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Gilles Lebeau

Publications and source records attributed to Gilles Lebeau.

18 recordsLinked to original sources
arXiv2023

Dispersion for the wave equation inside strictly convex domains II: the general case

We consider the wave equation on a manifold $(Ω,g)$ of dimension $d\geq 2$ with smooth strictly convex boundary $\partialΩ\neq\emptyset$, with Dirichlet boundary conditions. We construct a sharp local in time parametrix and then proceed to obtain dispersion estimates: our fixed time decay rate for the Green function exhibits a $t^{1/4}$ loss with respect to the boundary less case. We precisely describe where and when these losses occur and relate them to swallowtail type singularities in the wave front set, proving that our decay is optimal. Moreover, we derive better than expected Strichartz estimates, balancing lossy long time estimates at a given incidence with short time ones with no loss: for $d=3$, it heuristically means that, on average the decay loss is only $t^{1/6}$.

math.AP↗
arXiv2021

Strichartz estimates for the wave equation on a 2d model convex domain

We prove better Strichartz type estimates than expected from the (optimal) dispersion we obtained in our earlier work on a 2d convex model. This follows from taking full advantage of the space-time localization of caustics in the parametrix we obtain, despite their number increasing like the inverse square root of the distance from the source to the boundary. As a consequence, we improve known Strichartz estimates for the wave equation. Several improvements on our previous parametrix construction are obtained along the way and are of independent interest for further applications.

math.AP↗
arXiv2021

Sufficient Stability Conditions for Time-varying Networks of Telegrapher's Equations or Difference Delay Equations

We give a sufficient condition for exponential stability of a network of lossless telegrapher's equations, coupled by linear time-varying boundary conditions. The sufficient conditions is in terms of dissipativity of the couplings, which is natural for instance in the context of microwave circuits. Exponential stability is with respect to any $L^p$-norm, $1\leq p\leq\infty$. This also yields a sufficient condition for exponential stability to a special class of linear time-varying difference delay systems which is quite explicit and tractable. One ingredient of the proof is that $L^p$ exponential stability for such difference delay systems is independent of $p$, thereby reproving in a simpler way some results from [Y. Chitour, G. Mazanti, and M. Sigalotti, $\it {Netw. Heterog. Media}$, 11 (2016), pp. 563--601].

math.DS↗
arXiv2019

Spectral Inequalities for the Schr{ö}dinger operator

In this paper we deal with the so-called "spectral inequalities", which yield a sharp quantification of the unique continuation for the spectral family associated with the Schrödinger operator in $ \mathbb{R}^d$ \begin{equation*} H_{g,V} = Δ_g + V(x), \end{equation*} where $Δ_g$ is the Laplace-Beltrami operator with respect to an analytic metric $g$, which is a perturbation of the Euclidean metric, and $V(x)$ a real valued analytic potential vanishing at infinity.

math.AP↗
arXiv2018

Smeared Coulomb potential orbitals for the electron-nucleus mean field configuration interaction method

We propose to use the eigenfunctions of a one-electron model Hamiltonian to perform electron-nucleus mean field configuration interaction (EN-MFCI) calculations. The potential energy of our model Hamiltonian corresponds to the Coulomb potential of an infinite wire with charge $Z$ distributed according to a Gaussian function. The time independent \sch equation for this Hamiltonian is solved perturbationally in the limit of small amplitude vibration (Gaussian function width close to zero).

physics.chem-ph↗
arXiv2017

Diffusion approximation for Fokker Planck with heavy tail equilibria : a spectral method in dimension 1

This paper is devoted to the diffusion approximation for the 1-d Fokker Planck equation with a heavy tail equilibria of the form (1+v^2)^{-β/2}, in the range beta\in ]1,5[. We prove that the limit diffusion equation involves a fractional Laplacian kappa|Δ|^{\frac{β+1}{6}}, and we compute the value of the diffusion coefficient kappa. This extends previous results of E. Nasreddine and M. Puel in the case beta>5, and of P. Cattiaux, E. Nasreddine and M. Puel in the case beta=5.

math.AP↗
arXiv2017

The Green function for waves on the $2$-regular Bethe lattice

In this paper, we compute an explicit analytic expression for the Green function of the wave operator on the $2$-regular lattice called the "Bethe lattice" equipped with its standard metric. In particular, we exhibit a phenomena of abnormal speed of propagation for waves: the effective speed of propagation of energy for large time is $c_*=2\sqrt 2/3 <1$, and there exists a true propagation at any speed $c<c_*$.

math.AP↗
arXiv2017

Dispersion for the wave equation outside a ball and counterexamples

The purpose of this note is to prove dispersive estimates for the wave equation outside a ball in R^d. If d = 3, we show that the linear flow satisfies the dispersive estimates as in R^3. In higher dimensions d $\ge$ 4 we show that losses in dispersion do appear and this happens at the Poisson spot.

