Some Historical Remarks on the Positivity of Boundary Integral Operators

Costabel, Martin

doi:10.1007/978-3-540-47533-0_1

Cited by 36 publications

(31 citation statements)

References 28 publications

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“…The normal trace γ n defines an isomorphism between W and the space H It is known that 1 2 + K 0 is a positive selfadjoint operator in H − 1 2 * (∂Ω) that differs at most by an operator of finite rank from a contraction. This follows from estimates similar to those in [6], see [4]. If ∂Ω is smooth, it is well known that K 0 is compact in H − 1 2 * (∂Ω).…”

Section: The Essential Spectrum Of the Electric Volume Integral Operatormentioning

confidence: 58%

Electromagnetic Scattering

Bruno¹

2002

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To cite this version:Martin Costabel, Eric Darrigrand, Hamdi Sakly. The essential spectrum of the volume integral operator in electromagnetic scattering by a homogeneous body. Comptes rendus de l'Académie des sciences. Série I, Mathématique, Elsevier, 2012, 350, pp.193-197. <10.1016/j.crma.2012 The essential spectrum of the volume integral operator in electromagnetic scattering by a homogeneous body AbstractWe study the strongly singular volume integral operator that describes the scattering of time-harmonic electromagnetic waves. For the case of piecewise constant material coefficients and smooth interfaces, we determine the essential spectrum. We show that it is a finite set and that the operator is Fredholm of index zero in H(curl) if and only if the relative permeability and permittivity are both different from 0 and −1. Version française abrégéeNous nous intéressonsà la diffraction d'ondesélectromagnétiques harmoniques en temps par Ω, un domaine borné de R 3 . Lorsque Ω est un objet pénétrable, le problème est régi par leséquations de Maxwell (4) satisfaites dans R 3 au sens des distributions, complétées par une condition de rayonnement a l'infini.Afin de ramener le problèmeà un domaine borné, nous considérons sa formulation par uneéquation intégrale volumique (VIE) (3) dans laquelle apparaissent deux opérateurs fortement singuliers A k et B k , Nous supposons ici que les coefficients ε (permittivitéélectrique) et µ (perméabilité magnétique) prennent des valeurs constantes dans Ω et dans son complément, voir (1). C'est la situation opposéè a celle de coefficients continus dans tout l'espace pour laquelle il est connu que la VIE estéquivalenteà uneéquation de seconde espèce faiblement singulière [3], le spectre essentielétant réduit au seul point {1}. Pour le cas de coefficients discontinus, une conjecture basée sur des observations numériques aété enoncée [2,10], mais nous démontrons qu'elle n'est pas vraie. Le cas d'un problème magnétostatique a déjà etéétudié en 1985 [6]. Dans notre formalisme, ceci correspond au casélectrique analysé dans la section 3. Dans les travaux [7,8], la VIE estétudiée pour des coefficients plus généraux, dans H(curl, Ω) et dans L 2 (Ω), sans pourtant répondreà la question du spectre essentiel. Les deux opérateurs A k et B k montrent des propriétés assez différentes : L'opérateur A k està une perturbation compacte près un opérateur autoadjoint borné dans L 2 (Ω) qui permet une description simple par des sous-espaces invariants. La partie "pertinente" de son spectre essentiel est reliéeà un opérateur intégral de frontière scalaire qui est bien connu de l'analyse harmonique. L'opérateur B k ne permet pas d'extension de H(curl)à L 2 , et son spectre essentiel est déterminé par un système d'opérateurs intégraux de frontière. Ce système (7) contient des opérateurs que l'on connaît de la méthode deséquations intégrales de frontière pour le problème de diffraction, sans pour autant provenir de cette méthode. Malgré cette différence, pour un bord régulier les spectres essentiels des op...

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Section: The Essential Spectrum Of the Electric Volume Integral Operatormentioning

confidence: 58%

Electromagnetic Scattering

Bruno¹

2002

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“…This method therefore proves that the standard first-and second-kind integral operators used to solve the Helmholtz equation are compact perturbations of coercive operators (see [10] and [24, §1.4] for overviews of this method).…”

Section: Upper Bounds For More General Domainsmentioning

confidence: 69%

Bounding acoustic layer potentials via oscillatory integral techniques

Spence

2014

Bit Numer Math

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We consider the Helmholtz single-layer operator (the trace of the single-layer potential) as an operator on L 2 (Γ ) where Γ is the boundary of a 3-d obstacle. We prove that if Γ is C 2 and has strictly positive curvature then the norm of the single-layer operator tends to zero as the wavenumber k tends to infinity. This result is proved using a combination of (i) techniques for obtaining the asymptotics of oscillatory integrals, and (ii) techniques for obtaining the asymptotics of integrals that become singular in the appropriate parameter limit. This paper is the first time such techniques have been applied to bounding norms of layer potentials. The main motivation for proving this result is that it is a component of a proof that the combined-field integral operator for the Helmholtz exterior Dirichlet problem is coercive on such domains in the space L 2 (Γ ).

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“…We are interested in their behaviours as X → ∞ since, as has been shown in [28], [30], [31], [34], the Gauss variational problem for the noncompact condenser A = ( 1 , 2 ) can in general be nonsolvable, and then the infimum G f (A, a, g) is attained at γ ∈ E α (A) with 2 g dγ 2 < a 2 , whereas λ X → γ vaguely and strongly as X → ∞. According to [30,Theorems 4,8], under our particular assumptions, such a phenomenon of nonsolvability occurs for A = ( 1 , 2 ) with 2 being infinitely long, if and only if C α ( 2 ) = ∞ while 2 is α-thin at ∞ R 3 , the latter by [4], [5] means that the inverse of 2 relative to the unit sphere is α-irregular at the origin x = 0. In the case r (x) = exp(−x), both these conditions hold true for α = 2 (hence, also for α close to 2), so that then lim X →∞ X > 0, (10.3) while in the case r (x) = 1/(1 + x), 2 is not α-thin at ∞ R 3 for any α ∈ (1, 2], so that for this geometry lim X →∞ X = 0.…”

Section: Numerical Resultsmentioning

confidence: 87%

Riesz minimal energy problems on Ck−1,1-manifolds

2013

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In Rn, n⩾2, we study the constructive and numerical solution of minimizing the energy relative to the Riesz kernel |boldx−boldy|α−n, where 1<α(α−1)/2, each Γℓ being charged with Borel measures with the sign αℓ:=±1 prescribed. We show that the Gauss variational problem over a convex set of Borel measures can alternatively be formulated as a minimum problem over the corresponding set of surface distributions belonging to the Sobolev–Slobodetski space H−ɛ/2(Γ), where ɛ:=α−1 and Γ:=⋃ℓ∈LΓℓ. An equivalent formulation leads in the case of two manifolds to a nonlinear system of boundary integral equations involving simple layer potential operators on Γ. A corresponding numerical method is based on the Galerkin–Bubnov discretization with piecewise constant boundary elements. Wavelet matrix compression is applied to sparsify the system matrix. Numerical results are presented to illustrate the approach.

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Some Historical Remarks on the Positivity of Boundary Integral Operators

Cited by 36 publications

References 28 publications

Electromagnetic Scattering

Electromagnetic Scattering

Bounding acoustic layer potentials via oscillatory integral techniques

Riesz minimal energy problems on Ck−1,1-manifolds

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