5 Spectral geometry of the Steklov problem

Abstract

The Steklov problem is an eigenvalue problem with the spectral parameter in the boundary conditions, which has various applications. Its spectrum coincides with that of the Dirichlet-to-Neumann operator. Over the past years, there has been a growing interest in the Steklov problem from the viewpoint of spectral geometry. While this problem shares some common properties with its more familiar Dirichlet and Neumann cousins, its eigenvalues and eigenfunctions have a number of distinctive geometric features, which makes the subject especially appealing. In this survey we discuss some recent advances and open questions, particularly in the study of spectral asymptotics, spectral invariants, eigenvalue estimates, and nodal geometry.2010 Mathematics Subject Classification. 58J50, 35P15, 35J25. Key words and phrases. Steklov eigenvalue problem, Dirichlet-to-Neumann operator, Riemannian manifold.

Boundary Integrals and Approximations of Harmonic Functions

“…The results hold for arbitrary rectangles in the plane upon scaling and possibly a rotation. In their recent survey, Girouard and Polterovich [11] outline similar results for the case of a square.…”

“…A proof that these eigenfunctions provide a basis for the set of all Steklov eigenfunctions for the case of a square is outlined in Section 3.1 of [11]. The argument provided there generalizes to the case of a rectangle in a straightforward manner.…”

“…Also, the boundary is Lipschitz but not C 1 within this article, so the very general methods used in [12] are not applicable. The recent review article [11] discusses a number of features of Steklov eigenproblems that change significantly when the boundary is no longer smooth. For our analysis, we also require some mild regularity conditions on and .…”

Gluing Formula for Casimir Energies

“…It is hoped that using the vast amount of literature on the Dirichlet-to-Neumann map, (see, e.g., [43][44][45][46][47][48][49]), new insights can be gained on what the decisive topological and geometrical factors are that lead to attractive or repulsive Casimir forces.…”

Steklov approximations of harmonic boundary value problems on planar regions

“…When the region Ω is a rectangle, the Steklov eigenfunctions are known explicitly see Auchmuty and Cho [6] or Girouard and Polterovich [14] where a completeness proof may be found. The paper [6] described the generalization of the mean value theorem to rectangles and to cases where Robin data on ∂Ω is known.…”