Nodal counts for the Robin problem on Lipschitz domains

TL;DR

This work studies nodal counts for the Robin Laplacian on bounded Lipschitz domains, proving Pleijel's theorem for Robin eigenvalues with an improved version that leverages a quantitative Faber–Krahn inequality. It develops a Neumann-Rayleigh bound on Robin nodal domains via outward-pointing vector fields and a DFV-style nodal-domain analysis, enabling sharp asymptotics and geometric refinements. A key contribution is an explicit geometric upper bound for the number of Courant-sharp Robin eigenvalues on convex $C^2$ domains, expressed in terms of volume, isoperimetric ratio, curvature radii, and the Robin parameter $H$, including the possibility of negative $h$. The results generalize prior Dirichlet/Neumann findings to the Robin setting with minimal boundary regularity assumptions and provide practical bounds via 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.

Theorems & Definitions (41)

• Definition 2.1
• Proposition 2.2
• proof
• Remark 2.3
• Proposition 2.4
• proof
• Definition 2.5
• Remark 2.6
• Example 2.7
• Theorem 3.1