On the inequality ${\mu }_3<{\lambda }_1$ for the eigenvalues of the Laplace operator

Abstract

In 1952, Pólya proved the inequality ${\mu }_2<{\lambda }_1$ for the eigenvalues of Dirichlet and Neumann problems in domain $\Omega \subset {\mathbb{R}}^d$, $d\ge 2$. In 2026, Tang and Zhang proved the inequality ${\mu }_3<{\lambda }_1$ in the two-dimensional case. We show that ${\mu }_3<{\lambda }_1$ for $d\ge 3$ as well.

arXiv commentTheorem 1.2 from the first version is removed. Reference [16] is added