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St. Petersburg Mathematical Journal

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Eigenvalues of the Neumann-Poincaré operator in dimension 3: Weyl's law and geometry


Authors: Y. Miyanishi and G. Rozenblum
Original publication: Algebra i Analiz, tom 31 (2019), nomer 2.
Journal: St. Petersburg Math. J. 31 (2020), 371-386
MSC (2010): Primary 47A75; Secondary 58J50
DOI: https://doi.org/10.1090/spmj/1602
Published electronically: February 4, 2020
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Abstract: Asymptotic properties of the eigenvalues of the Neumann-Poincaré ( $ \operatorname {NP}$) operator in three dimensions are treated. The region $ \Omega \subset \mathbb{R}^3$ is bounded by a compact surface $ \Gamma =\partial \Omega $, with certain smoothness conditions imposed. The $ \operatorname {NP}$ operator $ \mathcal {K}_{\Gamma }$, called often `the direct value of the double layer potential', acting in $ L^2(\Gamma )$, is defined by

$\displaystyle \smash [t]{\mathcal {K}_{\Gamma }[\psi ](\mathbf {x}) \colonequal... ...le }{\vert\mathbf {x}-\mathbf {y}\vert^3}\psi (\mathbf {y})\,dS_{\mathbf {y}},}$    

where $ dS_{\mathbf {y}}$ is the surface element and $ \mathbf {n}(\mathbf {y})$ is the outer unit normal on $ \Gamma $. The first-named author proved in [27] that the singular numbers $ s_j(\mathcal {K}_\Gamma )$ of $ \mathcal {K}_{\Gamma }$ and the ordered moduli of its eigenvalues $ \lambda _j(\mathcal {K}_\Gamma )$ satisfy the Weyl law

$\displaystyle s_j(\mathcal {K}(\Gamma ))\sim \vert\lambda _j(\mathcal {K}_\Gamm... ...ac {3W(\Gamma )-2\pi \chi (\Gamma )}{128\pi }\right \}^{\frac 12}j^{-\frac 12},$    

under the condition that $ \Gamma $ belongs to the class $ C^{2, \alpha }$ with $ \alpha >0,$ where $ W(\Gamma )$ and $ \chi (\Gamma )$ denote, respectively, the Willmore energy and the Euler characteristic of the boundary surface $ \Gamma $. Although the $ \operatorname {NP}$ operator is not selfadjoint (and therefore no general relationships between eigenvalues and singular numbers exists), the ordered moduli of the eigenvalues of $ \mathcal {K}_\Gamma $ satisfy the same asymptotic relation.

The main purpose here is to investigate the asymptotic behavior of positive and negative eigenvalues separately under the condition of infinite smoothness of the boundary $ \Gamma $. These formulas are used, in particular, to obtain certain answers to the long-standing problem of the existence or finiteness of negative eigenvalues of $ \mathcal {K}_\Gamma $. A more sophisticated estimate allows us to give a natural extension of the Weyl law for the case of a smooth boundary.


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Additional Information

Y. Miyanishi
Affiliation: Center for Mathematical, Modeling and Data Science, Osaka University, Japan
Email: miyanishi@sigmath.es.osaka-u.ac.jp

G. Rozenblum
Affiliation: Chalmers University of Technology; The University of Gothenburg, Sweden; Deptartment of Math. Physics, St.Petersburg State University, St.Petersburg, Russia
Email: grigori@chalmers.se

DOI: https://doi.org/10.1090/spmj/1602
Keywords: Neumann--Poincar\'e operator, eigenvalues, Weyl's law, pseudodifferential operators, Willmore energy
Received by editor(s): December 3, 2018
Published electronically: February 4, 2020
Dedicated: To Volodya Maz’ya, an outstanding mathematician
Article copyright: © Copyright 2020 American Mathematical Society