Higher order Cheeger inequalities for Steklov eigenvalues
| Autor: | Laurent Miclo, Asma Hassannezhad |
|---|---|
| Přispěvatelé: | Institut Mittag-Leffler, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), ANR-12-BS01-0019,STAB,Stabilité du comportement asymptotique d'EDP, de processus stochastiques et de leurs discrétisations.(2012), Laboratoire de Mathématiques et Physique Théorique (LMPT), Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université de Tours |
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Mathematics - Differential Geometry
Dirichlet–to–Neumann operator Spectral theory General Mathematics 01 natural sciences Upper and lower bounds Laplace-Beltrami operator higher order Cheeger inequalities Combinatorics Mathematics - Spectral Theory 0103 physical sciences FOS: Mathematics Order (group theory) 0101 mathematics Spectral Theory (math.SP) B- ECONOMIE ET FINANCE Dirichlet--to--Neumann operator 35P15 58J50 58J65 60J25 60J27 Mathematics MSC2010: 15A18 35P15 58J50 58J65 60J25 60J27 60J60 60J75 Dirichlet-to-Neumann operator Probability (math.PR) 010102 general mathematics Spectrum (functional analysis) eigenvalues higher or-der Cheeger inequalities Mathematics::Spectral Theory jump Markov processes [MATH.MATH-PR]Mathematics [math]/Probability [math.PR] Steklov problems Differential Geometry (math.DG) Differential geometry Laplace–Beltrami operator Bounded function Steklov problem Brownian motion on Riemannian manifolds 010307 mathematical physics isoperimetric ratios finite Markov processes Isoperimetric inequality Mathematics - Probability [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP] |
| Zdroj: | Hassannezhad, A & Miclo, L 2020, ' Higher order Cheeger inequalities for Steklov eigenvalues ', Annales Scientifiques de l'École Normale Supérieure, vol. 53, no. 1, pp. 43-88 . https://doi.org/10.24033/asens.2417 Annales Scientifiques de l'École Normale Supérieure Annales Scientifiques de l'École Normale Supérieure, 2019 Annales Scientifiques de l'Ecole Normale Superieure Annales Scientifiques de l'École Normale Supérieure, Société mathématique de France, 2019 |
| ISSN: | 0012-9593 1873-2151 |
| DOI: | 10.24033/asens.2417 |
| Popis: | 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. Correcting few typos and doing some changes on pages 41 and 45 |
| Databáze: | OpenAIRE |
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