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Geometric Spectral Theory
Titre du projet
Geometric Spectral Theory
Description
The main topic of this proposal is
"spectral geometry on Riemannian manifolds", and more precisely the
study of lower and upper bounds for the spectrum of the Laplacian and of generalized Laplacians or analogous operators.
The classical Laplacian may be generalized in many directions, and during these last years, there were in particular a lot of investigations around the Laplacian on a weighted manifold, the Steklov (or Dirichlet-to-Neumann) operator and the magnetic Laplacian. In this proposal, I focus on a geometrical approach of the spectrum: I am mainly interested in finding uniform bounds which are of metric nature (a typical example being isoperimetric constants). Such are rather robust regarding variations of the Riemannian metric. I am also looking for uniform upper bounds as compatible as possible with the Weyl asymptotic formula.
In this context, I am asking for two fellowships, one for a doctoral student, Luc Pétiard, and one for a post-doc, Guillaume Roy-Fortin.
In the following, I present separately different projects, even if there will certainly be interactions between them.
With A. El Soufi, we will mainly investigate the Laplacian in a manifold with densities: we will study the problem $-\mbox{div}(\sigma \nabla u)=\lambda \rho u $ in the conformal class [g_0] of a Riemannian manifold (M,g_0), which was already adressed in [CES]. We will study the interaction between $\rho,\sigma$ and $[g_0]$, try to find good uniform upper bounds in special cases and a Cheeger type lower bound for the first nonzero eigenvalue. We will discuss the sharpness of the upper bounds and try to find extremal manifolds/domains in some special situations (first nonzero eigenvalue).
This will also be the context of the Ph.D. thesis of L. Pétiard.
In collaboration with A. Girouard, the aim is to investigate the spectrum of the Steklov operator. There are two main topics, the first being the rough discretization of the spectrum and application (also in collaboration with B. Raveendran). The second will be an investigation of the Steklov spectrum for a family of Riemannian manifolds with fixed boundary, in order to measure how the restriction of fixing the boundary implies some rigidity.
A more prospective part of the project will be to investigate the magnetic Laplacian in collaboration with A. Savo. There has been a lot of interest around the magnetic Laplacian in relation with physics. We are interested in a geometrical approach taking into account the geometry and the topology of the underlying manifold, which seems to be rather new in this context.
The collaboration with G. Roy-Fortin will focuss on the shape optimization for the Neumann problem and the study of generalized laplacian, such as the bi-Laplacian.
"spectral geometry on Riemannian manifolds", and more precisely the
study of lower and upper bounds for the spectrum of the Laplacian and of generalized Laplacians or analogous operators.
The classical Laplacian may be generalized in many directions, and during these last years, there were in particular a lot of investigations around the Laplacian on a weighted manifold, the Steklov (or Dirichlet-to-Neumann) operator and the magnetic Laplacian. In this proposal, I focus on a geometrical approach of the spectrum: I am mainly interested in finding uniform bounds which are of metric nature (a typical example being isoperimetric constants). Such are rather robust regarding variations of the Riemannian metric. I am also looking for uniform upper bounds as compatible as possible with the Weyl asymptotic formula.
In this context, I am asking for two fellowships, one for a doctoral student, Luc Pétiard, and one for a post-doc, Guillaume Roy-Fortin.
In the following, I present separately different projects, even if there will certainly be interactions between them.
With A. El Soufi, we will mainly investigate the Laplacian in a manifold with densities: we will study the problem $-\mbox{div}(\sigma \nabla u)=\lambda \rho u $ in the conformal class [g_0] of a Riemannian manifold (M,g_0), which was already adressed in [CES]. We will study the interaction between $\rho,\sigma$ and $[g_0]$, try to find good uniform upper bounds in special cases and a Cheeger type lower bound for the first nonzero eigenvalue. We will discuss the sharpness of the upper bounds and try to find extremal manifolds/domains in some special situations (first nonzero eigenvalue).
This will also be the context of the Ph.D. thesis of L. Pétiard.
