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Domain dependence of eigenvalues of elliptic type operators

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Abstract

The dependence on the domain for the Dirichlet eigenvalues of elliptic operators considered in bounded domains is studied. The proximity of domains is measured by a norm of the difference of two orthogonal projectors corresponding to the reference domain and the perturbed one; this allows to compare eigenvalues corresponding to domains that have non-smooth boundaries and different topology. The main result is an asymptotic formula in which the remainder is evaluated in terms of this quantity. Applications of this result are given. The results are new for the Laplace operator.

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Notes

  1. Two terms are lost in formula (5), [12], compare with (9) and (10). However, if \(\Omega _2\subset \Omega _1\), then formula (5), [12], is true.

  2. We note that the spectral problem (17) and (4), and also (18) and (5) have the same eigenvalues and eigenvectors.

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  1. Department of Mathematics, Linköping University, Linköping, 581 83, Sweden

    Vladimir Kozlov

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  1. Vladimir Kozlov
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Correspondence to Vladimir Kozlov.

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The author was supported by the Swedish Research Council (VR).

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Kozlov, V. Domain dependence of eigenvalues of elliptic type operators. Math. Ann. 357, 1509–1539 (2013). https://doi.org/10.1007/s00208-013-0947-9

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  • DOI: https://doi.org/10.1007/s00208-013-0947-9

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