Geometric eigenvalue estimates of Kuttler-Sigillito type on differential forms

TL;DR

This paper introduces a new biharmonic Steklov problem for differential forms, proving its ellipticity, spectrum discreteness, and deriving eigenvalue estimates linked to manifold curvature.

Contribution

It presents a novel biharmonic Steklov problem on differential forms with boundary conditions, establishing its spectral properties and geometric eigenvalue bounds.

Findings

The problem is elliptic and has a discrete spectrum.

Eigenvalue estimates relate to curvature quantities.

Variational characterizations of eigenvalues are provided.

Abstract

We introduce a new biharmonic Steklov problem on differential forms with Dirichlet-type boundary conditions and show that it is elliptic. We prove the existence of a discrete spectrum for this problem and give variational characterizations for eigenvalues associated to it. We establish eigenvalue estimates known as Kuttler-Sigillito inequalities, that connect the eigenvalues of different problems on differential forms with curvature quantities on the manifold.