Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Asymptotics of the eigenvalues of the Dirichlet-Laplace problem in a domain with thin tube excluded

Author: X. Claeys
Journal: Quart. Appl. Math. 74 (2016), 595-605
MSC (2010): Primary 35B25
DOI: https://doi.org/10.1090/qam/1436
Published electronically: June 17, 2016
MathSciNet review: 3539023
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Abstract: We consider a Laplace problem with Dirichlet boundary condition in a three dimensional domain containing an inclusion taking the form of a thin tube with small thickness $ \delta $. We prove convergence in operator norm of the resolvent of this problem as $ \delta \to 0$, establishing that the perturbation induced by the inclusion on the resolvent is not greater than $ O(\vert \ln \delta \vert ^{-\gamma })$ for some $ \gamma >0$. We deduce convergence of the eigenvalues of the perturbed operator toward the limit operator.

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  • [1] H. Ammari, An introduction to mathematics of emerging biomedical imaging, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 62, Springer, Berlin, 2008. MR 2440857 (2010j:44002)
  • [2] H. Ammari and H. Kang, Reconstruction of small inhomogeneities from boundary measurements, Lecture Notes in Mathematics, vol. 1846, Springer-Verlag, Berlin, 2004. MR 2168949 (2006k:35295)
  • [3] I. I. Argatov and F. J. Sabina, Acoustic diffraction by a thin soft torus, Wave Motion 45 (2008), no. 6, 846-856. MR 2418141 (2009f:76151), https://doi.org/10.1016/j.wavemoti.2008.03.001
  • [4] A. L. Dontchev and R. T. Rockafellar, Implicit functions and solution mappings, A view from variational analysis, 2nd ed., Springer Series in Operations Research and Financial Engineering, Springer, New York, 2014. MR 3288139
  • [5] 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 (82d:35027)
  • [6] M. V. Fedoryuk.
    Theory of cubature formulas and the applications of functionnal analysis to problems of mathematical physics, volume 126 of Trans. Math. Monogr., chapter ``The Dirichlet problem for the Laplace operator in the exterior of a thin body of revolution'',
    American Mathematical Society, 1985.
  • [7] J. Geer, The scattering of a scalar wave by a slender body of revolution, SIAM J. Appl. Math. 34 (1978), no. 2, 348-370. MR 0492887 (58 #11940)
  • [8] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988. Reprint of the 1952 edition. MR 944909 (89d:26016)
  • [9] A. M. Ilin, Matching of asymptotic expansions of solutions of boundary value problems, Translated from the Russian by V. Minachin [V. V. Minakhin], Translations of Mathematical Monographs, vol. 102, American Mathematical Society, Providence, RI, 1992. MR 1182791 (93g:35016)
  • [10] A. M. Ilin, Study of the asymptotic behavior of the solution of an elliptic boundary value problem in a domain with a small hole, Trudy Sem. Petrovsk. 6 (1981), 57-82 (Russian, with English summary). MR 630701 (83e:35045)
  • [11] D. S. Jones, Methods in electromagnetic wave propagation, The Clarendon Press, Oxford University Press, New York, 1979. MR 571011 (81k:78001)
  • [12] T. Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452 (96a:47025)
  • [13] V. A. Kozlov, V. G. Mazya, and J. Rossmann, Elliptic boundary value problems in domains with point singularities, Mathematical Surveys and Monographs, vol. 52, American Mathematical Society, Providence, RI, 1997. MR 1469972 (98f:35038)
  • [14] V. Mazya, S. Nazarov, and B. Plamenevskij, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vol. I, translated from the German by Georg Heinig and Christian Posthoff, Operator Theory: Advances and Applications, vol. 111, Birkhäuser Verlag, Basel, 2000. MR 1779977 (2001e:35044a)
  • [15] V. G. Mazya, 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 (83e:35046b)
  • [16] V. G. Mazya, 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 (86b:35152)
  • [17] S. A. Nazarov, Asymptotic conditions at a point, selfadjoint extensions of operators, and the method of matched asymptotic expansions, Proceedings of the St. Petersburg Mathematical Society, Vol. V, Amer. Math. Soc. Transl. Ser. 2, vol. 193, Amer. Math. Soc., Providence, RI, 1999, pp. 77-125. MR 1736907 (2000m:35016)
  • [18] S. A. Nazarov and J. Sokołowski, Asymptotic analysis of shape functionals, J. Math. Pures Appl. (9) 82 (2003), no. 2, 125-196 (English, with English and French summaries). MR 1976204 (2004c:35015), https://doi.org/10.1016/S0021-7824(03)00004-7
  • [19] S. A. Nazarov and J. Sokołowski, Self-adjoint extensions for the Neumann Laplacian and applications, Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 3, 879-906. MR 2220182 (2007a:49073), https://doi.org/10.1007/s10114-005-0652-z
  • [20] A. A. Novotny and J. Sokołowski, Topological derivatives in shape optimization, Interaction of Mechanics and Mathematics, Springer, Heidelberg, 2013. MR 3013681
  • [21] M. Yu. Planida, On the convergence of solutions of singularly perturbed boundary value problems for the Laplacian, Mat. Zametki 71 (2002), no. 6, 867-877 (Russian, with Russian summary); English transl., Math. Notes 71 (2002), no. 5-6, 794-803. MR 1933107 (2003g:35006), https://doi.org/10.1023/A:1015820928854
  • [22] G. V. Zhdanova, The Dirichlet problem for the Helmholtz operator in the exterior of a thin body of revolution, Differentsialnye Uravneniya 20 (1984), no. 8, 1403-1411 (Russian). MR 759595 (86a:35043)

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Additional Information

X. Claeys
Affiliation: Laboratoire Jacques-Louis Lions, UPMC University of Paris 6 and CNRS UMR 7598, 75005, Paris, France;; INRIA-Paris-Rocquencourt, EPC Alpines, Le Chesnay Cedex, France
Email: claeys@ann.jussieu.fr

DOI: https://doi.org/10.1090/qam/1436
Received by editor(s): February 25, 2015
Published electronically: June 17, 2016
Article copyright: © Copyright 2016 Brown University

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