LEBEAU , Gilles

On introduit les bases du calcul h-pseudodifférentiel. On applique ce calcul pour prouver des estimations de Carleman elliptiques, ou paraboliques. En utilisant les inégalités de Carleman elliptiques, on prouve une inégalité de type spectral sur les combinaisons linéaires de fonctions propres du laplacien. Comme application de cette inégalité...

International audience

On introduit les bases du calcul h-pseudodifférentiel. On applique cecalcul pour prouver des estimations de Carleman elliptiques, ou paraboliques. En utilisantles inégalités de Carleman elliptiques, on prouve une inégalité de type spectralsur les combinaisons linéaires de fonctions propres du laplacien. Comme applicationde cette inégalité...

Until now, the correspondence between the Alexander-Kolmogorov Complex, and the De Rham one, by means of a small scale parameter, has not gone that far as passing to the limit of the resolvent of the associated Laplacian, when the small parameter tends towards zero. In this line, a result proving a complete Hodge decomposition was missing. We...

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 R^d H_{g,V} = -\Delta_g + V (x), where \Delta_g is the Laplace-Beltrami operator with respect to an analytic metric g, which is a perturbation of the...

International audience

In this paper, we study the following problem: let $\Omega$ be a half-space of $\mathbb{R}^N$, defined by $\Omega = \{x = (x', x_N) \in\mathbb{R}^/x_N > \}$ where $x' = (x,\ldots, x_{N-1})$ is the usual notation, and let there be given functions $u_0\in H^1(\Omega)$ and $u_1 \in L^2(\Omega)$. We assume that $u_0|_{x_N=0}$ is nonnegative,...

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...

National audience

We consider the wave and Schrödinger equations with Dirichlet boundary conditions in the exterior of a ball in $R^d$. In dimension $d = 3$ we construct a sharp, global in time parametrix and then proceed to obtain sharp dispersive estimates, matching the $R^3$ case, for all frequencies (low and high). If $d ≥ 4$, we provide an explicit solution...

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...

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 ≥ 4 we show that losses in dispersion do appear and this happens at the Poisson spot.

International audience

We consider the wave and Schrödinger equations with Dirichlet boundary conditions in the exterior of a ball in $R^d$. In dimension $d = 3$ we construct a sharp, global in time parametrix and then proceed to obtain sharp dispersive estimates, matching the $R^3$ case, for all frequencies (low and high). If $d ≥ 4$, we provide an explicit solution...

We consider the problem of the numerical approximation of the linear controllability of waves. All our experiments are done in a bounded domain Ω of 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...

We consider the problem of the numerical approximation of the linear controllability of waves. All our experiments are done in a bounded domain $\Omega$ of 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...

We consider an 1-electron model Hamiltonian, whose potential energy corresponds to the Coulomb potential of an infinite wire with charge Z distributed according to a Gaussian function. The time independent Schrödinger equation for this Hamiltonian is solved perturbationally in the asymptotic limit of small amplitude vibration (Gaussian function...

We prove sharper Strichartz estimates than expected from theoptimal dispersion bounds.

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...

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