Spherical shells as maximizers for negative Robin eigenvalues

Establish whether, for every negative Robin parameter, a spherical shell is a maximizer of the lowest Robin eigenvalue among bounded Lipschitz domains of fixed volume, with the shell radii chosen to match the prescribed volume.

Background

Bareket’s conjecture that balls maximize the lowest Robin eigenvalue for negative boundary parameter is disproved in the paper for sufficiently large absolute values of the parameter: spherical shells outperform volume-equivalent balls. The authors state that it remains unknown whether spherical shells are the actual maximizers over all bounded Lipschitz domains. The proposed problem retains the fixed-volume constraint and asks for the precise extremal geometry.

References

It remains open to show that spherical shells are the maximisers. \begin{OProblem} For every $\alpha < 0$, there exist positive numbers $R_1<R_2$ such that \begin{equation*} \max_{|\Omega|=const} \lambda_1\alpha(\Omega) = \lambda_1\alpha(A_{R_1,R_2}) \,, \end{equation*} where the maximum is taken over all bounded Lipschitz domains $\Omega \subset Rd$ of a fixed volume $|\Omega|=const$ and $A_{R_1,R_2}$ denotes a spherical shell of the same volume as~$\Omega$ (i.e. $|A_{R_1,R_2}|=|\Omega| = const$). \end{OProblem}

— Spectral geometry: old questions and new answers  (2609.28602 - Krejcirik, 23 Sep 2026) in Open Problem following Theorem 5.4, Section 5.4 (page unavailable in source)