Liste des cours
Colette Anné (Université de Nantes, France)
Titre: Perturbations spectrales singulières. (6h)
To study the effect of small (wild) perturbations like making small holes or adding thin and short handles on the spectra of the Laplace operator, we choose the approach by the study of the Energy form.
1. We introduce the notion of unbounded operators on Hilbert spaces, and the relation with closed quadratic forms.
2. Spectral theory of selfadjoint operators, resolvant.
3. Sobolev embedding theorem and the Rellich theorem.
4. Elliptic estimates and Garding inequality.
5. Spectral perturbation by excision
6. Spectral perturbation by addition
Virginie Bonnaillie-Noël (ENS Paris, France)
Titre: Méthodes numériques pour le calcul des modes propres. (6h)
In this course, we present a survey of some methods to compute eigenmodes. The main part of the lecture concerns the computation of the eigenvalues of a matrix. First, we give the results about the location of the eigenvalues. Then we present the power method and methods specific for symmetric matrix.
The second part of the lectures deals with the approximation of the eigenmodes of some second order operator. We will gives some illustrations or applications along the lectures.
Bruno Colbois (Université de Neuchatel, Swisse)
Titre : Introduction à la géométrie spectrale. (6h)
We will mainly consider the Laplace operator acting on functions, and we will investigate its eigenvalues and eigenfunctions.
1. The Dirichlet spectrum of a bounded domain. Statement of the Faber-Krahn inequality and its developments.
2. The Neumann spectrum of a bounded domain. The variational characterization, the min-max and applications.
3. Nodal domains, Chaldni plates, Courant and Pleijel Theorem.
4. Construction of domains with small eigenvalues for the Neumann problem. The Cheeger constant and the statement of the Cheeger inequality.
5. Extremal problems. Examples, Poly'a conjecture.
6. The case of surfaces.
Ali Fardoun (Université de Brest, France)
Titre: Courbure Gaussienne prescrite. (4h)
We first recall Sobolev Theorems on riemannian manifolds. Next, we introduce the prescribed gaussian curvature and next we give the analityc tools (such as the Moser Trudinger inequality) to solve the second order equation associated to this problem depending on the Euler Characteristic of the manifold.
Alexandre Girouard (Université de Laval, Canada)
Titre: Géométrie spectrale pour le problème de Steklov. (6h)
. The Dirichlet-to-Neumann map and its applications to inverse problems.
. Physical interpretation of the Steklov problems (fluid mecanics, stationary heat).
. Some simple examples (Intervals; Disks, balls and cylinders using separation of variables).
. Typical questions.
2. The spectral theorem for the Steklov problem
. Compactness of trace operators on bounded regular domains.
. The spectral theorem for coercive quadratic forms.
. The structure of the proof of the spectral theorem.
3. Min-max variational characterization of Steklov eigenvalues
. Min-max and Rayleigh quotient.
. Loss of compactness: domains with thin channels and cusps.
. Comparison with mixed problems.
. Cylindrical boundaries.
4. Isoperimetric inequalities for planar domains
. Conformal invariance of the Dirichlet energy in dimension 2.
. Renormalization of center of mass à la Hersch.
. The Weinstock inequality for simply connected planar domains of prescribed perimeter.
. The spectrum of an annulus.
. Brock's inequality for domains with prescribed area.
5. Surfaces with large eigenvalues
. The Steklov spectrum of a graph
. Comparison estimates
. Expander graphs
6. Perspective and open problems
Asma Hassannezhed (Université de Bristol, United Kingdom)
Titre : Un tour sur les moyennes de Riesz et les bornes des valeurs propres dans les espaces euclidiens. (6h)
This course aims to give a selective overview onresults and techniques in the study of asymptotically sharp bounds onLaplace eigenvalues and the Riesz means on bounded domains in the Euclidean space. We first start by recalling the Weyl asymptotic lawforthe Laplace eigenvalues. Then we talk about the Polya conjecture andsome main results in this direction,namely Li-Yau and Kröger inequalities.
Ali Wehbe (Université Libanaise, Liban)
Titre : Méthodes variationnelles pour les problèmes elliptiques. (6h)
1. Sobolev spaces the Sobolev embedding theorem and the Rellich- Kondrachov theorem.
2. Elliptic operators : We study second order elliptic operators with main goal to apply the results to the Laplace operator. We will present in particular the regularity theorems and the maximum principle.