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Dégénérescence et problèmes extrémaux pour les valeurs propres du laplaciens sur les surfaces

Abstract : The main topic of the present thesis is spectral geometry of surfaces. The spectrum of a closed surface (Σ, g) is a sequence of numbers 0 = λ0 < λ1 (g) ≤ λ2 (g) ≤ · · · called the eigenvalues. From the viewpoint of the theory of sound, each eigenvalue represents a frequency of vibration of the surface. We study the dependence of the eigenvalues on the geometric properties of the surface. This is a classical subject in spectral geometry, originated in the works of Lord Rayleigh [51], Faber [17], Krahn [32, 33], Pólya [49, 48], Szegö [54], Hersch [27], and many others. This thesis is a collection of three papers. The first one [23], "Fundamental tone, concentration of density to points and conformal degeneration on surfaces", is presented in Chapter 1. The influence on the fundamental tone (i.e., the first positive eigenvalue of the Laplacian) of two types of degeneration is studied : concentration of mass on any surface, and conformal degeneration on the torus and on the Klein bottle. In both cases, I prove that in the limit the fundamental tone is bounded above by the fundamental tone of a round sphere of the same area. The second paper [24] presented in the thesis is a joint work with Nikolai Nadirashvili and Iosif Polterovich. It is entitled "Maximization of the second positive Neumann eigenvalue for planar domains". The Neumann spectrum of a planar domain Ω ⊂ R2 is a sequence of numbers 0 = μ0 < μ1 (Ω) ≤ μ2 (Ω) ≤ · · ·. A classical result due to G. Szegö states that for each simply connected regular planar domain Ω, μ1 (Ω) Area(Ω) ≤ μ1 (D)π where D is the unit disk. The main result of the paper is a sharp upper bound on the second eigenvalue : μ2 (Ω) Area(Ω) ≤ 2 μ1 (D) π. This bound is attained in the limit by a family of domains degenerating to a disjoint union of two identical disks. In particular, this result implies the Pólya conjecture for μ2 . Our approach is based on a combination of analytic and topological arguments. A similar method leads to an upper bound on the second eigenvalue for conformally round spheres of odd dimension. The subject of the third paper Relative Homological Linking in Critical Point Theory [22] is not directly related to spectral geometry. It is an extension of my M.Sc. thesis written under the direction of Marlène Frigon. A homological linking for a pair of subspaces is introduced. It is used in combination with elementary Morse theory to detect the critical points of a functional. In particular, it is proved that homological linking implies homotopical linking.
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Alexandre Girouard. Dégénérescence et problèmes extrémaux pour les valeurs propres du laplaciens sur les surfaces. Mathématiques [math]. Université de Montréal, 2008. Français. ⟨tel-00576283⟩

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