Karpukhin, Mikhail A; (2019) The Steklov Problem on Differential Forms. Canadian Journal of Mathematics , 71 (2) pp. 417-435. 10.4153/CJM-2018-028-6.

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Abstract

In this paper we study spectral properties of the Dirichlet-to-Neumann map on differential forms obtained by a slight modification of the definition due to Belishev and Sharafutdinov. The resulting operator$\unicode[STIX]{x039B}$is shown to be self-adjoint on the subspace of coclosed forms and to have purely discrete spectrum there. We investigate properties of eigenvalues of$\unicode[STIX]{x039B}$and prove a Hersch–Payne–Schiffer type inequality relating products of those eigenvalues to eigenvalues of the Hodge Laplacian on the boundary. Moreover, non-trivial eigenvalues of$\unicode[STIX]{x039B}$are always at least as large as eigenvalues of the Dirichlet-to-Neumann map defined by Raulot and Savo. Finally, we remark that a particular case of$p$-forms on the boundary of a$2p+2$-dimensional manifold shares many important properties with the classical Steklov eigenvalue problem on surfaces.

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