Quantitative Kr\"{o}ger inequalities for Neumann eigenvalues of convex domains

TL;DR

This paper refines upper bounds for Neumann eigenvalues of convex domains, establishing inequalities involving domain geometry and providing explicit constants in the planar case.

Contribution

It introduces new inequalities that improve existing bounds on Neumann eigenvalues by incorporating geometric parameters of convex domains.

Findings

Established inequalities relating eigenvalues to domain geometry.

Provided explicit constants for the planar case.

Refined previous bounds by Kröger (1999).

Abstract

Refining the sharp upper bounds $\mu_{k,d}^* $ obtained by Kr\"oger (1999) for the $k$-th Neumann eigenvalue of a convex domain $\Omega \subset \mathbb{R}^d$, we prove the following inequalities: for any $k\in \mathbb{N}$ there exists a constant $C(k,d) >0$ such that $$D_{\Omega}^2 \mu_k(\Omega) \leq \mu_{k,d}^* - C(k,d) a_2(\Omega)^2/D_{\Omega}^2$$ where $D_{\Omega}$ is the diameter of $\Omega$ and $a_2(\Omega)$ is the second largest semiaxis of the John ellipsoid of $\Omega$. In the planar case, for $k=1$ we also give an explicit value of the constant $C(1,2)$.