Nodal counts for the Robin problem on Lipschitz domains

TL;DR

This paper extends Pleijel's theorem to Robin eigenvalues on Lipschitz domains, showing only finitely many are Courant-sharp and providing explicit bounds based on geometric properties.

Contribution

It proves a generalized Pleijel's theorem for Robin Laplacian eigenvalues on Lipschitz domains and offers explicit bounds for convex sets with smooth boundaries.

Findings

Finiteness of Courant-sharp Robin eigenvalues on Lipschitz domains

An improved Pleijel's theorem for Robin eigenvalues

Explicit upper bounds based on geometric quantities

Abstract

We consider the Courant-sharp eigenvalues of the Robin Laplacian for bounded, connected, open sets in $\mathbb{R}^n$, $n \geq 2$, with Lipschitz boundary. We prove Pleijel's theorem which implies that there are only finitely many Courant-sharp eigenvalues in this setting as well as an improved version of Pleijel's theorem, extending previously known results that required more regularity of the boundary. In addition, we obtain an upper bound for the number of Courant-sharp Robin eigenvalues of a bounded, connected, convex, open set in $\mathbb{R}^n$ with $C^2$ boundary that is explicit in terms of the geometric quantities of the set and the norm sup of the negative part of the Robin parameter.