Eigenvalues of the Neumann-Poincaré operator in dimension 3: Weyl’s law and geometry. (English) Zbl 1513.47041
St. Petersbg. Math. J. 31, No. 2, 371-386 (2020) and Algebra Anal. 31, No. 2, 248-268 (2019).
Summary: Asymptotic properties of the eigenvalues of the Neumann-Poincaré (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 NP operator \(\mathcal{K}_{\Gamma } \), called often “the direct value of the double layer potential”, acting in \(L^2(\Gamma )\), is defined by \[\mathcal{K}_\Gamma [\psi ](\mathbf{x}) := \frac{1}{4\pi}\int_\Gamma \frac{\langle\mathbf{y}-\mathbf{x},\mathbf{n}(\mathbf{y})\rangle}{\vert \mathbf{x}-\mathbf{y}\vert^3} \psi (\mathbf{y})dS_{\mathbf{y}},\] 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 [Adv. Math. 406, Article ID 108547, 19 p. (2022; Zbl 07567808)] 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 \[s_j(\mathcal{K}(\Gamma ))\sim \vert\lambda_j(\mathcal{K}_\Gamma)\vert\sim\left\{ \frac{3W(\Gamma )-2\pi \chi (\Gamma )}{128\pi }\right\}^{\frac{1}{2}} j^{-\frac{1}{2}},\] 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 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.
MSC:
| 47A75 | Eigenvalue problems for linear operators |
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
Keywords:
Neumann-Poincaré operator; eigenvalues; Weyl’s law; pseudodifferential operators; Willmore energyCitations:
Zbl 07567808References:
| [1] | Agrano-book M. S. Agranovich, N. N. Voitovich, B. Z. Katsenelenbaum, and A. N. Sivov, Generalized method of eigenoscillations in diffraction Theory, Nauka, Moscow, 1977; English transl., Wiley-VCH Verlag, Berlin, 1999. (2000j:35002) · Zbl 0929.65097 |
| [2] | Ahl L. V. Ahlfors, Remarks on the Neumann-Poincar\'e integral equation, Pacific. J. Math. 2 (1952), 271-280. · Zbl 0047.07907 |
| [3] | Ah2 J. F. Ahner, On the eigenvalues of the electrostatic integral operator. II, J. Math. Anal. Appl. 181 (1994), no. 2, 328-334. · Zbl 0791.47008 |
| [4] | ACKLM H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, Spectral theory of a Neumann-Poincar\'e-type operator and analysis of cloaking due to anomalous localized resonance, Arch. Ration. Mech. Anal. 208 (2013), no. 2, 667-692. · Zbl 1282.78004 |
| [5] | AKL H. Ammari, G. Ciraolo, H. Kang, and H. Lee, Layer potential techniques in spectral analysis, Math. Surveys Monogr., vol. 153, Amer. Math. Soc., Providence, RI, 2009. · Zbl 1167.47001 |
| [6] | AK-JMAA-16 K. Ando and H. Kang, Analysis of plasmon resonance on smooth domains using spectral properties of the Neumann-Poincar\'e operator, J. Math. Anal. Appl. 435 (2016), no. 1, 162-178. · Zbl 1327.81193 |
| [7] | AKM2 K. Ando, H. Kang, and Y. Miyanishi, Exponential decay estimates of the eigenvalues for the Neumann-Poincar\'e operator on analytic boundaries in two dimensions, J. Integral Equations Appl. 30 (2018), no. 4, 473-489; arXiv:1606.01483 · Zbl 1407.35147 |
| [8] | AKMU K. Ando, H. Kang, Y. Miyanishi, and E. Ushikoshi, The first Hadamard variation of Neumann-Poincar\'e eigenvalues, Proc. Amer. Math. Soc. 147 (2019), no. 3, 1073-1080 · Zbl 1541.45001 |
| [9] | BS M. Birman and M. Solomyak, Asymptotic behavior of the spectrum of pseudodifferential operators with anisotropically homogeneous symbols, Vestnik Leningrad. Univ. Mat. Meh. Astronom. 1977, vyp. 3, 13-21. (Russian) · Zbl 0377.47033 |
| [10] | BirYaf M. Birman and D. Yafaev, Asymptotic behavior of the spectrum of the scattering matrix. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 110 (1981), 3-29; English transl., J. Sov. Math. 25 (1984), no. 1, 793-814. · Zbl 0531.35062 |
| [11] | Bl W. Blaschke, Voresungen \"Uber Differentialgeometrie III, Springer, Berlin, 1929. · JFM 55.0422.01 |
| [12] | BT-ARMA-13 E. Bonnetier and F. Triki, On the spectrum of Poincar\'e variational problem for two close-to-touching inclusions in 2D, Arch. Ration. Mech. Anal. 209 (2013), no. 2, 541-567. · Zbl 1280.49029 |
| [13] | Carleman T. Carleman, Uber das Neumann-Poincar\'esche Problem f\"ur ein Gebiet mit Ecken, Almquist and Wiksells, Uppsala, 1916. · JFM 46.0732.04 |
| [14] | Grieser D. Grieser, The plasmonic eigenvalue problem, Rev. Math. Phys. 26 (2014), no. 3, 1450005. · Zbl 1294.78004 |
| [15] | JiKang Y. Ji and H. Kang, A concavity condition for existence of a negative Neumann-Poincar\'e eigenvalue in three dimensions, Proc. Amer. Math. Soc. 147 (2019), no. 8, 3431-3438. · Zbl 07081339 |
