Perturbation formula of eigenvalues in a singularly perturbed domain

Shuichi JIMBO

doi:10.2969/jmsj/04520339

Shuichi JIMBO

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JOURNAL FREE ACCESS

1993 Volume 45 Issue 2 Pages 339-356

DOI https://doi.org/10.2969/jmsj/04520339

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Published: 1993 Received: May 06, 1992 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) C. Anné, Spectre du Laplacien et écrasement d'andes, Ann. Sci. École Norm. Sup., 20 (1987), 271-280.
2) J. M. Arrieta, J. Hale and Q. Han, Eigenvalue problems for nonsmoothly perturbed domains, J. Differential Equations, 91 (1991), 24-52.
3) I. Babuska and R. Výborný, Continuous dependence of the eigenvalues on the domains, Czechoslovak Math. J., 15 (1965), 169-178.
4) J. T. Beale, Scattering frequencies of resonators, Comm. Pure Appl. Math., 26 (1973), 549-563.
5) Y. Colin de Verdière, Sur la multiplicité de la première valeur propre nonnulle du Laplacien, Comment. Math. Helv., 61 (1986), 254-270.
6) I. Chavel and D. Feldman, Spectra of manifolds with small handles, Comment. Math. Helv., 56 (1981), 83-102.
7) R. Courant and D. Hilbert, Method of Mathematical Physics, Vol. I, Wiley-Inter-science, New York, 1953.
8) E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations, J. Differential Equations, 74 (1988), 120-156.
9) Q. Fang, Asymptotic behavior and domain-dependency of solutions to a class of reaction-diffusion systems with large diffusion coefficients, Hiroshima Math. J., 20 (1990), 549-571.
10) D. Fujiwara, A Remark on the Hadamard Variational Formula, Proc. Japan Acad., 55 (1979), 180-184.
11) K. Fukaya, Collapsing of Riemannian manifolds and eigenvalues of Laplace operator, Invent. Math., 87 (1987), 517-547.
12) J. K. Hale and J. M. Vegas, A Nonlinear Parabolic Equation with Varying Domain, Arch. Rational Mech. Anal., 86 (1984), 99-123.
13) J. K. Hale and G. Raugel, Reaction diffusion equation on thin domains, to appear in J. Math. Pures Appl..
14) 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.
15) S. Jimbo, Singular Perturbation of Domains and the Semilinear Elliptic Equation II, J. Differential Equations, 75 (1988), 265-289.
16) S. Jimbo, The singularly perturbed domain and the characterization for the eigenfunctions with Neumann boundary conditions, J. Differential Equations, 77 (1989), 322-350.
17) S. Jimbo, A construction of the perturbed solution of semilinear elliptic equation in the singularly perturbed domain, J. Fac. Sci. Univ. Tokyo, 36 (1989), 163-185.
18) S. Jimbo and Y. Morita, Remarks on the behavior of certain eigenvalues in the singularly perturbed domain with several thin channels, Comm. Partial Differential Equations, 17 (1992), 523-552.
19) M. Lobo-Hidalgo and E. Sanchez-Palencia, Sur certaines propriétés spectrales des perturbations du domaine dans les problèmes aux limites, Comm. Partial Differential Equations, 4 (1979), 1085-1098.
20) H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci., Kyoto Univ., 19 (1979), 401-454.
21) H. Matano and M. Mimura, Pattern formation in competition diffusion systems in nonconvex domains, Publ. Res. Inst. Math. Sci., Kyoto Univ., 19 (1983), 1049-1079.
22) Y. Morita, Reaction Diffusion Systems in Nonconvex Domains; Invariant Manifold and Reduced Form, J. Dynamics Differential Equations, 2 (1990), 69-115.
23) Y. Morita and S. Jimbo, ODEs on Inertial Manifolds for Reaction Diffusion Systems in a Singularly Perturbed Domain with Several Thin Channels, J. Dynamics Differential Equations, 4 (1992), 65-98.
24) S. Ozawa, Singular variations of domains and eigenvalues of the Laplacian, Duke Math. J., 48 (1981), 769-778.
25) S. Ozawa, Electrostatic capacity and eigenvalues of the Laplacian, J. Fac. Sci. Univ. Tokyo, 30 (1983), 53-62.
26) S. Ozawa, Spectra of domains with small spherical Neumann boundary, J. Fac. Sci. Univ. Tokyo, 30 (1983), 259-277.
27) J. Rauch and M. Taylor, Potential and scattering theory on wildly perturbed domains, J. Funct. Anal., 18 (1975), 27-59.
28) M. Taylor, Pseudo differential operators, Princeton University Press, 1981.
29) J. M. Vegas, Bifurcation caused by perturbing the domain in an elliptic equation, J. Differential Equations, 48 (1983), 189-226.

