Journal of the Korean Society for Industrial and Applied Mathematics

  • 제27권2호
  • /
  • Pages.87-108
  • /
  • 2023
  • /
  • 1226-9433(pISSN)
  • /
  • 1229-0645(eISSN)
한국산업응용수학회 (Korean Society for Industrial and Applied Mathematics)
권호 표지
DOI QR코드

DOI QR Code

SPECTRAL PROPERTIES OF THE NEUMANN-POINCARÉ OPERATOR AND CLOAKING BY ANOMALOUS LOCALIZED RESONANCE: A REVIEW

  • SHOTA FUKUSHIMA (DEPARTMENT OF MATHEMATICS AND INSTITUTE OF APPLIED MATHEMATICS, INHA UNIVERSITY) ;
  • YONG-GWAN JI (SCHOOL OF MATHEMATICS, KOREA INSTITUTE FOR ADVANCED STUDY) ;
  • HYEONBAE KANG (DEPARTMENT OF MATHEMATICS AND INSTITUTE OF APPLIED MATHEMATICS, INHA UNIVERSITY) ;
  • YOSHIHISA MIYANISHI (DEPARTMENT OF MATHEMATICAL SCIENCES, FACULTY OF SCIENCE, SHINSHU UNIVERSITY)
  • 투고 : 2023.05.08
  • 심사 : 2023.06.20
  • 발행 : 2023.06.25
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

This is a review paper on recent development on the spectral theory of the Neumann-Poincaré operator. The topics to be covered are convergence rate of eigenvalues of the Neumann-Poincaré operator and surface localization of the single layer potentials of its eigenfunctions. Study on these topics is motivated by their relations with the cloaking by anomalous localized resonance. We review on this topic as well.

키워드

과제정보

This work is partially supported by National Research Foundation (of S. Korea) grants No. 2022R1A2B5B01001445 and by a KIAS Individual Grant (MG089001) at Korea Institute for Advanced Study.

