Uniqueness and reconstruction in the three-dimensional Calderón inverse conductivity problem can be reduced to the study of the inverse boundary problem of a Schrödinger operator with a real potential. Due to the lack of a rigorous definition, the Born approximation has been relegated to a marginal place in the reconstruction problem. In this talk we will introduce the Born approximation for Schrödinger operators in the ball, which amounts to studying the linearization of the inverse problem. We first analyze this approximation for real and radial potentials in any dimension >2. We show that this approximation satisfies a closed formula that only involves the spectrum of the Dirichlet-to-Neumann map, and which is closely related to a particular moment problem.
Séminaire d'analyse (archives)
Breathers are temporally periodic and spatially localized solutions of evolutionary PDEs. They are known to exist for integrable PDEs such as the sine-Gordon equation, but are believed to be rare for general nonlinear PDEs. When the spatial dimension is equal to one, exchanging the roles of time and space variables (in the so-called spatial dynamics framework), breathers can be interpreted as homoclinic solutions to steady solutions and thus arise from the intersections of the stable and unstable manifolds of the steady states. In this talk, we shall study the nonlinear Klein-Gordon equation and show that small amplitude breathers cannot exist (under certain conditions).
This is a conjecture on weighted estimates for the classical Fourier extension operators of harmonic analysis. In particular, let E be the extension operator associated to some surface, and g be a function on that surface. If we 'erase' part of Eg, how well can we control the 2-norm of the remaining piece? The Mizohata-Takeuchi conjecture claims some remarkable control on this quantity, involving the X-ray transform of the part of the support of Ef that we kept. In this talk we will discuss the basics and history of the problem, as well as some small progress. This is joint work with Anthony Carbery and Hong Wang.
Dans ce séminaire je présenterai un travail en collaboration avec Oana Ivanovici. Nous considérons l’équation des ondes, avec conditions de Dirichlet, à l’extérieur d’un cylindre en dimension trois, nous construisons une parametrice globale et nous en déduisons des estimations dispersives optimales pour les solutions.
Dans cet exposé, on s'intéressera à un problème de scattering par des obstacles dans le plan et plus particulièrement, à l'étude des résonances du Laplacien en dehors de ces obstacles (ce sont des valeurs propres généralisées). On présentera un résultat nouveau qui établit l'existence d'un trou spectral. Après quelques rebonds, on se retrouvera très vite au pays des fractales, ce qui nous amènera à faire une excursion dans le monde des surfaces hyperboliques. On y évoquera un outil récemment développé dans ce contexte et central dans la preuve du trou spectral : un principe d'incertitude fractal. Enfin, si le temps le permet, nous finirons chez le boulanger (et sa transformation) pour tâcher d'expliquer sur un modèle jouet les tenants de la preuve.
In this talk, I will present new results concerning the study of the resolvent of the damped-wave operator associated with the sub-elliptic Laplacian known as Baouendi-Grushin operator on the two-dimensional flat torus. From different hypothesis on the geometry of the damping region and the Hölder regularity of the damping term, I will show sharp resolvent estimates of the associated non-selfadjoint operator on the real axis. As an application, sharp energy-decay-rates of the damped-wave equation are obtained. The proofs are based on the study of two-microlocal semiclassical measures, normal form reductions and constructions of quasimodes in different parts of the phase-space.
This work has been done in collaboration with Chenmin Sun. Reference: arXiv:2201.08189.
We present some results on the spectral analysis of the semiclassical Neumann magnetic Laplacian on a smooth bounded domain in dimension three. When the magnetic field is constant and in the semiclassical limit, we establish a four-term asymptotic expansion of the low-lying eigenvalues, involving a geometric quantity along the apparent contour of the domain in the direction of the field. In particular, we prove that they are simple.