Some inequalities and asymptotic formulas for eigenvalues on Riemannian manifolds

TL;DR

This paper derives sharp inequalities and asymptotic formulas for eigenvalues on Riemannian manifolds, addressing classical problems and providing a negative answer to the Payne conjecture in one dimension.

Contribution

It introduces new sharp inequalities for eigenvalues and asymptotic formulas for buckling and clamped plate problems on Riemannian manifolds, and resolves the Payne conjecture in one dimension.

Findings

Established sharp inequalities for four classical eigenvalues.

Derived asymptotic formulas for buckling and clamped plate eigenvalues.

Provided a negative answer to the Payne conjecture in one dimension.

Abstract

In this paper, we establish sharp inequalities for four kinds of classical eigenvalues on a bounded domain of a Riemannian manifold. We also establish asymptotic formulas for the eigenvalues of the buckling and clamped plate problems. In addition, we give a negative answer to the Payne conjecture for the one-dimensional case.