistics , finance , repressor , artificial intelligence , computer science , transcription factor , economics , gene
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 spaceH 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.