On the number of critical points of the second eigenfunction of the Laplacian in convex planar domains

In this paper we consider the second eigenfunction of the Laplacian with Dirichlet boundary conditions in convex domains. If the domain has large eccentricity then the eigenfunction has exactly two nondegenerate critical points (of course they are one maximum and one minimum). The proof uses some estimates proved by Jerison ([13]) and GrieserJerison ([10]) jointly with a topological degree argument. Analogous results for higher order eigenfunctions are proved in rectangular-like domains considered in [11].(c) 2022 Elsevier Inc. All rights reserved.