Estimates on eigenvalues for the biharmonic operator on a bounded domain in H^n(-1)

TL;DR

This paper derives universal bounds for eigenvalues of the biharmonic operator on bounded domains in hyperbolic space, providing insights into spectral properties independent of specific domain shapes.

Contribution

It introduces domain-independent bounds for the eigenvalues of the biharmonic operator in hyperbolic spaces, advancing spectral analysis in non-Euclidean geometries.

Findings

Universal bounds on the (k+1)th eigenvalue in terms of the first k eigenvalues

Eigenvalue estimates are independent of the specific domain shape

Results extend spectral theory to hyperbolic space contexts

Abstract

In this paper, we consider eigenvalues of the Dirichlet biharmonic operator on a bounded domain in a hyperbolic space. We obtain universal bounds on the $(k+1)$th eigenvalue in terms of the first $k$th eigenvalue independent of the domains.