On the Spectra of Elastostatic and Hydrostatic Layer Potentials on Curvilinear Polygons

Mitrea, Irina

doi:10.1007/s00041-002-0022-5

Cited by 28 publications

(36 citation statements)

References 17 publications

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“…In contrast with the spectral properties of K : L p → L p [16], any non-essential point in the spectrum of K : H 1/2 → H 1/2 is an isolated eigenvalue of finite multiplicity (see Corollary 3.5), owing to the special symmetry features that K exhibits on H 1/2 . In modern computational applications involving K, or more general double layer potential operators such as those associated with the Helmholtz equation, such points in the discrete spectrum are easy to recognize.…”

Section: Introductionmentioning

confidence: 99%

Spectral bounds for the Neumann-Poincaré operator on planar domains with corners

Perfekt

Putinar

2014

JAMA

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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 transform bearing similarities to the Beurling-Ahlfors transform in 2D. As an application, bounds for the spectrum of the Neumann-Poincaré operator are derived from recent results in quasi-conformal mapping theory, in the case of planar curves with corners.

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Section: Introductionmentioning

confidence: 99%

Spectral bounds for the Neumann-Poincaré operator on planar domains with corners

Perfekt

Putinar

2014

JAMA

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“…[36]. If p ≥ p 0 we can repeat the reasoning from the proof of Lemma 24 using results in [29], analogical to the result in [52], used in the proof.…”

Section: Medková I E O Tmentioning

confidence: 99%

The Oblique Derivative Problem for the Laplace Equation in a Plain Domain

Medková

2004

Integral Equations and Operator Theory

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The oblique derivative problem for the Laplace equation is studied in a planar multiply connected domain. The boundary condition has a form (n + βτ ) · ∇u = g, where n is the unit normal vector, τ is the unit tangential vector and β is a fixed real number. If g is a Hölderian function and the corresponding domain has Ljapunov boundary then the classical problem is studied. If g ∈ Lp on the boundary and the domain has a locally Lipschitz boundary then a solution, which fulfils the boundary condition in the sense of a nontangential limit, is studied. If g is a real measure on the boundary and the domain has bounded cyclic variation then a solution in a sense of distributions is studied. The solution is looked for in a form of a linear combination of a single layer potential and an angular potential. Mathematics Subject Classification (2000). Primary 35J05; Secondary 31A10.

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“…We would like to point out that the case under consideration is both physically relevant and, from a technical standpoint, the most challenging among all Neumann-type boundary problems for the system of elastostatics. Indeed, the spectral analysis of K undertaken here is considerably more difficult and subtle than the one for the layer potential operator associated with the pseudo-stress conormal derivative, considered previously in [17,18].…”

Section: Introductionmentioning

confidence: 98%

Interval analysis techniques for boundary value problems of elasticity in two dimensions

Mitrea

Tucker

2007

Journal of Differential Equations

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In this paper we prove that the L 2 spectral radius of the traction double layer potential operator associated with the Lamé system on an infinite sector in R 2 is within 10 −2 from a certain conjectured value which depends explicitly on the aperture of the sector and the Lamé moduli of the system. This type of result is relevant to the spectral radius conjecture]. The techniques employed in the paper are a blend of classical tools such as Mellin transforms, and Calderón-Zygmund theory, as well as interval analysis-resulting in a computer-aided proof.

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On the Spectra of Elastostatic and Hydrostatic Layer Potentials on Curvilinear Polygons

Cited by 28 publications

References 17 publications

Spectral bounds for the Neumann-Poincaré operator on planar domains with corners

Spectral bounds for the Neumann-Poincaré operator on planar domains with corners

The Oblique Derivative Problem for the Laplace Equation in a Plain Domain

Interval analysis techniques for boundary value problems of elasticity in two dimensions

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