eprintid: 26251 rev_number: 26 eprint_status: archive userid: 1482 importid: 105 dir: disk0/00/02/62/51 datestamp: 2018-09-13 07:20:13 lastmod: 2021-07-08 09:37:44 status_changed: 2021-07-08 09:37:44 type: article metadata_visibility: show creators_name: Hassannezhad, Asma creators_name: Miclo, Laurent creators_idrefppn: 057258279 creators_affiliation: Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS creators_halaffid: 1954 title: Higher order Cheeger inequalities for Steklov eigenvalues ispublished: pub subjects: subjects_ECO 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. date: 2020 date_type: published publisher: Société de Mathématiques id_number: 10.24033/asens.2417 official_url: http://tse-fr.eu/pub/32936 faculty: tse divisions: IMT divisions: tse language: en has_fulltext: FALSE doi: 10.24033/asens.2417 view_date_year: 2020 full_text_status: none publication: Annales scientifiques de l’École normale supérieure volume: vol. 53 number: n° 1 place_of_pub: Paris pagerange: 43-88 refereed: TRUE issn: 0012-9593 oai_identifier: oai:tse-fr.eu:32936 harvester_local_overwrite: date_type harvester_local_overwrite: issn harvester_local_overwrite: faculty harvester_local_overwrite: publisher harvester_local_overwrite: place_of_pub harvester_local_overwrite: title harvester_local_overwrite: pending harvester_local_overwrite: note harvester_local_overwrite: creators_affiliation harvester_local_overwrite: publish_to_hal harvester_local_overwrite: divisions harvester_local_overwrite: number harvester_local_overwrite: volume harvester_local_overwrite: creators_idrefppn harvester_local_overwrite: creators_halaffid oai_lastmod: 2021-06-17T15:52:09Z oai_set: tse site: ut1 publish_to_hal: FALSE citation: Hassannezhad, Asma and Miclo, Laurent (2020) Higher order Cheeger inequalities for Steklov eigenvalues. Annales scientifiques de l’École normale supérieure, vol. 53 (n° 1). pp. 43-88.