Steklov problem on differential forms

2017/05/24 by Karpukhin, Mikhail · 2 citations
#Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1705.08951

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 Λ 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 Λ 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 Λ 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.

Cited by

• A weighted Reilly type integral formula for differential forms and its applications
• Biharmonic Steklov problems with Neumann boundary conditions and spectral inequalities on differential forms

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