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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
Email: Luc.Hillairet@math.univ-nantes.fr

Chris Judge
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47401
Email: cjudge@indiana.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-04943-8
PII: S 0002-9947(2010)04943-8
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.