1,721,057 research outputs found
Neumann Eigenvalues of the Biharmonic Operator on Domains: Geometric Bounds and Related Results
Full text linkWe study an eigenvalue problem for the biharmonic operator with Neumann boundary conditions on domains of Riemannian manifolds. We discuss the weak formulation and the classical boundary conditions, and we describe a few properties of the eigenvalues. Moreover, we establish upper bounds compatible with the Weyl’s law under a given lower bound on the Ricci curvature
Lower bounds for the first eigenvalue of the magnetic Laplacian
Full text linkWe consider a Riemannian cylinder endowed with a closed potential -form and study the magnetic Laplacian with magnetic Neumann boundary conditions associated with those data. We establish a sharp lower bound for the first eigenvalue and show that the equality characterizes the situation where the metric is a product. We then look at the case of a planar domain bounded by two closed curves and obtain an explicit lower bound in terms of the geometry of the domain. We finally discuss sharpness of this last estimate
Eigenvalues upper bounds for the magnetic Schrödinger operator
Full text linkWe study the eigenvalues λk(HA,q) of the magnetic Schro ̈dinger
operator HA,q associated with a magnetic potential A and a scalar
potential q, on a compact Riemannian manifold M, with Neu-
mann boundary conditions if ∂M ̸= ∅. We obtain various bounds
on λ1(HA,q),λ2(HA,q) and, more generally on λk(HA,q). Some of
them are sharp. Besides the dimension and the volume of the man-
ifold, the geometric quantities which plays an important role in
these estimates are: the first eigenvalue λ′′ (M) of the Hodge-de 1,1
Rham Laplacian acting on co-exact 1-forms, the mean value of the scalar potential q, the L2-norm of the magnetic field B = dA, and the distance, taken in L2, between the harmonic component of A and the subspace of all closed 1-forms whose cohomology class is in- tegral (that is, having integral flux around any loop). In particular, this distance is zero when the first cohomology group H1(M,R) is trivial. Many other important estimates are obtained in terms of the conformal volume, the mean curvature and the genus (in dimension 2). Finally, we also obtain estimates for sum of eigen- values (in the spirit of Kro ̈ger estimates) and for the trace of the heat kernel
A reverse Faber-Krahn inequality for the magnetic Laplacian
Full text linkWe consider the first eigenvalue of the magnetic Laplacian in a bounded and simply connected planar domain, with uniform magnetic field and Neumann boundary conditions. We investigate the reverse Faber-Krahn inequality conjectured by S. Fournais and B. Helffer, stating that this eigenvalue is maximized by the disk for a given area. Using the method of level lines, we prove the conjecture for small enough values of the magnetic field (those for which the corresponding eigenfunction in the disk is radial).27 pages. Accepted pre-publication versio
Inégalités géométriques pour des valeurs propres de Steklov de graphes et de surfaces
Full text linkAcceptée sur proposition du jury :
Prof. Bruno Colbois, directeur de thèse, Université de Neuchâtel, Suisse
Prof. Alexandre Girouard, rapporteur, Université Laval, Québec, Canada
Prof. Asma Hassannezhad, rapporteure, University of Bristol, Royaume-Uni
Prof. Alain Valette, Université de Neuchâtel, Suisse
Soutenue le 1er juin 2023
No de thèse : 3043Cette 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
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Eigenvalue control for a Finsler-Laplace operator
Full text linkUsing the definition of a Finsler-Laplacian given by the first author, we show that two bi-Lipschitz Finsler metrics have a controlled spectrum. We deduce from that several generalizations of Riemannian results. In particular, we show that the spectrum on Finsler surfaces is controlled above by a constant depending on the topology of the surface and on the quasireversibility constant of the metric. In contrast to Riemannian geometry, we then give examples of highly non-reversible metrics on surfaces with arbitrarily large first eigenvalu
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