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Transactions of the American Mathematical Society

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Rates of eigenvalues on a dumbbell domain. Simple eigenvalue case


Author: José M. Arrieta
Journal: Trans. Amer. Math. Soc. 347 (1995), 3503-3531
MSC: Primary 35P15; Secondary 35J05, 35P20
DOI: https://doi.org/10.1090/S0002-9947-1995-1297521-1
MathSciNet review: 1297521
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Abstract: We obtain the first term in the asymptotic expansion of the eigenvalues of the Laplace operator in a typical dumbbell domain in $ {\mathbb{R}^2}$. This domain consists of two disjoint domains $ {\Omega ^L}$, $ {\Omega ^R}$ joined by a channel $ {R_\varepsilon }$ of height of the order of the parameter $ \varepsilon $. When an eigenvalue approaches an eigenvalue of the Laplacian in $ {\Omega ^L} \cup {\Omega ^R}$, the order of convergence is $ \varepsilon $, while if the eigenvalue approaches an eigenvalue which comes from the channel, the order is weaker: $ \varepsilon \left\vert {{\text{ln}}\varepsilon } \right\vert$. We also obtain estimates on the behavior of the eigenfunctions.


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  • [1] J. M. Arrieta, Spectral properties of Schrödinger operators under perturbations of the domain, Doctoral Dissertation, Georgia Institute of Technology, 1991.
  • [2] -, Neumann eigenvalue problems on exterior perturbations of the domain, J. Differential Equations 117 (1995). MR 1329403 (96c:35133)
  • [3] J. M. Arrieta, J. K. Hale and Q. Han, Eigenvalue problems for nonsmoothly perturbed domains, J. Differential Equations 91 (1991), 24-52. MR 1106116 (92f:35110)
  • [4] I. Babuska and R. Vyborny, Continuous dependence of eigenvalues on the domains, Czechoslovak Math. J. 15 (1965), 169-178. MR 0182799 (32:281)
  • [5] J. T. Beale, Scattering frequencies of resonators, Comm. Pure Appl. Math. 26 (1973), 549-563. MR 0352730 (50:5217)
  • [6] R. Brown, P. D. Hislop and A. Martinez, Eigenvalues and resonances for domains with tubes: Neumann boundary conditions, preprint. MR 1310941 (96c:35141)
  • [7] I. Chavel and E. A. Feldman, Spectra of domains in compact manifolds, J. Funct. Anal. 30 (1978), 198-222. MR 515225 (80c:58027)
  • [8] R. Courant and D. Hilbert, Methods of mathematical physics, Vol. I, Wiley-Interscience, New York, 1953. MR 0065391 (16:426a)
  • [9] K. Fukaya, Collapsing of Riemannian manifolds and eigenvalues of Laplace operator, Invent. Math. 87 (1987), 517-547. MR 874035 (88d:58125)
  • [10] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, New York and Berlin, 1977. MR 0473443 (57:13109)
  • [11] J. K. Hale and G. Raugel, Reaction-diffusion equations on thin domains, J. Math. Pures Appl. 71 (1992), 33-95. MR 1151557 (93a:35083)
  • [12] J. K. Hale and J. M. Vegas, A nonlinear parabolic equation with varying domain, Arch. Rational Mech. Anal. 86 (1984), 99-123. MR 751304 (87b:35060)
  • [13] R. Hempel, L. Seco and B. Simon, The essential spectrum of Neumann Laplacians on some bounded singular domains, J. Funct. Anal. 102 (1991), 448-483. MR 1140635 (93h:35144)
  • [14] P. D. Hislop and A. Martinez, Scattering resonances of a Helmholtz resonator, Indiana Univ. Math. J. 40 (1991), 767-788. MR 1119196 (92k:35170)
  • [15] S. Jimbo, The singularly perturbed domain and the characterization for the eigenfunctions with Neumann boundary conditions, J. Differential Equations 77 (1989), 322-350. MR 983298 (90c:35158)
  • [16] -, Perturbation formula of eigenvalues in a singularly perturbed domain, J. Math. Soc. Japan (2) 45 (1993), 339-356. MR 1206658 (94c:35136)
  • [17] S. Jimbo and Y. Morita, Remarks on the behavior of certain eigenvalues on a singularly perturbed domain with several thin channels, Comm. Partial Differential Equations 17 (1992), 523-552. MR 1163435 (93d:58167)
  • [18] M. Lobo-Hidalgo and E. Sanchez-Palencia, Sur certaines propriétés spectrales des perturbations du domaine dans les problémes aux limites, Comm. Partial Differential Equations 4 (1979), 1085-1098. MR 544883 (80h:35108)
  • [19] S. Ozawa, Singular variations of domains and eigenvalues of the Laplacian, Duke Math. J. 48 (1981), 769-778. MR 782576 (86k:35117)
  • [20] -, Spectra of domains with small spherical Neumann boundary, J. Fac. Sci. Univ. Tokyo 30 (1983), 259-277. MR 722497 (85k:35174)
  • [21] J. Rauch and M. Taylor, Potential and scattering theory on wildly perturbed domains, J. Funct. Anal. 18 (1975), 27-59. MR 0377303 (51:13476)
  • [22] J. M. Vegas, Bifurcation caused by perturbing the domain in an elliptic equation, J. Differential Equations 48 (1983), 189-226. MR 696867 (84g:35020)

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DOI: https://doi.org/10.1090/S0002-9947-1995-1297521-1
Article copyright: © Copyright 1995 American Mathematical Society

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