# Inequalities for eigenvalues of fourth order elliptic operators in divergence form on Riemannian manifolds

## TL;DR

This paper establishes a general inequality for eigenvalues of fourth order elliptic operators in divergence form on Riemannian manifolds, with applications to eigenvalues on submanifolds in Euclidean space.

## Contribution

It introduces a new inequality for eigenvalues of fourth order elliptic operators on Riemannian manifolds, extending previous results and applying to submanifold eigenvalue problems.

## Key findings

- Derived a general eigenvalue inequality for fourth order elliptic operators.
- Applied the inequality to eigenvalues on submanifolds in Euclidean space.
- Provided insights into spectral properties of elliptic operators on manifolds.

## Abstract

In this paper, we study eigenvalue of linear fourth order elliptic operators in divergence form with Dirichlet boundary condition on a bounded domain in a compact Riemannian manifolds with boundary (possibly empty) and find a general inequality for them. As an application, by using this inequality, we study eigenvalues of this operator on compact domains of complete submanifolds in a Euclidean space.