Spectral geometry of the Steklov problem (Survey article)

  • Alexandre Girouard

    Université Laval, Québec, Canada
  • Iosif Polterovich

    Université de Montréal, Canada

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.

Cite this article

Alexandre Girouard, Iosif Polterovich, Spectral geometry of the Steklov problem (Survey article). J. Spectr. Theory 7 (2017), no. 2, pp. 321–359

DOI 10.4171/JST/164