| [16] | AKJiKKM K. Ando, Y. Ji, H. Kang, D. Kawagoe, and Y. Miyanishi, Spectral structure of the Neumann-Poincar\'e operator on tori, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 36 (2019), no. 7, 1817-1828. · Zbl 07129764 |
| [17] | KLY H. Kang, M. Lim, and S. Yu, Spectral resolution of the Neumann-Poincar\'e operator on intersecting disks and analysis of plasmon resonance, Arch. Ration. Mech. Anal. 226 (2017), no. 1, 83-115. · Zbl 1386.35285 |
| [18] | KPS D. Khavinson, M. Putinar, and H. S. Shapiro, Poincar\'e’s variational problem in potential theory, Arch. Ration. Mech. Anal. 185 (2007), no. 1, 143-184. · Zbl 1119.31001 |
| [19] | KyFan Ky Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Nat. Acad. Sci., U. S. A. 37 (1951), 760-766. · Zbl 0044.11502 |
| [20] | MFZ-PR-05 I. D. Mayergoyz, D. R. Fredkin, and Z. Zhang, Electrostatic (plasmon\rm ) resonances in nanoparticles, Phys. Rev. B 72 (2005), 155412. |
| [21] | MN F. C. Marques and A. Neves, Min-max theory and the Willmore conjecture, Anal. of Math. 179 (2014), no. 2, 683-782. · Zbl 1297.49079 |
| [22] | Ma E. Martensen, A spectral property of the electrostatic integral operator, J. Math. Anal. Appl. 238 (1999), no. 2, 551-557. · Zbl 0939.45001 |
| [23] | Maz V. Maz\textprime ya, Boundary integral equations, Itogi Nauki i Techniki. Sovrem. Probl. Mat. Fund. Naprav., vol. 27, VINITI, Moscow, 1988, p. 131-288; English transl., Analysis. IV, Encyclopaedia Math. Sci., vol. 27, Springer, Berlin, 1991. · Zbl 0691.45002 |
| [24] | MazBook V. Maz\textprime ya and A. Soloviev, Boundary integral equations on contours with peaks, Oper. Theory Adv. Appl., vol. 196, Birkhauser, Basel, 2010. · Zbl 1179.45001 |
| [25] | MR2263683 G. W. Milton and N.-A. P. Nicorovici, On the cloaking effects associated with anomalous localized resonance, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 462 (2006), no. 2074, 3027-3059. · Zbl 1149.00310 |
| [26] | Miyanishi:2015aa Y. Miyanishi and T. Suzuki, Eigenvalues and eigenfunctions of double layer potentials. Trans. Amer. Math. Soc. 369 (2017), 8037-8059 · Zbl 1460.47022 |
| [27] | Miyanishi:Weyl 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. |
| [28] | Neumann-87 C. Neumann, \"Uber die Methode des arithmetischen Mittels, Hezzel, Leipzig, 1887/88. |
| [29] | PP-arXiv K. Perfekt and M. Putinar, The essential spectrum of the Neumann-Poincar\'e operator on a domain with corners, Arch. Ration. Mech. Anal. 223 (2017), no. 2, 1019-1033. · Zbl 1377.35203 |
| [30] | Poincare-AM-87 H. Poincar\'e, La m\'ethode de Neumann et le probl\`eme de Dirichlet, Acta Math. 20 (1897), 59-152. · JFM 27.0316.01 |
| [31] | Ri2 S. Ritter, 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., vol. 40, Peter Lang, Frankfurt am Main, 1995, pp. 157-167. · Zbl 0817.45003 |
| [32] | RT G. Rozenblum and G. Tashchiyan, Eigenvalue asymptotics for potential type operators on Lipschitz surfaces. Russ. J. Math. Phys. 13 (2006), no. 3, 326-339. · Zbl 1123.35037 |
| [33] | MR0104934 M. Schiffer, The Fredholm eigenvalues of plane domains, Pacific J. Math. 7 (1957), 1187-1225. · Zbl 0138.30003 |
| [34] | Si B. Simon, Trace ideals and their applications, 2nd ed., Math. Surveys Monogr., vol. 120, Amer. Math. Soc., Providence, RI, 2005. · Zbl 1074.47001 |
| [35] | SW O. Steinbach and W. L. Wendland, On C. Neumann’s method for second-order elliptic systems in domains with non-smooth boundaries, J. Math. Anal. Appl. 262 (2001), no. 2, 733-748. · Zbl 0998.35014 |
| [36] | Ta M. E. Taylor, Tools for PDE. Pseudodifferential operators, paradifferential operators, and layer potentials, Math. Surveys Monogr., vol. 81, Amer. Math. Soc., Providence, RI, 2000. · Zbl 0963.35211 |
| [37] | Verch-JFA-84 G. C. Verchota, Layer potentials and boundary value problems for Laplace’s equation in Lipschitz domains, J. Funct. Anal. 59 (1984), no. 3, 572-611. · Zbl 0589.31005 |
| [38] | Wh J. H. White, A global invariant of conformal mappings in space, Proc. Amer. Math. Soc. 38 (1973), 162-164. · Zbl 0256.53008 |
| [39] | Zaremba S. Zaremba, Les fonctions fondamentales de M. Poincar\'e et la m\'ethode de Neumann pour une fronti\`ere compos\'e de polygones curvilignes, J. Math. Pures Appl. 10 (1904), 395-444. · JFM 35.0355.03 |
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