# Eigenvalues of elliptic operators with density

## TL;DR

This paper investigates how the eigenvalues of high-order elliptic operators with Neumann boundary conditions depend on the mass density, establishing bounds under fixed total mass or density norm, with results influenced by operator order and space dimension.

## Contribution

It provides new insights into eigenvalue bounds for elliptic operators considering density variations, emphasizing the role of operator order and dimension in these bounds.

## Key findings

- Existence of eigenvalue bounds depends on fixed mass or density norm.
- The order of the operator and space dimension critically influence bounds.
- Characterization of upper and lower eigenvalue bounds under density constraints.

## Abstract

We consider eigenvalue problems for elliptic operators of arbitrary order $2m$ subject to Neumann boundary conditions on bounded domains of the Euclidean $N$-dimensional space. We study the dependence of the eigenvalues upon variations of mass density and in particular we discuss the existence and characterization of upper and lower bounds under both the condition that the total mass is fixed and the condition that the $L^{\frac{N}{2m}}$-norm of the density is fixed. We highlight that the interplay between the order of the operator and the space dimension plays a crucial role in the existence of eigenvalue bounds.