SPECTRAL GEOMETRY OF THE STEKLOV PROBLEM

By A. Girouard, Iosif Polterovich. 2014

315 citations recorded by the source corpus.

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 eigenval- ues 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.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f5396e5a1b0cb70f022d57d8d7a59eb19c4c7526

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In Computational Theory and Mathematics

1. Kolesnikov Alexander
2. Milman Emanuel
3. A. V. Kolesnikov
4. Peter J. McGrath
5. Emanuel Milman

In Mathematical Physics

1. Emanuel Milman
2. Kolesnikov Alexander
3. Milman Emanuel
4. A. V. Kolesnikov
5. Peter J. McGrath

Citations in the corpus

• Local $L^p$-Brunn-Minkowski inequalities for $p<1$ (2017, background): “It is related to the classical Steklov (or Dirichlet-to-Neumann) 1st order elliptic pseudo-differential operator S [2, 24].”
• A Characterization of the Critical Catenoid (2016, background): “, Z ∂Σ u = 0, and u is piecewise C1}. If u ∈ C achieves equality in the Rayleigh quotient (2), then u is an eigenfunction with eigenvalue σ1. Hereafter, we refer to such u as first eigenfunctions. See [9] for more information about the Steklov eigenvalue problem. Definition 2.1. Suppose u is a Steklov”
• A few shape optimization results for a biharmonic Steklov problem (2015, unknown): “We refer to [30] for related problems, and to [20] for a recent survey on the subject.”
• Local Lp-Brunn-Minkowski inequalities for p<1 (2017, methodology): “x,ν∂K(x)i dx ≤ BH(K) Z K ∇2u 2 dx, (7.1) which we will use for calculating BH(Bn 2 ). It is related to the classical Steklov (or Dirichlet-to-Neumann) 1st order elliptic pseudo-differential operator S [2, 27]. 7.1 Second Steklov operator Let us assume for simplicity that ∂K is C∞ smooth, and denote b”