The weak Pleijel theorem with geometric control

Pierre Bérard, Bernard Helffer
2016 Journal of Spectral Theory  
Let Ω ⊂ R d , d ≥ 2, be a bounded open set, and denote by λj (Ω), j ≥ 1, the eigenvalues of the Dirichlet Laplacian arranged in nondecreasing order, with multiplicities. The weak form of Pleijel's theorem states that the number of eigenvalues λj (Ω), for which there exists an associated eigenfunction with precisely j nodal domains (Courant-sharp eigenvalues), is finite. The purpose of this note is to determine an upper bound for Courant-sharp eigenvalues, expressed in terms of simple geometric
more » ... nvariants of Ω. We will see that this is connected with one of the favorite problems considered by Y. Safarov.
doi:10.4171/jst/138 fatcat:kqvvhgygenc53ksnke7icptbu4