참고문헌

  1. K. Ando, H. Kang, Y. Miyanishi and M. Putinar, Spectral analysis of Neumann-Poincare operator, Rev. Roumaine Math. Pures Appl, Vol. LXVI (2021), 545-575.
  2. H. Kang, Spectral Geometry and Analysis of the Neumann-Poincare Operator, a Review, In: Kang, NG., Choe, J., Choi, K., Kim, Sh. (eds) Recent Progress in Mathematics. KIAS Springer Series in Mathematics, vol 1. Springer, Singapore, 2022. https://doi.org/10.1007/978-981-19-3708-8_4
  3. D. Khavinson, M. Putinar and H.S. Shapiro, Poincare's variational problem in potential theory ' , Arch. Rational Mech. Anal, 185 (2007), 143-184. https://doi.org/10.1007/s00205-006-0045-1
  4. S. Fukushima, H. Kang and Y. Miyanishi, Decay rate of the eigenvalues of the Neumann-Poincare operator, arXiv:2304.04772.
  5. K. Ando and H. Kang, Analysis of plasmon resonance on smooth domains using spectral properties of the Neumann-Poincare operator, Jour. Math. Anal. Appl, 435 (2016), 162-178. https://doi.org/10.1016/j.jmaa.2015.10.033
  6. H. Ammari, G. Ciraolo, H. Kang, H. Lee and G.W. Milton, Spectral theory of a Neumann-Poincare-type operator and analysis of cloaking due to anomalous localized resonance, Arch. Rational Mech. Anal, 208 (2013), 667-692. https://doi.org/10.1007/s00205-012-0605-5
  7. R. C. McPhedran and G. W. Milton, A review of anomalous resonance, its associated cloaking, and superlensing, C. R. Phy, 21 (2020), 409-423. https://doi.org/10.5802/crphys.6
  8. H. Ammari and H. Kang, Polarization and moment tensors with applications to inverse problems and effective medium theory, Applied Mathematical Sciences, Vol. 162, Springer-Verlag, New York, 2007.
  9. K. Ando, H. Kang and Y. Miyanishi, Exponential decay estimates of the eigenvalues for the Neumann-Poincare operator on analytic boundaries in two dimensions, J. Integr. Equ. Appl, 30 (2018), 473-489. https://doi.org/10.1216/JIE-2018-30-4-473
  10. E. Bonnetier and H. Zhang, Characterization of the essential spectrum of the Neumann-Poincare operator in 2D domains with corner via Weyl sequences, Rev. Mat. Iberoam, 35 (2019), 925-948. https://doi.org/10.4171/rmi/1075
  11. J. Helsing and K.-M. Perfekt, On the Polarizability and Capacitance of the Cube, Applied and Computational Harmonic Analysis, 34 (2013), 445-468.
  12. J. Helsing and K.-M. Perfekt, The spectra of harmonic layer potential operators on domains with rotationally symmetric conical points, J. Math. Pures Appl, 118 (2018), 235--287. https://doi.org/10.1016/j.matpur.2017.10.012
  13. H. Kang, M. Lim and S. Yu, Spectral resolution of the Neumann-Poincare operator on intersecting disks and analysis of plasmon resonance, Arch. Rational Mech. Anal, 226(1) (2017), 83-115. https://doi.org/10.1007/s00205-017-1129-9
  14. K.-M. Perfekt and M. Putinar, Spectral bounds for the Neumann-Poincare operator on planar domains with corners, J. d'Analyse Math, 124 (2014), 39-57. https://doi.org/10.1007/s11854-014-0026-5
  15. K.M. Perfekt and M. Putinar, The essential spectrum of the Neumann-Poincare operator on a domain with corners, Arch. Rational Mech. Anal, 223 (2017), 1019-1033. https://doi.org/10.1007/s00205-016-1051-6
  16. K.-M. Perfekt. Plasmonic eigenvalue problem for corners: limiting absorption principle and absolute continuity in the essential spectrum, J. Math. Pures Appl, 145 (2021), 130-162. https://doi.org/10.1016/j.matpur.2020.07.001
  17. Y. Miyanishi, Weyl's law for the eigenvalues of the Neumann-Poincare operators in three dimensions: Will-more energy and surface geometry, Adv. Math, 406 (2022), 108547.
  18. Y. Miyanishi and G. Rozenblum, Eigenvalues of the Neumann-Poincare operator in dimension 3: Weyl's law and geometry, Algebra i Analiz 31(2) (2019), 248-268; reprinted in St. Petersburg Math. J, 31(2) (2020), 371-386. https://doi.org/10.1090/spmj/1602
  19. Y. Jung and M. Lim, A decay estimate for the eigenvalues of the Neumann-Poincare operator using the Grunsky coefficients, Proc. Amer. Math. Soc, 148 (2020), 591-600. https://doi.org/10.1090/proc/14785
  20. Y. Miyanishi and T. Suzuki, Eigenvalues and eigenfunctions of double layer potentials, Trans. Amer. Math. Soc. 369 (2017), 8037-8059. https://doi.org/10.1090/tran/6913