In collaboration with A. Girouard, the aim is to investigate the spectrum of the Steklov operator. There are two main topics, the first being the rough discretization of the spectrum and application (also in collaboration with B. Raveendran). The second will be an investigation of the Steklov spectrum for a family of Riemannian manifolds with fixed boundary, in order to measure how the restriction of fixing the boundary implies some rigidity.
A more prospective part of the project will be to investigate the magnetic Laplacian in collaboration with A. Savo. There has been a lot of interest around the magnetic Laplacian in relation with physics. We are interested in a geometrical approach taking into account the geometry and the topology of the underlying manifold, which seems to be rather new in this context.
The collaboration with G. Roy-Fortin will focuss on the shape optimization for the Neumann problem and the study of generalized laplacian, such as the bi-Laplacian.
Chercheur principal
Pétiard, Luc
Statut
Completed
Date de début
1 Octobre 2015
Date de fin
30 Septembre 2018
Organisations
Identifiant interne
31100
identifiant
10 Résultats
Voici les éléments 1 - 10 sur 10
- PublicationAccès libreSteklov Eigenvalues of Submanifolds with Prescribed Boundary in Euclidean Space(2019-4-4)
; ; Gittins, KatieWe obtain upper and lower bounds for Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space. A very general upper bound is proved, which depends only on the geometry of the fixed boundary and on the measure of the interior. Sharp lower bounds are given for hypersurfaces of revolution with connected boundary: we prove that each eigenvalue is uniquely minimized by the ball. We also observe that each surface of revolution with connected boundary is Steklov isospectral to the disk. - PublicationAccès libreIsoperimetric Inequalities for the Magnetic Neumann and Steklov Problems with Aharonov–Bohm Magnetic Potential(2022-9-14)
; - PublicationAccès libreCompact manifolds with fixed boundary and large Steklov eigenvalues(2019-8-22)
; - PublicationAccès libreLower bounds for the first eigenvalue of the magnetic Laplacian(2018-5-17)
; Savo, AlessandroWe consider a Riemannian cylinder $\Omega$ endowed with a closed potential $1$-form $A$ and study the magnetic Laplacian $\Delta_A$ 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. - 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 libreSharp Steklov upper bound for submanifolds of revolution(2021-5-10)
; Verma, Sheela - PublicationAccès libreLower bounds for the first eigenvalue of the Laplacian with zero magnetic field in planar domains(2021-3-26)
; Savo, Alessandro - PublicationAccès libre
- PublicationAccès libreThe Steklov spectrum and coarse discretizations of manifolds with boundary(2018-8-22)
; ; Raveendran, BinoyGiven $\kappa, r_0>0$ and $n\in\N$, we consider the class $\mathcal{M}=\mathcal{M}(\kappa,r_0,n)$ of compact $n$-dimensional Riemannian manifolds with cylindrical boundary, Ricci curvature bounded below by $-(n-1)\kappa$ and injectivity radius bounded below by $r_0$ away from the boundary. For a manifold $M\in\mathcal{M}$ we introduce a notion of discretization, leading to a graph with boundary which is roughly isometric to $M$, with constants depending only on $\kappa,r_0,n$. In this context, we prove a uniform spectral comparison inequality between the Steklov eigenvalues of a manifold $M\in\mathcal{M}$ and those of its discretization. Some applications to the construction of sequences of surfaces with boundary of fixed length and with arbitrarily large Steklov spectral gap $\sigma_2-\sigma_1$ are given. In particular, we obtain such a sequence for surfaces with connected boundary. The applications are based on the construction of graph-like surfaces which are obtained from sequences of graphs with good expansion properties. - PublicationAccès libreEigenvalues of elliptic operators with density(2018-5-17)
; Provenzano, LuigiWe consider eigenvalue problems for elliptic operators of arbitrary order $2m$ subject to Neumann boundary conditions on bounded domains of the Euclidean $N$-dimensional space. We study the dependence of the eigenvalues upon variations of mass density. In particular we discuss the existence and characterization of upper and lower bounds under both the condition that the total mass is fixed and the condition that the $L^{\frac{N}{2m}}$-norm of the density is fixed. We highlight that the interplay between the order of the operator and the space dimension plays a crucial role in the existence of eigenvalue bounds.