## Notes of Oana Ivanovici’s Orsay lecture 28-06-2017

Geometry and analysis of waves in manifolds with boundary

The wave-front is a subset of the cotangent bundle, whose projection is the singular support. In all dimensions, in Euclidean space, it travels at constant speed along straight lines (Fermat,…, Hormander).

In general Riemannian manifolds without boundary, it travels along geodesics as long as time stays less than the injectivity radius (Duistermaat-Hormander).

We impose Dirichlet boundary conditions. Then transverse waves reflect according to Snell’s law of reflection (Chazarain). What about tangencies? Assume obstacle is convex. Do waves propagate in the shadow?

Melrose-Taylor 1975: if the boundary is ${C^\infty}$, no smooth singularities in the shadow region. However, analytic singularities occur.

Inside strictly convex domains, waves reflect a large number of times. The wave shrinks in size between two reflections, it refocusses, therefore its maximum increases. Caustics appear, together with swallowtail and cusp singularities.

In the non-convex case, especially if infinite order tangencies occur, one does not even know what the continuation of a ray should be (Taylor 1976).

1. Dispersive estimates

It is a measurement of the decay of amplitude of waves due to spreading out while energy is conserved.

In ${{\mathbb R}^d}$, after a high frequency cut-off around frequency ${\lambda}$, the maximum amplitude decays like ${\lambda^{(d+1)/2}t^{-(d-1)/2}}$. Indeed, the wave is concentrated in an annulus of width ${1/\lambda}$. The same holds in Riemannian manifolds without boundary.

In the presence of boundary, propagation of singularities has brought results in the 1980’s. Later on, people have tried a reduction to the boundary-less case with a Lipschitz metric: this requires no assumptions on the boundary, but ignores reflection and its refocussing effect.

1.1. Within convex domains

Theorem 1 (Ivanovici-Lascar-Lebeau-Planchon 2017) For strictly convex domains, dispersion is in

$\displaystyle \begin{array}{rcl} \lambda^d\max\{1,(\lambda t)^{-\frac{d-1}{2}+\frac{1}{4}}\}. \end{array}$

This follows from a detailed description of the wave-front, including swallow-tails. It takes into account infinitely many reflections. It is sharp.

1.2. Outside convex obstacles

The Poisson spot. This is a place where diffracted light waves interfere. It is in the shadow area, but much more light concentrates there. This was confirmed experimentally by Arago, following a debate launched by Fresnel who did not believe in the wave description of light. It should exist if one believes in Fermat’s principle that light rays follow geodesics, including those which creep along the boundary surface (Keller’s conjecture). In 1994, Hargé and Lebeau proved that, when light creeps along the bounday, it decays like ${e^{-\lambda^{1/3}}}$.

Theorem 2 (Ivanovici-Lebeau 2017) For strictly convex obstacles,

1. if ${d=3}$, dispersion estimates hold like in ${{\mathbb R}^3}$,
2. if ${d\geq 4}$, they fail at the Poisson spot.

The reason is that a ${d-2}$-dimensional surface lits the Poisson point.