2014
DOI: 10.1007/s11854-014-0026-5
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Spectral bounds for the Neumann-Poincaré operator on planar domains with corners

Abstract: Abstract. The boundary double layer potential, or the Neumann-Poincaré operator, is studied on the Sobolev space of order 1/2 along the boundary, coinciding with the space of charges giving rise to double layer potentials with finite energy in the whole space. Poincaré's program of studying the spectrum of the boundary double layer potential is developed in complete generality, on closed Lipschitz hypersurfaces in Euclidean space. Furthermore, the Neumann-Poincaré operator is realized as a singular integral tr… Show more

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Cited by 49 publications
(67 citation statements)
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“…If the domain has corners, the corresponding NP operator may exhibit a continuous spectrum (as well as eigenvalues). For recent development in this direction we refer to [28,30,43,44]. If the domain has the smooth boundary, then the spectrum consists of eigenvalues converging to 0.…”
Section: Introduction and Resultsmentioning
confidence: 99%
“…If the domain has corners, the corresponding NP operator may exhibit a continuous spectrum (as well as eigenvalues). For recent development in this direction we refer to [28,30,43,44]. If the domain has the smooth boundary, then the spectrum consists of eigenvalues converging to 0.…”
Section: Introduction and Resultsmentioning
confidence: 99%
“…The same spectral problem arises in several other areas, such as in the theory of composite media [9]; the plasmonic spectrum is also closely linked to the specturm of the Neumann-Poincaré integral operator [10][11][12]. There is an enormous body of literature on this problem, with recent developments including analyses of strongly interacting particles using separation of variables [13,14], transformation optics [15], multipole methods [16], matched asymptotic expansions [17,18] and layer-potential techniques [12,19]; analysis of corners [20,21]; application to stimulated emission [22] and second-harmonic generation [23]; regular shape perturbations [24,25]; extensions incorporating nonlocality [26][27][28][29] and retardation [30][31][32][33][34]; and high-mode-number asymptotics [35,36].…”
Section: Introductionmentioning
confidence: 92%
“…Without loss of generality, let the potential in the the boundary layer be O(1). The form of the governing equations (19)- (20) then suggests the asymptotic expansion…”
Section: B Slowly Varying Surface Plasmonsmentioning
confidence: 99%
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“…If Ω is merely a Lipschitz domain, then the corresponding K * ∂Ω admits the continuous spectrum as well as the eigenvalues. Various studies are underway to investigate the spectral properties of the NP operator for cornered domains; see for examples [11,15,20,26,27].…”
Section: Introductionmentioning
confidence: 99%