Biharmonic Steklov problems with Neumann boundary conditions and spectral inequalities on differential forms

TL;DR

This paper introduces new biharmonic Steklov problems on differential forms with Neumann boundary conditions, establishing their spectral properties, variational characterizations, and inequalities relating their eigenvalues to classical problems.

Contribution

It presents three novel biharmonic Steklov problems with Neumann boundary conditions on differential forms, proving their ellipticity, spectral discreteness, and deriving eigenvalue inequalities.

Findings

Proved ellipticity of the new problems

Established existence of discrete spectra

Derived eigenvalue inequalities relating to classical problems

Abstract

We introduce three biharmonic Steklov problems on differential forms with Neumann boundary conditions and show that they are elliptic. We prove the existence of a discrete spectrum for each of those problems and give associated variational characterizations for their eigenvalues. We establish eigenvalue estimates known as Kuttler-Sigillito inequalities, relating the eigenvalues of these problems with the eigenvalues of the Steklov, Dirichlet and Neumann standard problems as well as the biharmonic Steklov problem with Dirichlet boundary conditions on differential forms.