Eigenvalues of the Neumann–Poincaré operator in dimension 3: Weyl’s law and geometry

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 \begin{equation*} [t]{\mathcal {K}_{\Gamma }[\psi ](\mathbf {x}) \coloneq \frac {1}{4\pi }\int _\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 [Y.~Miyanishi, Weyl's law for the eigenvalues of the Neumann--Poincar\'e operators in three dimensions: Willmore energy and surface geometry, arXiv:1806.03657] 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 \}^{\frac 12}j^{-\frac 12}, \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 $\operatorname {NP}$ 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.