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On Regular and Singular Perturbations of the Eigenelements of the Laplacian

In: Integral Methods in Science and Engineering, Volume 1

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  • R. R. Gadyl’shin

    (Bashkir State Pedagogical University)

Abstract

We discuss the concept of singularly perturbed eigenvalue problems for the Laplace operator. Typical problems are the boundary value problems for the eigenvalue equations in bounded domains and the possible perturbations are a small parameter at higher derivatives, small holes, thin slits, thin appendices, frequent alternation of boundary conditions, etc. The main feature of these problems is that there exists no change of variables reducing them to problems in a fixed domain with a regularly perturbed operator. At the same time, the eigenvalues of such singularly perturbed boundary value problems converge to those of certain limiting problems. This is why, in the sense of convergence, the eigenvalues behave in the regular way.

Suggested Citation

  • R. R. Gadyl’shin, 2010. "On Regular and Singular Perturbations of the Eigenelements of the Laplacian," Springer Books, in: Christian Constanda & M.E. Pérez (ed.), Integral Methods in Science and Engineering, Volume 1, chapter 14, pages 135-148, Springer.
    • Handle: RePEc:spr:sprchp:978-0-8176-4899-2_14
    DOI: 10.1007/978-0-8176-4899-2_14
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