  21. J. Delgado and M. Ruzhansky, Schatten classes on compact manifolds: kernel conditions, J. Funct. Anal, 267(3) (2014), 772-798. https://doi.org/10.1016/j.jfa.2014.04.016
  22. S. Fukushima and H. Kang, Spectral structure of the Neumann-Poincare operator on axially symmetric functions, in preparation.
  23. K. Ando, H. Kang, Y. Miyanishi and T. Nakazawa, Surface localization of plasmons in three dimensions and convexity, SIAM J. Appl. Math, 81 (2021), 1020-1033. https://doi.org/10.1137/20M1373530
  24. Y. Ji and H. Kang, A concavity condition for existence of a negative value in Neumann-Poincare spectrum in three dimensions, Proc. Amer. Math. Soc, 147 (2019), 3431-3438. https://doi.org/10.1090/proc/14467
  25. M. S. Birman and M. Z. Solomjak, Asymptotic behavior of the spectrum of pseudodifferential operators with anisotropically homogeneous symbols, Vestnik Leningrad. Univ, (13 Mat. Meh. Astronom. vyp. 3), 169 (1977), 13-21.
  26. G.W. Milton and N.-A.P. Nicorovici, On the cloaking effects associated with anomalous localized resonance, Proc. R. Soc. A, 462 (2006), 3027-3059. https://doi.org/10.1098/rspa.2006.1715
  27. D. Chung, H. Kang, K. Kim and H. Lee, Cloaking due to anomalous localized resonance in plasmonic structures of confocal ellipses, SIAM J. Appl. Math, 74 (2014), 1691-1707. https://doi.org/10.1137/140956762
  28. H. Ammari, G. Ciraolo, H. Kang, H. Lee and G.W. Milton, Spectral theory of a Neumann-Poincare-type operator and analysis of anomalous localized resonance II, Contemporary Math, 615 (2014), 1-14. https://doi.org/10.1090/conm/615/12244
  29. H. Ammari, H. Kang, and H. Lee, A boundary integral method for computing elastic moment tensors for ellipses and ellipsoids, J. Comp. Math, 25 (1) (2007), 2-12.
  30. V.D. Kupradze, Potential methods in the theory of elasticity, Daniel Davey & Co., New York, 1965.
  31. K. Ando, Y. Ji, H. Kang, K. Kim and S. Yu, Spectral properties of the Neumann-Poincare operator and cloaking by anomalous localized resonance for the elasto-static system, Euro. J. Appl. Math, 29 (2018), 189-225. https://doi.org/10.1017/S0956792517000080
  32. B.E.J. Dahlberg, C.E. Kenig and G.C. Verchota, Boundary value problems for the systems of elastostatics in Lipschitz domains, Duke Math. J, 57(3) (1988), 795-818. https://doi.org/10.1215/S0012-7094-88-05735-3
  33. S. Fukushima, Y.-G. Ji, and H. Kang, A decomposition theorem of surface vector fields and spectral structure of the Neumann-Poincare operator in elasticity, arXiv:2211.15879.
  34. N.I. Muskhelishvili, Singular integral equations. Boundary problems of function theory and their application to mathematical physics, Translated from the second (1946) Russian edition and with a preface by J. R. M. Radok, Noordhoff International Publishing-Leyden, 1977.
  35. A.P. Calderon, Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sc. USA, 74 (1977), 1324-1327. https://doi.org/10.1073/pnas.74.4.1324
  36. L. Escauriaza, E. B. Fabes, and G. Verchota, On a regularity theorem for weak solutions to transmission problems with internal Lipchitz boundaries, Proc. Amer. Math. Soc. 115 (4) (1992), 1069-1076. https://doi.org/10.1090/S0002-9939-1992-1092919-1
  37. G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains, J. Funct. Anal, 59(3) (1984), 572-611. https://doi.org/10.1016/0022-1236(84)90066-1
  38. K. Ando, H. Kang and Y. Miyanishi, Elastic Neumann-Poincare operators on three dimensional smooth domains: Polynomial compactness and spectral structure, Int. Math. Res. Notices, 12 (2019), 3883-3900. https://doi.org/10.1093/imrn/rnx258
  39. K. Ando, H. Kang and Y. Miyanishi, Convergence rate for eigenvalues of the elastic Neumann-Poincare operator on smooth and real analytic boundaries in two dimensions, Jour. Math. Pures Appl, 140 (2020), 211-229. https://doi.org/10.1016/j.matpur.2020.06.008
  40. G. Rozenblum, The Discrete Spectrum of the Neumann-poincare Operator in 3D Elasticity, J. Pseudo-Differ. Oper. Appl. 14 (2023), article number 26. https://doi.org/10.1007/s11868-023-00520-y.