Some remarks on the isoperimetric problem for the higher eigenvalues of the Robin and Wentzell Laplacians

TL;DR

This paper investigates the minimization of higher eigenvalues of the Robin and Wentzell Laplacians under volume constraints, revealing optimal domain shapes and dependencies on boundary parameters for various eigenvalues.

Contribution

It proves the second eigenvalue is minimized by two equal-volume disjoint balls and explores how minimizers depend on boundary conditions for higher eigenvalues.

Findings

Second eigenvalue minimized by two equal-volume disjoint balls

Minimizers for higher eigenvalues depend on boundary parameters

Results extend to Wentzell boundary conditions

Abstract

We consider the problem of minimising the $k$th eigenvalue, $k \geq 2$, of the ($p$-)Laplacian with Robin boundary conditions with respect to all domains in $\mathbb{R}^N$ of given volume $M$. When $k=2$, we prove that the second eigenvalue of the $p$-Laplacian is minimised by the domain consisting of the disjoint union of two balls of equal volume, and that this is the unique domain with this property. For $p=2$ and $k \geq 3$, we prove that in many cases a minimiser cannot be independent of the value of the constant $\alpha$ in the boundary condition, or equivalently of the volume $M$. We obtain similar results for the Laplacian with generalised Wentzell boundary conditions $\Delta u + \beta \frac{\partial u}{\partial \nu} + \gamma u = 0$.