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THE EIGENVALUES OF THE LAPLACIAN ON DOMAINS WITH SMALL SLITS

LUC HILLAIRET and CHRIS JUDGE
Transactions of the American Mathematical Society
Vol. 362, No. 12 (DECEMBER 2010), pp. 6231-6259
Stable URL: http://www.jstor.org/stable/40997202
Page Count: 29
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THE EIGENVALUES OF THE LAPLACIAN ON DOMAINS WITH SMALL SLITS
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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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