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

TL;DR

This paper proves that in convex planar domains with large eccentricity, the second eigenfunction of the Laplacian has exactly two nondegenerate critical points, using estimates and topological degree arguments.

Contribution

It establishes a precise count of critical points for the second eigenfunction in convex domains with large eccentricity, extending understanding of eigenfunction topology.

Findings

Second eigenfunction has exactly two critical points in elongated convex domains.

Critical points are one maximum and one minimum, nondegenerate.

Results build on estimates by Jerison and Grieser-Jerison, using topological degree.

Abstract

In this paper we consider the second eigenfunction of the Laplacian with Dirichlet boundary conditions in convex domains. If the domain has \emph{large eccentricity} then the eigenfunction has \emph{exactly} two nondegenerate critical points (of course they are one maximum and one minimum). The proof uses some estimates proved by Jerison ([Jer95a]) and Grieser-Jerison ([GJ96]) jointly with a topological degree argument. Analogous results for higher order eigenfunctions are proved in rectangular-like domains considered in [GJ09].