math.AP↗
arXiv2017

Geometric control condition for the wave equation with a time-dependent observation domain

We characterize the observability property (and, by duality, the controllability and the stabilization) of the wave equation on a Riemannian manifold $Ω,$ with or without boundary, where the observation (or control) domain is time-varying. We provide a condition ensuring observability, in terms of propagating bicharacteristics. This condition extends the well-known geometric control condition established for fixed observation domains. As one of the consequences, we prove that it is always possible to find a time-dependent observation domain of arbitrarily small measure for which the observability property holds. From a practical point of view, this means that it is possible to reconstruct the solutions of the wave equation with only few sensors (in the Lebesgue measure sense), at the price of moving the sensors in the domain in an adequate way.We provide several illustrating examples, in which the observationdomain is the rigid displacement in $Ω$ of a fixed domain, withspeed $v,$ showing that the observability property depends both on $v$and on the wave speed. Despite the apparent simplicity of some of ourexamples, the observability property can depend on nontrivial arithmeticconsiderations.

math.AP↗
arXiv2013

Spectral Analysis of hypoelliptic random walks

We study the spectral theory of a reversible Markov chain associated to a hypoelliptic random walk on a manifold M. This random walk depends on a parameter h which is roughly the size of each step of the walk. We prove uniform bounds with respect to h on the rate of convergence to equilibrium, and the convergence when h goes to zero to the associated hypoelliptic diffusion.

math.AP↗
arXiv2013

Dispersion for the wave equation inside strictly convex domains I: the Friedlander model case

We consider a model case for a strictly convex domain of dimension $d\geq 2$ with smooth boundary and we describe dispersion for the wave equation with Dirichlet boundary conditions. More specifically, we obtain the optimal fixed time decay rate for the smoothed out Green function: a $t^{1/4}$ loss occurs with respect to the boundary less case, due to repeated occurrences of swallowtail type singularities in the wave front set.

math.AP↗
arXiv2011

Injections de Sobolev probabilistes et applications

In this article, we give probabilistic versions of Sobolev embeddings on any Riemannian manifold $(M,g)$. More precisely, we prove that for natural probability measures on $L^2(M)$, almost every function belong to all spaces $L^p(M)$, $p<+\infty$. We then give applications to the study of the growth of the $L^p$ norms of spherical harmonics on spheres $\mathbb{S}^d$: we prove (again for natural probability measures) that almost every Hilbert base of $L^2(\mathbb{S}^d)$ made of spherical harmonics has all its elements uniformly bounded in all $L^p(\mathbb{S}^d), p<+\infty$ spaces. We also prove similar results on tori $\mathbb{T}^d$. We give then an application to the study of the decay rate of damped wave equations in a frame-work where the geometric control property on Bardos-Lebeau-Rauch is not satisfied. Assuming that it is violated for a measure 0 set of trajectories, we prove that there exists almost surely a rate. Finally, we conclude with an application to the study of the $H^1$-supercritical wave equation, for which we prove that for almost all initial data, the weak solutions are strong and unique, locally in time.

math.AP↗
arXiv2011

Gibbs/Metropolis algorithms on a convex polytope

This paper gives sharp rates of convergence for natural versions of the Metropolis algorithm for sampling from the uniform distribution on a convex polytope. The singular proposal distribution, based on a walk moving locally in one of a fixed, finite set of directions, needs some new tools. We get useful bounds on the spectrum and eigenfunctions using Nash and Weyl-type inequalities. The top eigenvalues of the Markov chain are closely related to the Neuman eigenvalues of the polytope for a novel Laplacian.

math.SP↗
arXiv2009

Experimental Study of the HUM Control Operator for Linear Waves

We consider the problem of the numerical approximation of the linear controllability of waves. All our experiments are done in a bounded domain Ωof the plane, with Dirichlet boundary conditions and internal control. We use a Galerkin approximation of the optimal control operator of the continuous model, based on the spectral theory of the Laplace operator in Ω. This allows us to obtain surprisingly good illustrations of the main theoretical results available on the controllability of waves, and to formulate some questions for the future analysis of optimal control theory of waves.

math.OC↗
arXiv2006

Global existence for energy critical waves in 3-D domains

We prove that the defocusing quintic wave equation, with Dirichlet boundary conditions, is globally well posed on $H^1_0(Ω) \times L^2(Ω)$ for any smooth (compact) domain $Ω\subset \mathbb{R}^3$. The main ingredient in the proof is an $L^5$ spectral projector estimate, obtained recently by Smith and Sogge, combined with a precise study of the boundary value problem.

math.AP↗