Neumann eigenvalues of the biharmonic operator on domains: geometric bounds and related results

TL;DR

This paper investigates Neumann eigenvalues of the biharmonic operator on Riemannian domains, providing geometric bounds and properties, with results compatible with Weyl's law under curvature constraints.

Contribution

It introduces new upper bounds for eigenvalues of the biharmonic operator with Neumann conditions on Riemannian domains, linking geometry and spectral properties.

Findings

Eigenvalues satisfy specific properties under Neumann boundary conditions.

Established upper bounds compatible with Weyl's law.

Results depend on lower bounds of Ricci curvature.

Abstract

We study an eigenvalue problem for the biharmonic operator with Neumann boundary conditions on domains of Riemannian manifolds. We discuss the weak formulation and the classical boundary conditions, and we describe a few properties of the eigenvalues. Moreover, we establish upper bounds compatible with the Weyl's law under a given lower bound on the Ricci curvature.