Biharmonic Steklov operator on differential forms

TL;DR

This paper introduces a new biharmonic Steklov problem on differential forms, characterizes its spectrum, and establishes estimates and inequalities relating its eigenvalues to classical boundary value problems.

Contribution

It defines the biharmonic Steklov problem on differential forms and derives spectral properties and eigenvalue estimates, connecting them to existing boundary problems.

Findings

Characterization of the smallest eigenvalue.

Elementary properties of the spectrum.

New inequalities relating eigenvalues of different boundary problems.

Abstract

We introduce the biharmonic Steklov problem on differential forms by considering suitable boundary conditions. We characterize its smallest eigenvalue and prove elementary properties of the spectrum. We obtain various estimates for the first eigenvalue, some of which involve eigenvalues of other problems such as the Dirichlet, Neumann, Robin and Steklov ones. Independently, new inequalities relating the eigenvalues of the latter problems are proved.