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Transactions of the Moscow Mathematical Society

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Asymptotics of the eigenvalues of boundary value problems for the Laplace operator in a three-dimensional domain with a thin closed tube


Author: S. A. Nazarov
Translated by: V. E. Nazaikinskii
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 76 (2015), vypusk 1.
Journal: Trans. Moscow Math. Soc. 2015, 1-53
MSC (2010): Primary 35J25; Secondary 35B25, 35B40, 35B45, 35P20, 35S05
DOI: https://doi.org/10.1090/mosc/243
Published electronically: November 17, 2015
MathSciNet review: 3467259
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Abstract: We construct and justify asymptotic representations for the eigenvalues and eigenfunctions of boundary value problems for the Laplace operator in a three-dimensional domain $ \Omega (\varepsilon )=\Omega \setminus \Bar {\Gamma }_\varepsilon $ with a thin singular set $ \Gamma _\varepsilon $ lying in the  $ c\varepsilon $-neighborhood of a simple smooth closed contour $ \Gamma $. We consider the Dirichlet problem, a mixed boundary value problem with the Neumann conditions on  $ \partial \Gamma _\varepsilon $, and also a spectral problem with lumped masses on  $ \Gamma _\varepsilon $. The asymptotic representations are of diverse character: we find an asymptotic series in powers of the parameter $ \vert{\ln \varepsilon }\vert^{-1}$ or  $ \varepsilon $. The most comprehensive and complicated analysis is presented for the lumped mass problem; namely, we sum the series in powers of $ \vert{\ln \varepsilon }\vert^{-1}$ and obtain an asymptotic expansion with the leading term holomorphically depending on  $ \vert{\ln \varepsilon }\vert^{-1}$ and with the remainder $ O(\varepsilon ^\delta )$, $ \delta \in (0,1)$. The main role in asymptotic formulas is played by solutions of the Dirichlet problem in $ \Omega \setminus \Gamma $ with logarithmic singularities distributed along the contour $ \Gamma $.


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  • 1. V. G. Maz′ya, S. A. Nazarov, and B. A. Plamenevskiĭ, On the asymptotic behavior of solutions of elliptic boundary value problems with irregular perturbations of the domain, Probl. Mat. Anal., vol. 8, Leningrad. Univ., Leningrad, 1981, pp. 72–153, 222–223 (Russian). MR 658154
  • 2. W. G. Mazja, S. A. Nasarow, and B. A. Plamenewski, Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten. I, Mathematische Lehrbücher und Monographien, II. Abteilung: Mathematische Monographien [Mathematical Textbooks and Monographs, Part II: Mathematical Monographs], vol. 82, Akademie-Verlag, Berlin, 1991 (German). Störungen isolierter Randsingularitäten. [Perturbations of isolated boundary singularities]. MR 1101139
  • 3. V. G. Maz′ya, S. A. Nazarov, and B. A. Plamenevskiĭ, Asymptotic expansions of eigenvalues of boundary value problems for the Laplace operator in domains with small openings, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 2, 347–371 (Russian). MR 740795
  • 4. A. Campbell and S. Nazarov, Une justification de la méthode de raccordement des développements asymptotiques appliquée à un problème de plaque en flexion. Estimation de la matrice d’impédance, J. Math. Pures Appl. (9) 76 (1997), no. 1, 15–54 (French, with English and French summaries). MR 1429996, https://doi.org/10.1016/S0021-7824(97)89944-8
  • 5. Alain Campbell and Sergueï Nazarov, An asymptotic study of a plate problem by a rearrangement method. Application to the mechanical impedance, RAIRO Modél. Math. Anal. Numér. 32 (1998), no. 5, 579–610 (English, with English and French summaries). MR 1643481
  • 6. G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, no. 27, Princeton University Press, Princeton, N. J., 1951. MR 0043486
  • 7. N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy; Die Grundlehren der mathematischen Wissenschaften, Band 180. MR 0350027
  • 8. S. A. Nazarov, Asymptotics of the eigenvalues of the Neumann problem under mass concentration on a thin toroidal set, Vestn. SPbGU Ser. 1 (2006), no. 3, 43-53. (Russian)
  • 9. M. V. Fedoryuk, The Dirichlet problem for the Laplace operator in the exterior of a thin solid of revolution, Theory of cubature formulas and the application of functional analysis to problems of mathematical physics, Trudy Sem. S. L. Soboleva, No. 1, vol. 80, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1980, pp. 113–131, 150 (Russian). MR 631695
