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Colbois, Bruno
Nom
Colbois, Bruno
Affiliation principale
Fonction
Professeur ordinaire
Email
Bruno.Colbois@unine.ch
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Voici les éléments 1 - 8 sur 8
- PublicationAccès libreInégalités géométriques pour des valeurs propres de Steklov de graphes et de surfaces(2023)
; Cette thèse est consacrée à l’obtention d’inégalités géométriques pour des valeurs propres de Steklov de variétés riemanniennes de dimension 2 et de graphes. Les résultats obtenus concernent différentes situations. D’un côté, je m’intéresse à la géométrie de la première valeur propre non nulle de Steklov σ1 d’un graphe à bord. Pour cette valeur propre, je donne une borne inférieure qui dépend d’une borne supérieure sur le diamètre extrinsèque du bord et d’une borne supérieure sur le nombre de sommets du bord. Un autre résultat est une borne supérieure pour certains sous-graphes d’un graphe de Cayley à croissance polynomiale, qui montre en particulier que σ1 tend vers 0 lorsque le nombre de sommets du sous-graphe tend vers l’infini et généralise ainsi un résultat de Han et Hua obtenu pour des sous-graphes de Zn. Un deuxième but de la thèse est d’obtenir des bornes inférieures pour la première valeur propre non nulle de Steklov σ1 d’une variété riemannienne M dont le bord a plusieurs composantes connexes. Dans ce cas, la géométrie de M loin du bord peut avoir une forte influence sur σ1. Afin de préciser la forme de cette relation on étudie les variétés riemanniennes dont le bord a un voisinage cylindrique. En dimension 2, en supposant que la courbure de Gauss de M est bornée inférieurement, je donne une borne inférieure qui dépend d’une borne supérieure sur le diamètre extrinsèque du bord, d’une borne supérieure sur la longueur du bord et d’une borne inférieure sur la rayon d’injectivité des points d’un certain sous-ensemble de M. Finalement, je donne des bornes inférieure et supérieure pour les premières valeurs propres de Steklov d’une surface hyperbolique à bord géodésique en fonction de la longueur de certaines familles de géodésiques qui séparent le bord. Ce résultat est similaire à un résultat classique de Schoen, Wolpert et Yau pour les valeurspropres du laplacien d’une surface hyperbolique fermée. Abstract The aim of this thesis is to obtain geometric inequalities for Steklov eigenvalues of 2-dimensional Riemannian manifolds and graphs. The results obtained relate to different situations. On the one hand, our interest focuses on the geometry of the first non-zero Steklov eigenvalue σ1 of a graph with boundary. For this eigenvalue, we give a lower bound which depends on an upper bound on the extrinsic diameter of the boundary and on an upper bound on the number of vertices of the boundary. Another result is an upper bound for some subgraphs of a Cayley graph with polynomial growth, which shows in particular that σ1 tends to 0 when the number of vertices of the subgraph tends to infinity and thus generalizes a result of Han and Hua obtained for subgraphs of Zn. A second goal of the thesis is to obtain lower bounds for the first non-zero Steklov eigenvalue σ1 of a Riemannian manifold M whose boundary has several connected components. In this case, the geometry of M far from the boundary can have a strong influence on σ1. In order to specify the form of this relation we study Riemannian manifolds whose boundary has a cylindrical neighborhood. In dimension 2, assuming that the Gaussian curvature of M is bounded below, we give a lower bound which depends on an upper bound on the extrinsic diameter of the boundary, an upper bound on the length of the boundary and a lower bound on the radius of injectivity at the points of a certain subset of M. Finally, we give lower and upper bounds for the first Steklov eigenvalues of hyperbolic surfaces with geodesic boundary, which depend on the length of some families of geodesics that separate the boundary.This result is similar to a classical result of Schoen, Wolpert and Yau for Laplace eigenvalues of a closed hyperbolic surface. - PublicationAccès libreSpectrum of the Laplacian with weights(2019-3-4)
; El Soufi, AhmadGiven a compact Riemannian manifold $(M,g)$ and two positive functions $\rho$ and $\sigma$, we are interested in the eigenvalues of the Dirichlet energy functional weighted by $\sigma$, with respect to the $L^2$ inner product weighted by $\rho$. Under some regularity conditions on $\rho$ and $\sigma$, these eigenvalues are those of the operator $-\rho^{-1} \mbox{div}(\sigma \nabla u)$ with Neumann conditions on the boundary if $\partial M\ne \emptyset$. We investigate the effect of the weights on eigenvalues and discuss the existence of lower and upper bounds under the condition that the total mass is preserved. - PublicationAccès libreBornes supérieures pour les valeurs propres des opérateurs naturels sur des variétés riemanniennes compactes(2012)
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The second aspect is to study the interplay of the extrinsic geometry and eigenvalues of the Laplace-Beltrami operator acting on compact submanifolds of RN and of CPN. We investigate an extrinsic invariant called the intersection index studied by Colbois, Dryden and El Soufi. For compact submanifolds of RN, we extend their results and obtain upper bounds which are stable under small perturbation. For compact submanifolds of CPN we obtain an upper bound depending only on the degree of submanifolds and which is sharp for the first eigenvalue.
