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arXiv · 2107.01989

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

Fabio De Regibus·Massimo Grossi
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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].

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Fabio De Regibus, Massimo Grossi. 2021-07-05. On the number of critical points of the second eigenfunction of the Laplacian in convex planar domains. https://arxiv.org/abs/2107.01989

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