  • 10. M. V. Fedorjuk, Asymptotic behavior of the solution of the Dirichlet problem for the Laplace and Helmholtz equations in the exterior of a thin cylinder, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), no. 1, 167–186, 240 (Russian). MR 607581
  • 11. V. G. Maz′ya, S. A. Nazarov, and B. A. Plamenevskiĭ, On the asymptotic behavior of the solutions to the Dirichlet problem in a three-dimensional domain with a cut-out thin body, Dokl. Akad. Nauk SSSR 256 (1981), no. 1, 37–39 (Russian). MR 599097
  • 12. V. G. Maz′ya, S. A. Nazarov, and B. A. Plamenevskiĭ, Asymptotics of the solutions of the Dirichlet problem in a domain with an excluded thin tube, Uspekhi Mat. Nauk 36 (1981), no. 5(221), 183–184 (Russian). MR 637449
    V. G. Maz′ya, S. A. Nazarov, and B. A. Plamenevskiĭ, The asymptotic behavior of solutions of the Dirichlet problem in a domain with a cut out thin tube, Mat. Sb. (N.S.) 116(158) (1981), no. 2, 187–217 (Russian). MR 637860
  • 13. I. S. Zorin and S. A. Nazarov, The strain-stress state of an elastic space with a thin toroidal inclusion, Mekh. Tverd. Tela (1985), no. 3, 79-86. (Russian)
  • 14. S. A. Nazarov, Averaging of boundary value problems in a domain that contains a thin cavity with a periodically changing cross section, Trudy Moskov. Mat. Obshch. 53 (1990), 98–129, 260 (Russian); English transl., Trans. Moscow Math. Soc. (1991), 101–134. MR 1097994
  • 15. S. A. Nazarov and M. V. Paukshto, Discrete models and homogenization in problems of the elastic theory, Leningrad University, Leningrad, 1984. (Russian)
  • 16. V. G. Maz′ya, S. A. Nazarov, and B. A. Plamenevskiĭ, Homogeneous solutions of the Dirichlet problem in the exterior of a thin cone, Dokl. Akad. Nauk SSSR 266 (1982), no. 2, 281–284 (Russian). MR 672858
  • 17. V. G. Maz′ya, S. A. Nazarov, and B. A. Plamenevskiĭ, Singularities of solutions of the Dirichlet problem in the exterior of a thin cone, Mat. Sb. (N.S.) 122(164) (1983), no. 4, 435–457 (Russian). MR 725451
  • 18. N. V. Movchan, Oscillations of elastic bodies with small holes, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. vyp. 1 (1989), 33–37, 123 (Russian, with English summary); English transl., Vestnik Leningrad Univ. Math. 22 (1989), no. 1, 50–55. MR 1021476
  • 19. Alain Campbell and SergueïA. Nazarov, Asymptotics of eigenvalues of a plate with small clamped zone, Positivity 5 (2001), no. 3, 275–295. MR 1836750, https://doi.org/10.1023/A:1011469822255
  • 20. E. Sánchez-Palencia, Perturbation of eigenvalues in thermoelasticity and vibration of systems with concentrated masses, Trends and applications of pure mathematics to mechanics (Palaiseau, 1983) Lecture Notes in Phys., vol. 195, Springer, Berlin, 1984, pp. 346–368. MR 755735, https://doi.org/10.1007/3-540-12916-2_66
  • 21. E. Sánchez-Palencia and P. Suquet, Friction and homogenization of a boundary, Free boundary problems: theory and applications, Vol. I, II (Montecatini, 1981) Res. Notes in Math., vol. 78, Pitman, Boston, MA, 1983, pp. 561–571. MR 714936
  • 22. O. A. Oleĭnik, J. Sanchez-Hubert, and G. A. Yosifian, On vibrations of a membrane with concentrated masses, Bull. Sci. Math. 115 (1991), no. 1, 1–27 (English, with French summary). MR 1086936
  • 23. S. A. Nazarov, On a problem of Sánchez-Palencia with Neumann boundary conditions, Izv. Vyssh. Uchebn. Zaved. Mat. 11 (1989), 60–66 (Russian); English transl., Soviet Math. (Iz. VUZ) 33 (1989), no. 11, 73–78. MR 1045104
  • 24. Yu. D. Golovatyĭ, S. A. Nazarov, and O. A. Oleĭnik, Asymptotic expansions of eigenvalues and eigenfunctions of problems on oscillations of a medium with concentrated perturbations, Trudy Mat. Inst. Steklov. 192 (1990), 42–60 (Russian). Translated in Proc. Steklov Inst. Math. 1992, no. 3, 43–63; Differential equations and function spaces (Russian). MR 1097888
  • 25. S. A. Nazarov, Interaction of concentrated masses in a harmonically oscillating spatial body with Neumann boundary conditions, RAIRO Modél. Math. Anal. Numér. 27 (1993), no. 6, 777–799 (English, with English and French summaries). MR 1246999
  • 26. J. Caínzos, E. Pérez, and M. Vilasánchez, Asymptotics for the eigenelements of the Neumann spectral problem with concentrated masses, Indiana Univ. Math. J. 56 (2007), no. 4, 1939–1987. MR 2354704, https://doi.org/10.1512/iumj.2007.56.3084
  • 27. M. Lobo and E. Pérez, Local problems for vibrating systems with concentrated masses: a review, C. R. Mecánique 331 (2003), no. 4, 303-317.