As a further application of the introduced method, we obtain an upper bound for the eigenvalues of the Steklov problem in a domain with C1 boundary in a complete Riemannian manifold in terms of the isoperimetric ratio of the domain and the min-conformal volume. A modification of our method also leads to have upper bounds for the eigenvalues of Schrödinger operators in terms of the min-conformal volume and integral quantity of the potential. As another application of our method, we obtain upper bounds for the eigenvalues of the Bakry-Emery Laplace operator depending on conformal invariants and properties of the weighted function. - PublicationMétadonnées seulementBounding the eigenvalues of the Laplace-Beltrami operator on compact submanifolds(2010-1-21)
; - PublicationMétadonnées seulementExtremal g-invariant eigenvalues of the Laplacian of g-invariant metrics(2008-12-21)
; - PublicationMétadonnées seulementEigenvalues of the laplacian acting on p-forms and metric conformal deformations(2006)
; El Soufi, AhmadLet (M, g) be a compact connected orientable Riemannian manifold of dimension n >= 4 and let lambda(k,p)(g) be the k-th positive eigenvalue of the Laplacian. Delta g,p = dd* + d* d acting on differential forms of degree p on M. We prove that the metric g can be conformally deformed to a metric g', having the same volume as g, with arbitrarily large lambda 1, p(g') for all p is an element of [2,n-2]. Note that for the other values of p, that is p = 0, 1, n-1 and n, one can deduce from the literature that, for all k > 0, the k-th eigenvalue lambda(k,p) is uniformly bounded on any conformal class of metrics of fixed volume on M. For p = 1, we show that, for any positive integer N, there exists a metric g(N) conformal to g such that, for all k - PublicationMétadonnées seulementExtremal eigenvalues of the Laplacian in a conformal class of metrics: The 'conformal spectrum'(2003-12-21)
; El Soufi, AhmadLet M be a compact connected manifold of dimension n endowed with a conformal class C of Riemannian metrics of volume one. For any integer k greater than or equal to 0, we consider the conformal invariant.c k( C) defined as the supremum of the k-th eigenvalue lambda(k)(g) of the Laplace-Beltrami operator Delta(g), where g runs over C. First, we give a sharp universal lower bound for lambda(k)(c)(C) extending to all k a result obtained by Friedlander and Nadirashvili for k = 1. Then, we show that the sequence {lambda(k)(c)(C)}, that we call 'conformal spectrum', is strictly increasing and satisfies, For Allk greater than or equal to 0, lambda(k+1)(c)(C)(n/2)-lambda(k)(c)(C)(n/2) greater than or equal to n(n/2) omega(n), where omega(n) is the volume of the n-dimensional standard sphere. When M is an orientable surface of genus gamma, we also consider the supremum zeta(k)(top) (gamma) of lambda(k)(g) over the set of all the area one Riemannian metrics on M, and study the behavior of lambda(k)(top)(gamma) in terms of gamma. - PublicationMétadonnées seulementOn the optimality of J. Cheeger and P. Buser inequalities(2003)
; Matei, Ana-MariaWe study the relationship between the first eigenvalue of the Laplacian and Cheeger constant when the Cheeger constant converges to zero, in the case of compact Riemannian manifolds and of finite graphs. (C) 2003 Elsevier B.V. All rights reserved.