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Transactions of the American Mathematical Society

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The eigenvalues of the Laplacian on domains with small slits

Authors: Luc Hillairet and Chris Judge
Journal: Trans. Amer. Math. Soc. 362 (2010), 6231-6259
MSC (2010): Primary 58C40; Secondary 58J37, 35P20
Published electronically: August 3, 2010
MathSciNet review: 2678972
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Abstract | References | Similar Articles | Additional Information

Abstract: We introduce a small slit into a planar domain and study the resulting effect upon the eigenvalues of the Laplacian. In particular, we show that as the length of the slit tends to zero, each real-analytic eigenvalue branch tends to an eigenvalue of the original domain. By combining this with our earlier work (2009), we obtain the following application: The generic multiply connected polygon has a simple spectrum.

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Additional Information

Luc Hillairet
Affiliation: Laboratoire de Mathématiques Jean Leray, UMR CNRS 6629, Université de Nantes, 2 rue de la Houssinière, BP 92 208, F-44 322 Nantes Cedex 3, France

Chris Judge
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47401

Received by editor(s): March 3, 2008
Published electronically: August 3, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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