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Статьи
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
Аннотация:
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 \{ \frac{3W(\Gamma)-2\pi\chi(\Gamma)}{128\pi}\}^{\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.
Ключевые слова:
Neumann–Poincaré operator, eigenvalues, Weyl's law, pseudodifferential operators, Willmore energy.
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Тип публикации:
Статья Поступила в редакцию: 03.12.2018
Язык публикации: английский
Образец цитирования:
Y. Miyanishi, G. Rozenblum, “Eigenvalues of the Neumann–Poincare operator in dimension 3: Weyl's law and geometry”, Алгебра и анализ, 31:2 (2019), 248–268
Цитирование в формате AMSBIB
\RBibitem{MiyRoz19}
\by Y.~Miyanishi, G.~Rozenblum
\paper Eigenvalues of the Neumann--Poincare operator in dimension 3: Weyl's law and geometry
\jour Алгебра и анализ
\yr 2019
\vol 31
\issue 2
\pages 248--268
\mathnet{http://mi.mathnet.ru/aa1648}
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http://mi.mathnet.ru/aa1648 http://mi.mathnet.ru/rus/aa/v31/i2/p248
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