Analysis ot the Geometrical Effects on Dispersive Equations – ANADEL

We are concerned with localization properties of solutions to hyperbolic PDEs, especially problems with a geometric component: how do boundaries and heterogeneous media influence spreading and concentration of solutions. While our first focus is on wave and Schrödinger equations on manifolds with boundary, strong connections exist with phase space localization for (clusters of) eigenfunctions, which are of independent interest. Motivations come from nonlinear dispersive models (in physically relevant settings), properties of eigenfunctions in quantum chaos (related to both physics of optic fiber design as well as number theoretic questions), or harmonic analysis on manifolds.

I expect to answer several open questions centering around how the geometry and especially the presence of a boundary affect waves concentration and which kind of waves may saturate corresponding bounds. We are concerned here with L^p estimates for eigenfunctions and dispersion / Strichartz type estimates in space-time, which represent fundamental tools in the study of nonlinear problems. For starter, to obtain sharp dispersive estimates on compact domains one needs to understand concentration phenomena that may occur for eigenfunctions near the boundary. Indeed, a loss in dispersion could be informally related to the presence of caustics, which occur when optical rays are non longer diverging from each other. Conversely, to prove sharp eigenfunctions (or spectral projectors) estimates one needs to understand how light rays can reflect and concentrate due to the geometry of the boundary. I am confident that my recent breakthroughs (with collaborators) on strictly convex domains will lead to a complete understanding of dispersion on bounded domains.

Waves propagation in real life physics occur in media which are neither homogeneous or spatially infinity. The birth of radar/sonar technologies (and the raise of computed tomography) greatly motivated numerous developments in microlocal analysis and the linear theory. Only recently toy nonlinear models have been studied on a curved background, sometimes compact or rough. Understanding how to extend such tools, dealing with wave dispersion or focusing, will allow us to significantly progress in our mathematical understanding of physically relevant models. There, boundaries appear naturally and most earlier developments related to propagation of singularities in this context have limited scope with respect to crucial dispersive effects. Despite great progress over the last decade, driven by the study of quasilinear equations, our knowledge is still very limited. Going beyond this recent activity requires new tools whose development is at the heart of this proposal, including good approximate solutions (parametrices) going over arbitrarily large numbers of caustics, sharp pointwise bounds on Green functions, development of efficient wave packets methods, quantitative refinements of propagation of singularities (with direct applications in control theory), only to name a few important ones.