Right : [1] C. Anné, Spectre du Laplacien et écrasement d'andes, Ann. Sci. École Norm. Sup., 20 (1987), 271-280.
[2] J. M. Arrieta, J. Hale and Q. Han, Eigenvalue problems for nonsmoothly perturbed domains, J. Differential Equations, 91 (1991), 24-52.
[3] I. Babuska and R. Výborný, Continuous dependence of the eigenvalues on the domains, Czechoslovak Math. J., 15 (1965), 169-178.
[4] J. T. Beale, Scattering frequencies of resonators, Comm. Pure Appl. Math., 26 (1973), 549-563.
[5] Y. Colin de Verdière, Sur la multiplicité de la première valeur propre nonnulle du Laplacien, Comment. Math. Helv., 61 (1986), 254-270.
[6] I. Chavel and D. Feldman, Spectra of manifolds with small handles, Comment. Math. Helv., 56 (1981), 83-102.
[7] R. Courant and D. Hilbert, Method of Mathematical Physics, Vol. I, Wiley-Interscience, New York, 1953.
[8] E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations, J. Differential Equations, 74 (1988), 120-156.
[9] Q. Fang, Asymptotic behavior and domain-dependency of solutions to a class of reaction-diffusion systems with large diffusion coefficients, Hiroshima Math. J., 20 (1990), 549-571.
[10] D. Fujiwara, A Remark on the Hadamard Variational Formula, Proc. Japan Acad., 55 (1979), 180-184.
[11] K. Fukaya, Collapsing of Riemannian manifolds and eigenvalues of Laplace operator, Invent. Math., 87 (1987), 517-547.
[12] J. K. Hale and J. M. Vegas, A Nonlinear Parabolic Equation with Varying Domain, Arch. Rational Mech. Anal., 86 (1984), 99-123.
[13] J. K. Hale and G. Raugel, Reaction diffusion equation on thin domains, to appear in J. Math. Pures Appl..
[14] 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.
[15] S. Jimbo, Singular Perturbation of Domains and the Semilinear Elliptic Equation II, J. Differential Equations, 75 (1988), 265-289.
[16] S. Jimbo, The singularly perturbed domain and the characterization for the eigenfunctions with Neumann boundary conditions, J. Differential Equations, 77 (1989), 322-350.
[17] S. Jimbo, A construction of the perturbed solution of semilinear elliptic equation in the singularly perturbed domain, J. Fac. Sci. Univ. Tokyo, 36 (1989), 163-185.
[18] S. Jimbo and Y. Morita, Remarks on the behavior of certain eigenvalues in the singularly perturbed domain with several thin channels, Comm. Partial Differential Equations, 17 (1992), 523-552.
[19] M. Lobo-Hidalgo and E. Sanchez-Palencia, Sur certaines propriétés spectrales des perturbations du domaine dans les problèmes aux limites, Comm. Partial Differential Equations, 4 (1979), 1085-1098.
[20] H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci., Kyoto Univ., 19 (1979), 401-454.
[21] H. Matano and M. Mimura, Pattern formation in competition diffusion systems in nonconvex domains, Publ. Res. Inst. Math. Sci., Kyoto Univ., 19 (1983), 1049-1079.
[22] Y. Morita, Reaction Diffusion Systems in Nonconvex Domains: Invariant Manifold and Reduced Form, J. Dynamics Differential Equations, 2 (1990), 69-115.
[23] Y. Morita and S. Jimbo, ODEs on Inertial Manifolds for Reaction Diffusion Systems in a Singularly Perturbed Domain with Several Thin Channels, J. Dynamics Differential Equations, 4 (1992), 65-98.
[24] S. Ozawa, Singular variations of domains and eigenvalues of the Laplacian, Duke Math. J., 48 (1981), 769-778.
[25] S. Ozawa, Electrostatic capacity and eigenvalues of the Laplacian, J. Fac. Sci. Univ. Tokyo, 30 (1983), 53-62.
[26] S. Ozawa, Spectra of domains with small spherical Neumann boundary, J. Fac. Sci. Univ. Tokyo, 30 (1983), 259-277.
[27] J. Rauch and M. Taylor, Potential and scattering theory on wildly perturbed domains, J. Funct. Anal., 18 (1975), 27-59.
[28] M. Taylor, Pseudo differential operators, Princeton University Press, 1981.
[29] J. M. Vegas, Bifurcation caused by perturbing the domain in an elliptic equation, J. Differential Equations, 48 (1983), 189-226.

Date of correction: October 20, 2006 Reason for correction: - Correction: PDF FILE Details: -

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