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Algebra i Analiz, 2019, Volume 31, Issue 2, Pages 248–268 (Mi aa1648)  

This article is cited in 17 scientific papers (total in 17 papers)

Research Papers

Eigenvalues of the Neumann–Poincare operator in dimension 3: Weyl's law and geometry

Y. Miyanishia, G. Rozenblumbcd

a Center for Mathematical Modeling and Data Science, Osaka University, Japan
b Dept. Math. Physics, St.Petersburg State University, St. Petersburg, Russia
c Chalmers University of Technology
d The University of Gothenburg, Sweden
References:
Abstract: We consider the asymptotic properties of the eigenvalues of the Neumann–Poincaré ($\mathrm{NP}$) operator in three dimensions. The region $\Omega\subset\mathbb{R}^3$ is bounded by a compact surface $\Gamma=\partial \Omega$, with certain smoothness conditions imposed. The $\mathrm{NP}$ operator $\mathcal{K}_{\Gamma}$, called often ‘the direct value of the double layer potential’, acting in $L^2(\Gamma)$, is defined by
\begin{equation*} \mathcal{K}_{\Gamma}[\psi](\mathbf{x}):=\frac{1}{4\pi}\int\limits_\Gamma\frac{\langle \mathbf{y}-\mathbf{x},\mathbf{n}(\mathbf{y})\rangle}{|\mathbf{x}-\mathbf{y}|^3}\psi(\mathbf{y})dS_{\mathbf{y}}, \end{equation*}
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
\begin{equation*} s_j(\mathcal{K}(\Gamma))\sim|\lambda_j(\mathcal{K}_{\Gamma})|\sim \left\{ \frac{3W(\Gamma)-2\pi\chi(\Gamma)}{128\pi}\right\}^{\frac12}j^{-\frac12}, \end{equation*}
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 $\mathrm{NP}$ operator is not selfadjoint (and therefore no general relationships between eigenvalues and singular number exists), the ordered moduli of the eigenvalues of $\mathcal{K}_{\Gamma}$ satisfy the same asymptotic relation.
Our 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 estimation allows us to give a natural extension of the Weyl law for the case of a smooth boundary.
Keywords: Neumann–Poincaré operator, eigenvalues, Weyl's law, pseudodifferential operators, Willmore energy.
Received: 03.12.2018
English version:
St. Petersburg Mathematical Journal, 2019, Volume 31, Issue 2, Pages 371–386
DOI: https://doi.org/10.1090/spmj/1602
Bibliographic databases:
Document Type: Article
MSC: Primary 47A75; Secondary 58J50
Language: English
Citation: Y. Miyanishi, G. Rozenblum, “Eigenvalues of the Neumann–Poincare operator in dimension 3: Weyl's law and geometry”, Algebra i Analiz, 31:2 (2019), 248–268; St. Petersburg Math. J., 31:2 (2019), 371–386
Citation in format AMSBIB
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\pages 248--268
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  • This publication is cited in the following 17 articles:
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