Abstract
This chapter is concerned with the behavior of the eigenvalues and eigenfunctions of the Laplace operator in bounded domains when the domain undergoes a perturbation. It is well known that if the boundary condition that we are imposing is of Dirichlet type, the kind of perturbations that we may allow in order to obtain the continuity of the spectra is much broader than in the case of a Neumann boundary condition. This is explicitly stated in the pioneer work of Courant and Hilbert [CoHi53], and it has been subsequently clarified in many works, see [BaVy65, Ar97, Da03] and the references therein among others. See also [HeA06] for a general text on different properties of eigenvalues and [HeD05] for a study on the behavior of eigenvalues and in general partial differential equations when the domain is perturbed.
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References
Arrieta, J.M.: Neumann eigenvalue problems on exterior perturbations of the domain. J. Differential Equations, 118, 54–103 (1995).
Arrieta, J.M.: Domain dependence of elliptic operators in divergence form. Resenhas IME-USP, 3, 107–123 (1997).
Arrieta, J.M., Carvalho, A.N.: Spectral convergence and nonlinear dynamics of reaction diffusion equations under perturbations of the domain. J. Differential Equations, 199, 143–178 (2004).
Babuška, I., Výborný, R.: Continuous dependence of eigenvalues on the domains. Czech. Math. J., 15, 169–178 (1965).
Courant, R., Hilbert, D.: Methods of Mathematical Physics, Vol. 1, Wiley-Interscience, New York (1953).
Daners, D.: Dirichlet problems on varying domains. J. Differential Equations, 188, 591–624 (2003).
Grieser, D.: Thin tubes in mathematical physics, global analysis and spectral geometry, in Analysis on Graphs and Its Applications, American Mathematical Society, Providence, RI, 565–594 (2008).
Henrot, A.: Extremum Problems for Eigenvalues of Elliptic Operators, Birkhäuser, Basel (2006).
Henry, D.: Perturbation of the Boundary in Partial Differential Equations, Cambridge University Press, Cambridge (2005).
Kovařík, H., Krejčiřík, D.: A Hardy inequality in a twisted Dirichlet-Neumann waveguide. Math. Nachr., 281, 1159–1168 (2008).
Krejčiřík, D.: Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions. ESAIM Control Optim. Calc. Var., http://dx.doi.org/10.1051/cocv:2008035.
Raugel, G.: Dynamics of partial differential equations on thin domain, in Lecture Notes in Math., 1609, Springer, Berlin-Heidelberg-New York, 208–315 (1995).
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Arrieta, J.M., Krejĉiřík, D. (2010). Geometric Versus Spectral Convergence for the Neumann Laplacian under Exterior Perturbations of the Domain. In: Constanda, C., Pérez, M. (eds) Integral Methods in Science and Engineering, Volume 1. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4899-2_2
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DOI: https://doi.org/10.1007/978-0-8176-4899-2_2
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