New Spectral Results for the Electrostatic Integral Operator

Ahner, John F.; Dyakin, V. V.; Raevskii, V. Ya.

doi:10.1006/jmaa.1994.1257

Cited by 13 publications

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“…Finally, it follows from the location of the point-spectrum of the harmonic double layer (cf. [1]) that for each mo0 and each 1opoN there exists a smooth, bounded domain O for which (the bounded domain version of) problem (1.1) is not well-posed. Our main result is as follows.…”

Section: Article In Pressmentioning

confidence: 98%

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Transmission problems and spectral theory for singular integral operators on Lipschitz domains

Escauriaza

Mitrea

2004

Journal of Functional Analysis

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We prove the well-posedness of the transmission problem for the Laplacian across a Lipschitz interface, with optimal non-tangential maximal function estimates, for data in Lebesgue and Hardy spaces on the boundary. As a corollary, we show that the spectral radius of the (adjoint) harmonic double layer potential K Ã in L p 0 ð@OÞ is less than 1 2 ; whenever O is a bounded convex domain and 1opp2: r

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Section: Article In Pressmentioning

confidence: 98%

“…Mðrw 7 ÞAL p ð@OÞ; w þ j @O ¼ w À j @O ; @ n w þ À m@ n w À ¼ g À ðÀ 1 2 I þ K Ã ÞðS À1 f ÞAL p ð@OÞ:…”

Section: Existence and Estimatesunclassified

Transmission problems and spectral theory for singular integral operators on Lipschitz domains

Escauriaza

Mitrea

2004

Journal of Functional Analysis

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“…and gives a provable lower bound to the reaction energy for geometries in which D * is known to not possess positive eigenvalues, for instance spheres and prolate spheroids [40]. It is not yet understood why BIBEE/P seems to provide a lower bound for a wide variety of surfaces [22], given that it is known that for some shapes, including oblate spheroids [41], D * can be proven to possess eigenvalues arbitrarily close to +1/2.…”

Section: Bibee/lb and Bibee/p: Lower Boundsmentioning

confidence: 99%

Rapid Bounds on Electrostatic Energies Using Diagonal Approximations of Boundary-integral Equations

Bardhan¹

2010

PIERS Online

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Life as we know it depends critically on electrostatic interactions within and between biological molecules such as proteins. One simple, but surprisingly effective, model for studying these interactions treats a biomolecule of interest as a dielectric continuum of homogeneous low permittivity with some embedded distribution of charges, and the aqueous solvent around it as another homogeneous dielectric with higher permittivity. This gives rise to a mixeddielectric Poisson problem, widely studied in the mathematics and electromagnetics communities. In this paper we describe some simple analytical approximations to a boundary-integral equation formulation of the mixed-dielectric problem. Remarkably, the approximations (which we call BIBEE, for boundary-integral-based electrostatics estimation) give provable upper and lower bounds for the actual electrostatic energy. Because BIBEE methods preserve interactions between components of the charge distribution, they may represent one approach to rapidly approximate the Green's function for the geometry of interest.

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“…In the BIE method, the eigenvalues of the electrostatic operator are related to plasmon resonances [28], with a straightforward eigenvalue-resonance relationship for Drude metals [36,37]. The mathematical literature offers us the whole set of eigenvalues and eigenfunctions for discs, ellipses and spheres [26], as well as for spheroids [38][39][40] and ellipsoids [41].…”

Section: Introductionmentioning

confidence: 99%

Analytical results regarding electrostatic resonances of surface phonon/plasmon polaritons: separation of variables with a twist

Voicu

Sandu

2017

Proc. R. Soc. A.

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The boundary integral equation (BIE) method ascertains explicit relations between localized surface phonon and plasmon polariton resonances and the eigenvalues of its associated electrostatic operator. We show that group-theoretical analysis of the Laplace equation can be used to calculate the full set of eigenvalues and eigenfunctions of the electrostatic operator for shapes and shells described by separable coordinate systems. These results not only unify and generalize many existing studies, but also offer us the opportunity to expand the study of phenomena such as cloaking by anomalous localized resonance. Hence, we calculate the eigenvalues and eigenfunctions of elliptic and circular cylinders. We illustrate the benefits of using the BIE method to interpret recent experiments involving localized surface phonon polariton resonances and the size scaling of plasmon resonances in graphene nanodiscs. Finally, symmetrybased operator analysis can be extended from the electrostatic to the full-wave regime. Thus, bound states of light in the continuum can be studied for shapes beyond spherical configurations.

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New Spectral Results for the Electrostatic Integral Operator

Cited by 13 publications

References 0 publications

Transmission problems and spectral theory for singular integral operators on Lipschitz domains

Transmission problems and spectral theory for singular integral operators on Lipschitz domains

Rapid Bounds on Electrostatic Energies Using Diagonal Approximations of Boundary-integral Equations

Analytical results regarding electrostatic resonances of surface phonon/plasmon polaritons: separation of variables with a twist

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