  • 28. Enrique Sánchez-Palencia, Nonhomogeneous media and vibration theory, Lecture Notes in Physics, vol. 127, Springer-Verlag, Berlin-New York, 1980. MR 578345
  • 29. J. Sanchez Hubert and E. Sánchez-Palencia, Vibration and coupling of continuous systems, Springer-Verlag, Berlin, 1989. Asymptotic methods. MR 996423
  • 30. O. A. Oleĭnik, G. A. Iosif′yan, and A. S. Shamaev, \cyr Matematicheskie zadachi teorii sil′no neodnorodnykh uprugikh sred, Moskov. Gos. Univ., Moscow, 1990 (Russian). MR 1115306
  • 31. S. A. Nazarov and B. A. Plamenevskiĭ, Selfadjoint elliptic problems: scattering and polarization operators on the edges of the boundary, Algebra i Analiz 6 (1994), no. 4, 157–186 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 6 (1995), no. 4, 839–863. MR 1304098
  • 32. Sergey A. Nazarov and Boris A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, De Gruyter Expositions in Mathematics, vol. 13, Walter de Gruyter & Co., Berlin, 1994. MR 1283387
  • 33. V. I. Smirnov, \cyr Kurs vyssheĭ matematiki. Tom II, 17th ed, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1961 (Russian). MR 0125182
  • 34. Ju. Nagel′, On equivalent norms in the function spaces 𝐻^{𝜇}, Vestnik Leningrad. Univ. 7 Mat. Meh. Astronom. Vyp. 2 (1974), 41–47, 171 (Russian, with English summary). MR 0355579
  • 35. Lars Hörmander, Linear partial differential operators, Third revised printing. Die Grundlehren der mathematischen Wissenschaften, Band 116, Springer-Verlag New York Inc., New York, 1969. MR 0248435
  • 36. M. Š. Birman and M. Z. Solomjak, \cyr Spektral′naya teoriya samosopryazhennykh operatorov v gil′bertovom prostranstve, Leningrad. Univ., Leningrad, 1980 (Russian). MR 609148
  • 37. I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products, Nauka, Moscow, 1971; English transl., Academic Press, New York, 1980.
  • 38. V. G. Maz'ya and B. A. Plamenevskii, On the ellipticity of boundary value problems in points with piecewise smooth boundary, in Proceedings of Symposium on Mechanics of Solids and Related Problems in Analysis, Tbilisi, Metsniereba, 1973, vol. 1, 171-181. (Russian)
  • 39. V. A. Kondrat′ev, Singularities of the solution of the Dirichlet problem for a second order elliptic equation in the neighborhood of an edge, Differencial′nye Uravnenija 13 (1977), no. 11, 2026–2032, 2109 (Russian). MR 0486987
  • 40. V. A. Nikiškin, Singularities of the solution of the Dirichlet problem for a second-order equation in the neighborhood of an edge, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2 (1979), 51–62, 103 (Russian, with English summary). MR 531648
  • 41. Vladimir Gilelevič Maz′ja and Jürgen Rossmann, Über die Asymptotik der Lösungen elliptischer Randwertaufgaben in der Umgebung von Kanten, Math. Nachr. 138 (1988), 27–53 (German). MR 975198, https://doi.org/10.1002/mana.19881380103
  • 42. S. A. Nazarov and B. A. Plamenevskiĭ, Selfadjoint elliptic problems with radiation conditions on the edges of the boundary, Algebra i Analiz 4 (1992), no. 3, 196–225 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 4 (1993), no. 3, 569–594. MR 1190778
  • 43. S. A. Nazarov and B. A. Plamenevskiĭ, Elliptic problems with radiation conditions on the edges of the boundary, Mat. Sb. 183 (1992), no. 10, 13–44 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math. 77 (1994), no. 1, 149–176. MR 1202790, https://doi.org/10.1070/SM1994v077n01ABEH003434
  • 44. V. A. Kondrat′ev, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obšč. 16 (1967), 209–292 (Russian). MR 0226187
  • 45. A. Pazy, Asymptotic expansions of solutions of ordinary differential equations in Hilbert space, Arch. Rational Mech. Anal. 24 (1967), 193–218. MR 0209618, https://doi.org/10.1007/BF00281343
  • 46. S. A. Nazarov, Polynomial property of selfadjoint elliptic boundary value problems, and the algebraic description of their attributes, Uspekhi Mat. Nauk 54 (1999), no. 5(329), 77–142 (Russian); English transl., Russian Math. Surveys 54 (1999), no. 5, 947–1014. MR 1741662, https://doi.org/10.1070/rm1999v054n05ABEH000204
