The eigenvalues of the Laplacian on domains with small slits

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 ear...

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Veröffentlicht in:	Transactions of the American Mathematical Society 2010-12, Vol.362 (12), p.6231-6259
Hauptverfasser:	Hillairet, Luc, Judge, Chris
Format:	Artikel
Sprache:	eng
Schlagworte:	Analysis of PDEs Boundary conditions Coordinate systems Eigenfunctions Eigenvalues Exact sciences and technology General mathematics General, history and biography Geometry Laplacians Mathematical theorems Mathematics Mathieu function Nonlinear algebraic and transcendental equations Numerical analysis Numerical analysis. Scientific computation Numerical linear algebra Polygons Research article Sciences and techniques of general use Spectral Theory Vertices
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Beschreibung
Zusammenfassung:	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.
ISSN:	0002-9947 1088-6850
DOI:	10.1090/S0002-9947-2010-04943-8