Asymptotic property and convergence estimation for the eigenelements of the Laplace operator

References

• Adams , R . 1975 . Sobolev Spaces , New York : Academic Press .
   Google Scholar
• Ammari , H and Triki , F . 2004 . Splitting of resonant and scattering frequencies under shape deformation . Journal of Differential equations , 202 : 231 – 255 .
   Google Scholar
• Bramble , JH and Osborn , JE . 1973 . Rate of convergence estimates for nonselfadjoint eigenvalue approximations . Mathametics of Computation , 27 : 525 – 549 .
   Google Scholar
• Bruno , OP and Reitich , F . 1992 . Solution of a boundary value problem for the Helmholtz equation via variations of the boundary into the complex domain . Proceedings of Royal Society of Edinburgh Section A , 122 : 317 – 340 .
   Google Scholar
• Bruno , OP and Reitich , F . 2001 . Boundary-variation solution of eigenvalue problems for elliptic operators . Journal of Fourier Analysis and Applications , 7 : 169 – 187 .
   Google Scholar
• Colton , D and Kress , R . 1983 . Integral Equation Methods in Scattering Theory , New York : John Wiley .
   Google Scholar
• Cox , SJ . 1995 . The Generalized gradient at a multiple eigenvalue . Journal of Functional. Analysis , 133 : 30 – 40 .
   Google Scholar
• Gadyl'shin , RR and Il'in , AM . 1998 . Asymptotic behavior of the eigenvalues of the Dirichlet problem in a domain with a narrow crack . Sbornik Mathametics , 189 : 503 – 526 .
   Google Scholar
• Gohberg , IT and Sigal , EI . 1971 . Operator extension of the logarithmic residue theorem and Rouché's theorem . Mathematics of USSR Sbornik , 84 : 607 – 642 .
   Google Scholar
• Hitrik , M . 2003 . Eigenfrequencies and expansions for damped wave equations . J. Methods Appl. Anal. , 10 ( 4 ) : 543 – 564 .
   Google Scholar
• Kato , T . 1980 . Perturbation Theory for Linear Operators , 2nd , Berlin : Springer-Verlag .
   Google Scholar
• Khelifi , A . 2005 . Asymptotic property of eigenvalues and eigenfunctions of the Laplace operator in domain with a perturbed boundary . Fractional Calculus & Appllied Analysis , 8 ( 3 ) : 277 – 298 .
   Google Scholar
• Lamberti , PD and Lanza de Cristoforis , M . 2004 . A real analyticity result for symmetric functions of the eigenvalues of a domain dependent Dirichlet problem for the Laplace operator . Journal of Nonlinear Convex Analysis , 5 ( 1 ) : 19 – 42 .
   Google Scholar
• Lamberti , PD and Lanza de Cristoforis , M . 2002 . An analyticity result for the dependence of multiple eigenvalues and eigenspaces of the Laplace operator upon perturbation of the domain . Glasgow Mathematical Journal , 1 : 29 – 43 .
   Google Scholar
• Lupo , D and Micheletti , AM . 1993 . On multiple eigenvalues of selfadjoint compact operators . Journal of Mathematical Analysis and Applications , 172 : 106 – 116 .
   Google Scholar
• Moskow , S and Vogelius , M . 2006 . First order correction to the homogenized eigenvalues of a periodic composite medium. The case of Neumann boundary conditions . Indiana Univ. Math. J. , to appear
   Google Scholar
• Nédélec , JC . 2001 . Acoustic and Electromagnetic Equations, Integral Representation for Harmonic Problem , New York : Springer-Verlag .
   Google Scholar
• Neymeyr , K . 2002 . A posteriori error estimation for elliptic eigenproblems . Numarical Linear Algebra Applications , 9 : 263 – 279 .
   Google Scholar
• Osborn , JE . 1975 . Spectral approximations for compact operators . Mathamatics of Computation , 29 : 712 – 725 .
   Web of Science ®Google Scholar
• Ozawa , S . 1996 . Eigenvalues of the Laplacian under singular variation of domains-the Robin problem with obstacle of general shape . Japan Academy Proceedings. Series A. Mathematical Sciences , 72 : 124 – 134 .
   Google Scholar
• Ozawa , S . 1985 . Asymptotic property of eigenfunction of the Laplacian under singular variation of domains-the Neumann condition . Osaka Journal of Mathematics , 22 : 639 – 655 .
   Google Scholar
• Ozawa , S . 1983 . An asymptotic formula for the eigenvalues of the Laplacian in a three-dimensional domain with a small hole . Journal of Faculty of Science. The University of Tokyo. Section IA , IA 30 : 243 – 257 .
   Google Scholar
• Reed , M and Simon , B . 1978 . Methods of Modern methematical physics , Vol. 4 , New York : Academic Press .
   Google Scholar
• Rellich , F . 1969 . Perturbation Theory of Eigenvalue Problems , New York : Gordon and Breach Science Publishers .
   Google Scholar
• Taylor , ME . 1996 . “ Partial Differential Equations II, Qualitative Studies of Linear Equations ” . In Applied Mathematical Sciences , Vol. 116 , New York : Springer-Verlag .
   Google Scholar