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

Y. Miyanishi ; G. Rozenblum

Algebra i analiz, Tome 31 (2019) no. 2, pp. 248-268.

Résumé

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.

Détail

Keywords: Neumann–Poincaré operator, eigenvalues, Weyl's law, pseudodifferential operators, Willmore energy.

Bibliographie

[1] Agranovich M. S., Voitovich N. N., Katsenelenbaum B. Z., Sivov A. N., Obobschennyi metod sobstvennykh kolebanii v teorii difraktsii, Nauka, M., 1977

[2] Ahlfors L. V., “Remarks on the Neumann–Poincaré integral equation”, Pacific. J. Math., 2 (1952), 271–280 | DOI | MR | Zbl

[3] Ahner J. F., “On the eigenvalues of the electrostatic integral operator. II”, J. Math. Anal. Appl., 181:2 (1994), 328–334 | DOI | MR | Zbl

[4] Ammari H., Ciraolo G., Kang H., Lee H., Milton G. W., “Spectral theory of a Neumann–Poincaré-type operator and analysis of cloaking due to anomalous localized resonance”, Arch. Ration. Mech. Anal., 208:2 (2013), 667–692 | DOI | MR | Zbl

[5] Ammari H., Kang H., Lee H., Layer potential techniques in spectral analysis, Math. Surveys Monogr., 153, Amer. Math. Soc., Providence, RI, 2009 | DOI | MR | Zbl

[6] Ando K., Kang H., “Analysis of plasmon resonance on smooth domains using spectral properties of the Neumann–Poincaré operator”, J. Math. Anal. Appl., 435:1 (2016), 162–178 | DOI | MR | Zbl

[7] Ando K., Kang H., Miyanishi Y., “Exponential decay estimates of the eigenvalues for the Neumann–Poincaré operator on analytic boundaries in two dimensions”, J. Integral Equations (to appear) , arXiv: 1606.01483 | MR

[8] Ando K., Kang H., Miyanishi Y., Ushikoshi E., “The first Hadamard variation of Neumann–Poincaré eigenvalues”, Proc. Amer. Math. Soc. (to appear) , arXiv: 1805.02414 | MR

[9] Birman M. Sh., Solomyak M. Z., “Asimptotika spektra psevdodifferentsialnykh operatorov s anizotropno-odnorodnymi simvolami”, Vestn. Leningrad. un-ta. Ser. mat., mekh., astronom., 1977, no. 13, 13–21

[10] Birman M. Sh., Yafaev D. R., “Asimptotika spektra matritsy rasseyaniya”, Zap. nauch. semin. LOMI, 110, 1981, 3–29 | Zbl

[11] Blaschke W., Vorlesungen Über Differentialgeometrie III, Springer, Berlin, 1929 | MR | Zbl

[12] Bonnetier E., Triki F., “On the spectrum of Poincaré variational problem for two close-to-touching inclusions in 2D”, Arch. Ration. Mech. Anal., 209:2 (2013), 541–567 | DOI | MR | Zbl

[13] Carleman T., Uber das Neumann–Poincarésche Problem für ein Gebiet mit Ecken, Almquist and Wiksells, Uppsala, 1916

[14] Grieser D., “The plasmonic eigenvalue problem”, Rev. Math. Phys., 26:3 (2014), 1450005 | DOI | MR | Zbl

[15] Ji Y., Kang H., A concavity condition for existence of a negative Neumann–Poincaré eigenvalue in three dimensions, arXiv: 1808.10621

[16] Ando K., Ji Y., Kang H., Kawagoe D., Miyanishi Y., Spectral structure of the Neumann–Poincaré operator on tori, arXiv: 1810.09693

[17] Kang H., Lim M., Yu S., Spectral resolution of the Neumann–Poincaré operator on intersecting disks and analysis of plasmon resonance, 2015, arXiv: 1501.02952 | MR

[18] Khavinson D., Putinar M., Shapiro H. S., “Poincaré's variational problem in potential theory”, Arch. Ration. Mech. Anal., 185:1 (2007), 143–184 | DOI | MR | Zbl

