Quantitative Kröger inequalities for Neumann eigenvalues of convex domains

Abstract

Refining the sharp upper bounds ${\mu }_{k,d}^{\ast }$ obtained by Kröger (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 $$D_{\Omega }^2{\mu }_k(\Omega )\le {\mu }_{k,d}^{\ast }-C(k,d)a_2(\Omega {)}^2/D_{\Omega }^2$$ 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 $C(1,2)$.