Restricted validity of Bareket’s conjecture

Determine whether the ball maximizes the lowest Robin eigenvalue for every negative Robin parameter among simply connected bounded Lipschitz planar domains and among convex bounded domains in dimensions at least three.

Background

The unrestricted negative-Robin isochoric maximization conjecture is false because shells can outperform balls. Nevertheless, the paper reports numerical and theoretical support for ball optimality in more restrictive geometric classes. The unresolved question concerns simply connected planar domains and convex domains in higher dimensions.

References

Does Conjecture~\ref{Bareket} hold within the class of simply connected bounded Lipschitz domains if $d=2$ and convex bounded domains if $d \geq 2$ ?

— 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)

It remains open to show that the extra geometric hypotheses are superfluous. \begin{OProblem} Extend Theorem~\ref{Thm.perimeter} to all bounded Lipschitz domains $\Omega \subset R3$ without assuming the convexity. \end{OProblem}

— Spectral geometry: old questions and new answers  (2609.28602 - Krejcirik, 23 Sep 2026) in Open Problem following Theorem 5.5, Section 5.4 (page unavailable in source)