Optimal bounds for Neumann eigenvalues in terms of the diameter (Q6618828)

The authors prove optimal upper bounds for the Neumann eigenvalues of the Laplace operator in an open, bounded and connected Lipschitz set \(\Omega\subset\mathbb R^d\). The bounds are of the form\N\[\ND(\Omega)^2\mu_k(\Omega)\leq C_k,\N\]\Nwhere \(D(\Omega)\) is the diameter of \(\Omega\) and \(C_k\) is a constant that depends on the label \(k\) of the eigenvalue. Furthermore, the bounds hold for an admissible class of domains that contains convex domains.\N\NInterestingly, the bounds are obtained via a clever link with \textit{relaxed} eigenvalues of a weighted Sturm-Liouville operator, by choosing the weight function as the profile function of \(\Omega\). One technical difficulty is that the \textit{relaxed} eigenvalues are defined as the terms of a min-max sequence, and in general they do not have corresponding eigenfunctions, unless the domain \(\Omega\) is convex.\N\NIf \(\Omega\) is convex, the authors recover the results in [\textit{P. Kröger}, Proc. Am. Math. Soc. 127, No. 6, 1665--1669 (1999; Zbl 0911.35079)], with a new proof.