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Search: MSC category 58J32 ( Boundary value problems on manifolds )

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1. CJM Online first

Karpukhin, Mikhail A.
Steklov problem on differential forms
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 $\Lambda$ 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 $\Lambda$ 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 $\Lambda$ 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 a lot of important properties with the classical Steklov eigenvalue problem on surfaces.

Keywords:Dirichlet-to-Neumann map, differential form, Steklov eigenvalue, shape optimization
Categories:58J50, 58J32, 35P15

2. CJM 2005 (vol 57 pp. 225)

Booss-Bavnbek, Bernhelm; Lesch, Matthias; Phillips, John
Unbounded Fredholm Operators and Spectral Flow
We study the gap (= ``projection norm'' = ``graph distance'') topology of the space of all (not necessarily bounded) self-adjoint Fredholm operators in a separable Hilbert space by the Cayley transform and direct methods. In particular, we show the surprising result that this space is connected in contrast to the bounded case. Moreover, we present a rigorous definition of spectral flow of a path of such operators (actually alternative but mutually equivalent definitions) and prove the homotopy invariance. As an example, we discuss operator curves on manifolds with boundary.

Categories:58J30, 47A53, 19K56, 58J32

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