On the multiplicity of the second eigenvalue of the Laplacian in non simply connected domains – with some numerics –

Article type: Research Article

Authors: Helffer, B.^{a; b; *} | Hoffmann-Ostenhof, T.^c | Jauberteau, F.^a | Léna, C.^d

Affiliations: [a] Laboratoire de Mathématique Jean Leray, Université de Nantes, France | [b] Laboratoire de Mathématiques d’Orsay, Unversité Paris Sud, France | [c] Institut für Theoretische Chemie, University of Wien, Austria | [d] Department of Mathematics, University of Stockholm, Sweden

Correspondence: [*] Corresponding author. E-mail: Bernard.Helffer@univ-nantes.fr.

Abstract: We revisit an interesting example proposed by Maria Hoffmann-Ostenhof, the second author and Nikolai Nadirashvili of a bounded domain in R2 for which the second eigenvalue of the Dirichlet Laplacian has multiplicity 3. We also analyze carefully the first eigenvalues of the Laplacian in the case of the disk with two symmetric cracks placed on a smaller concentric disk in function of their size.

Keywords: Spectral theory, Laplacian, multiplicity

DOI: 10.3233/ASY-191594

Journal: Asymptotic Analysis, vol. 121, no. 1, pp. 35-57, 2021