  • 47. J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR 0247243
    J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 2, Travaux et Recherches Mathématiques, No. 18, Dunod, Paris, 1968 (French). MR 0247244
  • 48. O. A. Ladyzhenskaya, The boundary value problems of mathematical physics, Applied Mathematical Sciences, vol. 49, Springer-Verlag, New York, 1985. Translated from the Russian by Jack Lohwater [Arthur J. Lohwater]. MR 793735
  • 49. A. M. Il′in, A boundary value problem for an elliptic equation of second order in a domain with a narrow slit. II. Domain with a small opening, Mat. Sb. (N.S.) 103(145) (1977), no. 2, 265–284 (Russian). MR 0442460
  • 50. M. A. Krasnosel′skiĭ, G. M. Vaĭnikko, P. P. Zabreĭko, Ya. B. Rutitskii, and V. Ya. Stetsenko, Approximate solution of operator equations, Wolters-Noordhoff Publishing, Groningen, 1972. Translated from the Russian by D. Louvish. MR 0385655
  • 51. Milton Van Dyke, Perturbation methods in fluid mechanics, Applied Mathematics and Mechanics, Vol. 8, Academic Press, New York-London, 1964. MR 0176702
  • 52. A. M. Il′in, \cyr Soglasovanie asimptoticheskikh razlozheniĭ resheniĭ kraevykh zadach, “Nauka”, Moscow, 1989 (Russian). With an English summary. MR 1007834
  • 53. M. I. Višik and L. A. Lyusternik, Regular degeneration and boundary layer for linear differential equations with small parameter, Uspehi Mat. Nauk (N.S.) 12 (1957), no. 5(77), 3–122 (Russian). MR 0096041
  • 54. S. A. Nazarov, Asymptotic analysis of thin plates and rods, vol. 1. Dimension reduction and integral estimates, Nauchn. kniga, Novosibirsk, 2001. (Russian)
  • 55. A. D. Aleksandrov and N. Yu. Netsvetaev, \cyr Geometriya, “Nauka”, Moscow, 1990 (Russian). MR 1129460
  • 56. I. V. Kamotskii and S. A. Nazarov, On eigenfunctions localized near the edge of a thin domain, Probl. Mat. Analiza 19 (1999), 105-148. (Russian)
  • 57. Serguei A. Nazarov, Localization effects for eigenfunctions near to the edge of a thin domain, Proceedings of EQUADIFF, 10 (Prague, 2001), 2002, pp. 283–292. MR 1981533
  • 58. L. Friedlander and M. Solomyak, On the spectrum of narrow periodic waveguides, Russ. J. Math. Phys. 15 (2008), no. 2, 238–242. MR 2410832, https://doi.org/10.1134/S1061920808020076
  • 59. Denis Borisov and Pedro Freitas, Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin planar domains, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 2, 547–560. MR 2504043, https://doi.org/10.1016/j.anihpc.2007.12.001
  • 60. S. A. Nazarov, Spectral properties of a thin layer with a doubly periodic family of thinning regions, Theoret. and Math. Phys. 174 (2013), no. 3, 343–359. Translation of Teoret. Mat. Fiz. 174 (2013), no. 3, 398–415. MR 3171515, https://doi.org/10.1007/s11232-013-0031-3
  • 61. S. A. Nazarov, Localization of the eigenfunctions of the Dirichlet problem in thin polyhedra near the vertices, Sibirsk. Mat. Zh. 54 (2013), no. 3, 655–672 (Russian, with Russian summary); English transl., Sib. Math. J. 54 (2013), no. 3, 517–532. MR 3112622, https://doi.org/10.1134/S0037446613030166
  • 62. S. A. Nazarov and E. Pérez, New asymptotic effects for the spectrum of problems on concentrated masses near the boundary, C. R. Mécanique 337 (2009), no. 8, 585-590.

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Additional Information

S. A. Nazarov
Affiliation: Laboratory of Nanomanufacturing, Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia; Mathematics and Mechanics Faculty, St. Petersburg State University, St. Petersburg, Russia; Laboratory of Mathematical Methods in Mechanics of Materials, Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, Russia
Email: srgnazarov@yahoo.co.uk

DOI: https://doi.org/10.1090/mosc/243
Keywords: Eigenvalue and eigenfunction asymptotics, convergence theorem, singular perturbation of a domain, thin toroidal cavity, Dirichlet and Neumann problems, lumped mass.
Published electronically: November 17, 2015
Additional Notes: Supported by grant no. 6.37.671.2013 from St. Petersburg State University.
Article copyright: © Copyright 2015 S. A. Nazarov