The weak Pleijel theorem with geometric control

Let $\mathrm{\Omega }\subset {\mathbb{R}}^d,d\ge 2$, be a bounded open set, and denote by $\lambda \mathrm{\_}j(\mathrm{\Omega }),j\ge 1$, 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 $\lambda \mathrm{\_}j(\mathrm{\Omega })$, 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 invariants of $\mathrm{\Omega }$. We will see that this is connected with one of the favorite problems considered by Y. Safarov.