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The Steklov Problem on Differential Forms

Part of: Partial differential equations on manifolds; differential operators Spectral theory and eigenvalue problems

Published online by Cambridge University Press:  07 January 2019

Mikhail A. Karpukhin
Mikhail A. Karpukhin*
Affiliation:
Department of Statistics and Mathematics, McGill University, Montreal QC H4A3J3, Canada Email: [email protected]
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Abstract

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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.

Keywords

Dirichlet-to-Neumann mapdifferential formSteklov eigenvalueshape optimization

MSC classification

Primary: 58J50: Spectral problems; spectral geometry; scattering theory
Secondary: 58J32: Boundary value problems on manifolds 35P15: Estimation of eigenvalues, upper and lower bounds
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 2 , April 2019 , pp. 417 - 435
Copyright
© Canadian Mathematical Society 2018 

References

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