[19] Ky Fan, “Maximum properties and inequalities for the eigenvalues of completely continuous operators”, Proc. Nat. Acad. Sci., U. S. A., 37 (1951), 760–766 | DOI | MR | Zbl

[20] Mayergoyz I. D., Fredkin D. R., Zhang Z., “Electrostatic (plasmon) resonances in nanoparticles”, Phys. Rev. B, 72 (2005), 155412 | DOI

[21] Marques F. C., Neves A., “Min-max theory and the Willmore conjecture”, Anal. of Math., 179:2 (2014), 683–782 | MR | Zbl

[22] Martensen E., “A spectral property of the electrostatic integral operator”, J. Math. Anal. Appl., 238:2 (1999), 551–557 | DOI | MR | Zbl

[23] Mazya V. G., “Granichnye integralnye uravneniya”, Itogi nauki i tekhn. Sovrem. probl. mat. Fundam. napravleniya, 27, VINITI, M., 1988, 131–228

[24] Maz'ya V., Soloviev A., Boundary integral equations on contours with peaks, Oper. Theory Adv. Appl., 196, Birkhauser, Basel, 2010 | MR | Zbl

[25] Milton G. W., Nicorovici N.-A. P., “On the cloaking effects associated with anomalous localized resonance”, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462:2074 (2006), 3027–3059 | DOI | MR | Zbl

[26] Miyanishi Y., Suzuki T., “Eigenvalues and eigenfunctions of double layer potentials”, Trans. Amer. Math. Soc., 369 (2017), 8037–8059 | DOI | MR | Zbl

[27] Miyanishi Y., Weyl's law for the eigenvalues of the Neumann–Poincaré operators in three dimensions: Willmore energy and surface geometry, arXiv: 1806.03657

[28] Neumann C., Über die Methode des arithmetischen Mittels, Hezzel, Leipzig, 1887/88 | MR

[29] Perfekt K., Putinar M., “The essential spectrum of the Neumann–Poincaré operator on a domain with corners”, Arch. Ration. Mech. Anal., 223:2 (2017), 1019–1033 | DOI | MR | Zbl

[30] Poincaré H., “La méthode de Neumann et le problème de Dirichlet”, Acta Math., 20 (1897), 59–152 | DOI | MR

[31] Ritter S., “The spectrum of the electrostatic integral operator for an ellipsoid”, Inverse Scattering and Potential Problems in Mathematical Physics, Lond. Ser. A Math. Phys. Eng. Sci. (Oberwolfach, 1993), Methoden Verfharen Math. Phys., 40, Peter Lang, Frankfurt am Main, 1995, 157–167 | MR | Zbl

[32] Rozenblum G., Tashchiyan G., “Eigenvalue asymptotics for potential type operators on Lipschitz surfaces”, Russ. J. Math. Phys., 13:3 (2006), 326–339 | DOI | MR | Zbl

[33] Schiffer M., “The Fredholm eigenvalues of plane domains”, Pacific J. Math., 7 (1957), 1187–1225 | DOI | MR | Zbl

[34] Simon B., Trace ideals and their applications, Math. Surveys Monogr., 120, 2nd ed., Amer. Math. Soc., Providence, RI, 2005 | MR | Zbl

[35] Steinbach O., Wendland W. L., “On C. Neumann's method for second-order elliptic systems in domains with non-smooth boundaries”, J. Math. Anal. Appl., 262:2 (2001), 733–748 | DOI | MR | Zbl

[36] Taylor M. E., Tools for PDE. Pseudodifferential operators, paradifferential operators, and layer potentials, Math. Surveys Monogr., 81, Amer. Math. Soc., Providence, RI, 2000 | MR | Zbl

[37] Verchota G. C., “Layer potentials and boundary value problems for Laplace's equation in Lipschitz domains”, J. Funct. Anal., 59:3 (1984), 572–611 | DOI | MR | Zbl

[38] White J. H., “A global invariant of conformal mappings in space”, Proc. Amer. Math. Soc., 38 (1973), 162–164 | DOI | MR | Zbl

[39] Zaremba S., “Les fonctions fondamentales de M. Poincaré et la méthode de Neumann pour une frontiére composé de polygones curvilignes”, J. Math. Pures Appl., 10 (1904), 395–444 | Zbl