A concavity condition for existence of a negative value in Neumann-Poincaré spectrum in three dimensions. (English) Zbl 07081339
Summary: It is proved that if a bounded domain in three dimensions satisfies a certain concavity condition, then the Neumann-Poincaré operator on either the boundary of the domain or its inversion in a sphere has a negative value in its spectrum. The concavity condition is quite simple, and is satisfied if there is a point on the boundary at which the Gaussian curvature is negative.
MSC:
| 47A45 | Canonical models for contractions and nonselfadjoint linear operators |
| 31B25 | Boundary behavior of harmonic functions in higher dimensions |
References:
| [1] | Ahner, John F., On the eigenvalues of the electrostatic integral operator. II, J. Math. Anal. Appl., 181, 2, 328-334 (1994) · Zbl 0791.47008 · doi:10.1006/jmaa.1994.1025 |
| [2] | Ahner, John F.; Arenstorf, Richard F., On the eigenvalues of the electrostatic integral operator, J. Math. Anal. Appl., 117, 1, 187-197 (1986) · Zbl 0593.45002 · doi:10.1016/0022-247X(86)90255-6 |
| [3] | Ammari, Habib; Ciraolo, Giulio; Kang, Hyeonbae; Lee, Hyundae; Milton, Graeme W., Spectral theory of a Neumann-Poincar\'{e}-type operator and analysis of cloaking due to anomalous localized resonance, Arch. Ration. Mech. Anal., 208, 2, 667-692 (2013) · Zbl 1282.78004 · doi:10.1007/s00205-012-0605-5 |
| [4] | Ammari, Habib; Kang, Hyeonbae, Polarization and moment tensors: With applications to inverse problems and effective medium theory, Applied Mathematical Sciences 162, x+312 pp. (2007), Springer, New York · Zbl 1220.35001 |
| [5] | Ammari, Habib; Millien, Pierre; Ruiz, Matias; Zhang, Hai, Mathematical analysis of plasmonic nanoparticles: the scalar case, Arch. Ration. Mech. Anal., 224, 2, 597-658 (2017) · Zbl 1375.35515 · doi:10.1007/s00205-017-1084-5 |
| [6] | AJKKM K. Ando, Y. Ji, H. Kang, D. Kawagoe, and Y. Miyanishi, Spectral structure of the Neumann-Poincar\'e operator on tori, arXiv:1810.09693. · Zbl 07129764 |
| [7] | BZ E. Bonnetier and H. Zhang, Characterization of the essential spectrum of the Neumann-Poincar\'e operator in 2D domains with corner via Weyl sequences, Revista Matematica Iberoamericana, to appear. · Zbl 1423.35268 |
| [8] | Feng, Tingting; Kang, Hyeonbae, Spectrum of the Neumann-Poincar\'{e} operator for ellipsoids and tunability, Integral Equations Operator Theory, 84, 4, 591-599 (2016) · Zbl 1336.35259 · doi:10.1007/s00020-016-2280-7 |
| [9] | Folland, Gerald B., Introduction to partial differential equations, xii+324 pp. (1995), Princeton University Press, Princeton, NJ · Zbl 0841.35001 |
| [10] | Gustafson, Karl E.; Rao, Duggirala K. M., Numerical range, Universitext, xiv+189 pp. (1997), Springer-Verlag, New York · Zbl 0874.47003 · doi:10.1007/978-1-4613-8498-4 |
| [11] | Helsing, Johan; Kang, Hyeonbae; Lim, Mikyoung, Classification of spectra of the Neumann-Poincar\'{e} operator on planar domains with corners by resonance, Ann. Inst. H. Poincar\'{e} Anal. Non Lin\'{e}aire, 34, 4, 991-1011 (2017) · Zbl 1435.35266 · doi:10.1016/j.anihpc.2016.07.004 |
| [12] | Helsing, Johan; Perfekt, Karl-Mikael, The spectra of harmonic layer potential operators on domains with rotationally symmetric conical points, J. Math. Pures Appl. (9), 118, 235-287 (2018) · Zbl 1400.31002 · doi:10.1016/j.matpur.2017.10.012 |
| [13] | Kang, Hyeonbae; Kim, Kyoungsun; Lee, Hyundae; Shin, Jaemin; Yu, Sanghyeon, Spectral properties of the Neumann-Poincar\'{e} operator and uniformity of estimates for the conductivity equation with complex coefficients, J. Lond. Math. Soc. (2), 93, 2, 519-545 (2016) · Zbl 1337.35047 · doi:10.1112/jlms/jdw003 |
| [14] | Kang, Hyeonbae; Lim, Mikyoung; Yu, Sanghyeon, Spectral resolution of the Neumann-Poincar\'{e} operator on intersecting disks and analysis of plasmon resonance, Arch. Ration. Mech. Anal., 226, 1, 83-115 (2017) · Zbl 1386.35285 · doi:10.1007/s00205-017-1129-9 |
| [15] | Khavinson, Dmitry; Putinar, Mihai; Shapiro, Harold S., Poincar\'{e}’s variational problem in potential theory, Arch. Ration. Mech. Anal., 185, 1, 143-184 (2007) · Zbl 1119.31001 · doi:10.1007/s00205-006-0045-1 |
| [16] | MacMillan, William Duncan, The theory of the potential, MacMillan’s Theoretical Mechanics, xiii+469 pp. (1958), Dover Publications, Inc., New York · Zbl 0088.15803 |
| [17] | Martensen, Erich, A spectral property of the electrostatic integral operator, J. Math. Anal. Appl., 238, 2, 551-557 (1999) · Zbl 0939.45001 · doi:10.1006/jmaa.1999.6538 |
| [18] | MFZ I. D. Mayergoyz, D. R. Fredkin, and Z. Zhang, Electrostatic (plasmon) resonances in nanoparticles, Phys. Rev. B 72 (2005), 155412. |
| [19] | Miya Y. Miyanishi, Weyl’s law for the eigenvalues of the Neumann-Poincar\'e operators in three dimensions: Willmore energy and surface geometry, arXiv:1806.03657v1. |
| [20] | Neumann-87 C. Neumann, \`“Uber die Methode des arithmetischen Mittels, Erste and zweite Abhandlung, Leipzig 1887/88, in Abh. d. Kgl. S\'”achs Ges. d. Wiss., IX and XIII. · JFM 19.1029.01 |
| [21] | Perfekt, Karl-Mikael; Putinar, Mihai, The essential spectrum of the Neumann-Poincar\'{e} operator on a domain with corners, Arch. Ration. Mech. Anal., 223, 2, 1019-1033 (2017) · Zbl 1377.35203 · doi:10.1007/s00205-016-1051-6 |
| [22] | Poincar\'{e}, H., La m\'{e}thode de Neumann et le probl\`eme de Dirichlet, Acta Math., 20, 1, 59-142 (1897) · JFM 27.0316.01 · doi:10.1007/BF02418028 |
| [23] | Ritter, S., The spectrum of the electrostatic integral operator for an ellipsoid. Inverse scattering and potential problems in mathematical physics, Oberwolfach, 1993, Methoden Verfahren Math. Phys. 40, 157-167 (1995), Peter Lang, Frankfurt am Main · Zbl 0817.45003 |
| [24] | Schiffer, M., The Fredholm eigen values of plane domains, Pacific J. Math., 7, 1187-1225 (1957) · Zbl 0138.30003 |
| [25] | Verchota, Gregory, Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, J. Funct. Anal., 59, 3, 572-611 (1984) · Zbl 0589.31005 · doi:10.1016/0022-1236(84)90066-1 |
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