Title: Steklov problem on differential forms,
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.