On the spectrum of the Dirichlet-to-Neumann operator acting on forms of a Euclidean domain

TL;DR

This paper computes the full spectrum of the Dirichlet-to-Neumann operator on differential forms for the Euclidean ball and establishes a new eigenvalue bound based on the domain's isoperimetric ratio.

Contribution

It provides the complete spectral characterization for the Euclidean ball and introduces a novel eigenvalue estimate for general domains using geometric measures.

Findings

Spectrum of the Dirichlet-to-Neumann operator on the Euclidean ball is fully determined.

A new upper bound for the first eigenvalue in terms of the isoperimetric ratio.

The eigenvalue bound links spectral properties to geometric domain features.

Abstract

We compute the whole spectrum of the Dirichlet-to-Neumann operator acting on differential p-forms on the unit Euclidean ball. Then, we prove a new upper bound for its first eigenvalue on a domain $\Omega$ in Euclidean space in terms of the isoperimetric ratio ${\rm Vol}(\bd\Omega)/{\rm Vol}(\Omega)$.