Higher order Cheeger inequalities for Steklov eigenvalues

Asma Hassannezhad, Laurent Miclo

Research output: Contribution to journalArticle (Academic Journal)

3 Downloads (Pure)

Abstract

We prove a lower bound for the k-th Steklov eigenvalues in terms of an isoperimetric constant called the k-th Cheeger-Steklov constant in three different situations: finite spaces, measurable spaces, and Riemannian manifolds. These lower bounds can be considered as higher order Cheeger type inequalities for the Steklov eigenvalues. In particular it extends the Cheeger type inequality for the first nonzero Steklov eigenvalue previously studied by Escobar in 1997 and by Jammes in 2015 to higher order Steklov eigenvalues. The technique we develop to get this lower bound is based on considering a family of accelerated Markov operators in the finite and mesurable situations and of mass concentration deformations of the Laplace-Beltrami operator in the manifold setting which converges uniformly to the Steklov operator. As an intermediary step in the proof of the higher order Cheeger type inequality, we define the Dirichlet-Steklov connectivity spectrum and show that the Dirichlet connectivity spectra of this family of operators converges to (or bounded by) the Dirichlet-Steklov spectrum uniformly. Moreover, we obtain bounds for the Steklov eigenvalues in terms of its Dirichlet-Steklov connectivity spectrum which is interesting in its own right and is more robust than the higher order Cheeger type inequalities. The Dirichlet-Steklov spectrum is closely related to the Cheeger-Steklov constants.
Original languageEnglish
Pages (from-to)43-88
Number of pages46
JournalAnnales Scientifiques de l'École Normale Supérieure
Volume53
Issue number1
DOIs
Publication statusPublished - 1 Jan 2020

Keywords

  • Dirichlet–to–Neumann operator
  • Steklov problems
  • eigenvalues
  • isoperimetric ratios
  • higher order Cheeger inequalities
  • finite Markov processes
  • jump Markov processes
  • Brownian motion on Riemannian manifolds
  • Laplace-Beltrami operator

Fingerprint Dive into the research topics of 'Higher order Cheeger inequalities for Steklov eigenvalues'. Together they form a unique fingerprint.

  • Cite this