# Steklov problem on differential forms

## TL;DR

This paper investigates the spectral properties of a modified Dirichlet-to-Neumann operator on differential forms, establishing its self-adjointness, discrete spectrum, and inequalities relating eigenvalues to boundary Laplacian eigenvalues, with connections to classical Steklov problems.

## Contribution

It introduces a modified Steklov problem for differential forms, analyzes its spectral properties, and establishes new inequalities linking eigenvalues to boundary Laplacian spectra.

## Key findings

- Operator is self-adjoint on coclosed forms
- Eigenvalues satisfy a Hersch-Payne-Schiffer type inequality
- Eigenvalues are bounded below by those of a related Dirichlet-to-Neumann map

## Abstract

In this paper we study spectral properties of Dirichlet-to-Neumann map on differential forms obtained by a slight modification of the definition due to Belishev and Sharafutdinov. The resulting operator $\Lambda$ is shown to be self-adjoint on the subspace of coclosed forms and to have purely discrete spectrum there.We investigate properies of eigenvalues of $\Lambda$ and prove a Hersch-Payne-Schiffer type inequality relating products of those eigenvalues to eigenvalues of Hodge Laplacian on the boundary. Moreover, non-trivial eigenvalues of $\Lambda$ are always at least as large as eigenvalues of Dirichlet-to-Neumann map defined by Raulot and Savo. Finally, we remark that a particular case of $p$-forms on the boundary of $2p+2$-dimensional manifold shares a lot of important properties with the classical Steklov eigenvalue problem on surfaces.