Estimates for Resonant Frequencies Under Boundary Deformation in Multi-dimensional Space PDF

[1] R. Adams, Sobolev Spaces, Academic Press, New York, 1975 .
[2] H. Ammari, and H. Kang, Reconstruction of small inhomogeneities from boundary measurements,
Lecture Notes in Mathematics, Vol.1846, Springer-Verlag, Berlin (2004).
[3] J.H. Bramble and J.E. Osborn, Rate of convergence estimates for nonselfadjoint eigenvalue
approximations , Math. Comp. Vol. 27 (1973), 525-549.
[4] O. P. Bruno and F. Reitich, Boundary-variation solution of eigenvalue problems for elliptic
operators, J. Fourier Anal. Appl. 7 (2001), 169-187.
[5] D. Colton and R. Kress, Integral equation methods in Scattering theory, New York: John
Wiley (1983).
[6] C. Conca, M. Duran and J. Rappaz, Rate of convergence estimates for the spectral approximation
of a generalized eigenvalue problem , Numer. Math. 79, no3, (1998), 349-369.
[7] S. J. Cox,The Generalized gradient at a multiple eigenvalue, J. Func. Anal, 133, (1995),
30-40.
[8] R. R. Gadyl’shin and A. M. Il’in, Asymptotic behavior of the eigenvalues of the Dirichlet
problem in a domain with a narrow Crack, Sbornik Math. 189 (1998), 503-526.
[9] T. Kato, Perturbation theory for linear operators, 2end edition, Springer-Verlag, Berlin
(1980).
[10] A. Khelifi, Diffraction d’ondes électromagnétiques par des inhomogénéités diélectriques,
PhD. Thesis, Applied Mathematics, Ecole Polytechnique - France, February (2002).
[11] A. Khelifi, The Integral Equation Methods for the Perturbed Helmholtz Eigenvalue Problems,
Int. J. Math. Math. Sc. 80,(2005), 1201-1220.
[12] P.D. Lamberti, andM. Lanza de Cristoforis, A real analyticity result for symmetric functions
of the eigenvalues of a domain dependent Dirichlet problem for the Laplace operator . J.
Nonlin. Con. Anal. Vol 5, no 1,(2004), 19-42.
[13] P.D. Lamberti, and M. Lanza de Cristoforis, An analyticity result for the dependence of multiple
eigenvalues and eigenspaces of the Laplace operator upon perturbation of the domain
. Glasgow Math J. no 1,(2002), 29-43.
[14] D. Lupo, and A. M. Micheletti, On multiple eigenvalues of selfadjoint compact operators,
J. Math. Anal. Appl. 172, (1993), 106-116.
[15] S. Moskow and M. Vogelius, First Order correction to the homogenized eigenvalues of a
periodic composite medium. The case of Neumann boundary conditions, to appear in J.
Indiana. Univ., (2006).
[16] J. C. Nédélec, Acoustic and Electromagnetic equations. Integral representation for Harmonic
problem, Springer-Verlag, New York, 2001.
[17] K. Neymeyr, A posteriori error estimation for elliptic eigenproblems, Num. Lin. Alg. Appl.
9 (2002), 263-279.
[18] J.E. Osborn, Spectral Approximations for Compact Operators, Math. Comp., Vol 29
(1975), 712-725.
[19] S. Ozawa, Eigenvalues of the Laplacian under singular variation of domains-the Robin
problem with obstacle of general shape, Proc.Japan Acad. Ser.A Math.Sci.72 (1996), 124-
134.
[20] S. Ozawa, Asymptotic property of eigenfunction of the Laplacian under singular variation
of domains-the Neumann condition, Osaka J.Math.22(1985), 639-655.
[21] S. Ozawa, An asymptotic formula for the eigenvalues of the Laplacian in a threedimensional
domain with a small hole, J. Fac. Sci. Univ. Tokyo Sect. IA 30 (1983), 243-
257.
[22] F. Rellich, Perturbation theory of eigenvalue problems, Gordon and Breach Science Publishers,
New York (1969).
[23] M. E. Taylor, Partial differential equations II, Qualitative studies of Linear equations, Applied
Mathematical Sciences 116, Springer- Verlag, (1996).