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EIGENVALUES OF THE NEUMANN-POINCARE OPERATOR FOR TWO INCLUSIONS WITH CONTACT OF ORDER m： A NUMericAL STUDY

Journal of Computational Mathematics 2018年第1期36卷 17-28页

作者： eric bonnetier Faouzi Triki Chun-Hsiang Tsou Universitd Grenoble-Alpes//CNRS Laboratoire Jean Kuntzraann UMR 5224 Grenoble F-38041 France

In a composite medium that contains close-to-touching inclusions, the pointwise values of the gradient of the voltage potential may blow up as the distance S between some inclusions tends to 0 and as the conductivity contrast degenerates. In a recent paper [9], we showed that the blow-up rate of the gradient is related to how the eigenvalues of the associated Neumann-Poincare operator converge to ±1/2 as δ→ 0, and on the regularity of the contact. Here, we consider two connected 2-D inclusions, at a distance 5 〉 0 from each other. When δ=0, the contact between the inclusions is of order m 〉 2. We numerically determine the asymptotic behavior of the first eigenvalue of the Neumann- Poincare operator, in terms of 5 and rn, and we check that we recover the estimates obtained in [10].

关键词： Elliptic equations Eigenvalues Numerical approximation.

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The Plasmonic Resonances of a Bowtie Antenna

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Analysis in Theory and Applications 2019年第1期35卷 85-116页

作者： eric bonnetier Charles Dapogny Faouzi Triki Hai Zhang Institut Fourier Université Grenoble-AlpesCS 4070038058 Grenoble Cedex 09France Laboratoire Jean Kuntzmann Unioersite Grenoble-Alpes700 Avenue Centrale38401 Domaine Universitaire de Saint-Martin-d'HèresFrance Department of Mathematics Hong Kong University of Science and TechnologyClear Water BayKowloonHong Kong

Metallic bowtie-shaped nanostructures are very interesting objects in optics,due to their capability of localizing and enhancing electromagnetic fields in the vicinity of their central *** this article,we investigate the electrostatic plasmonic resonances of two-dimensional bowtie-shaped domains by looking at the spectrum of their Poincare variational *** particular,we show that the latter only consists of essential spectrum and fills the whole interval[0,1].This behavior is very different from what occurs in the counterpart situation of a bowtie domain with only close-totouching wings,a case where the essential spectrum of the Poincare variational operator is reduced to an interval oess strictly containing in[0,1].We provide an explanation for this difference by showing that the spectrum of the Poincare variational operator of bowtie-shaped domains with close-to-touching wings has eigenvalues which densify and eventually fill the remaining parts of[0,1]\σess as the distance between the two wings tends to zero.

关键词： Neumann-Poincaréoperator corner singularity spectrum resonance

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