The eigenvalues of the Laplacian on domains with small slits

Luc Hillairet (1) , Chris Judge (2)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 Department of Mathematics, Indiana University

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 (arXiv:math/0703616), we obtain the following application: The generic multiply connected polygon has simple spectrum.

Domaines: Théorie spectrale [math.SP]; Equations aux dérivées partielles [math.AP]

Luc Hillairet, Chris Judge. The eigenvalues of the Laplacian on domains with small slits. Transactions of the American Mathematical Society, 2010, 362 (12), pp.6231-6259. ⟨10.1090/S0002-9947-2010-04943-8⟩. ⟨hal-00345667⟩