Abstract: The propagation of waves in periodic media has known a regain of interest for many important applications, particularly in optics for micro and nano-technology. Indeed, in some frequency ranges, periodic structures behave as insulators or filters: the corresponding monochromatic waves, also called Floquet modes, cannot propagate in the bulk. The study of these modes is, from a mathematical point of view, related to the spectrum of the underlying operator that presents so-called band structures: the spectrum may contain some forbidden frequency intervals, called band gaps. Using the Floquet theory, we will prove that the spectrum of periodic operators (i.e. operators with periodic coefficients) has this band structure. In 1D, it is well-known that a periodic operator has gaps unless it is constant. By contrast, in 2d and 3d, a periodic operator might or might not have gaps. Even if necessary conditions for the existence of band gaps are not known, in lots of papers sufficient conditions are proposed. We will see that playing with the (high) contrast of the coefficients or the shape of the boundary of the medium, gaps can be created.
In the presence of a boundary, an interface or more generally a lineic perturbation in a periodic medium, energy localization can be created. This is due to some monochromatic waves that propagate along the perturbation and are localized transversely to the perturbation, if the frequency is inside the spectral gap of the periodic medium. Such phenomena can be exploited in quantum, electronic or photonic device design. In the mathematical literature, sufficient conditions on the periodic media and the perturbations have been proposed in order to ensure the existence of such localized and guided waves. We will study various methods used to derive such conditions.
In general, the theoretical results will be illustrated by numerical results. We will describe the associated numerical methods if time permits.

Sonia Fliss (ENSTA PARISTECH)

Abstract: This course is an introduction to the question of unique continuation for differential operators, and some of its applications to control theory. The general question is whether a solution to P u=0 in a domain such that u vanishes on a subdomain has to vanish everywhere. The focus will be put on the wave operator.
During the course, we shall first review the classical local Hörmander theorem which holds under a pseudoconvexity condition. We shall discuss its advantages and weaknesses.
We shall then specialize our analysis to the case of a wave operator with time-independent coefficients. In this setting we shall prove that local unique continuation holds across any non-characteristic surface (as proved by Tataru, Robbiano-Zuily and Hörmander). This local result implies a global unique continuation statement which can be interpreted as a converse to finite propagation speed.
We shall finally discuss applications such as approximate controllability.

Matthieu Léautaud (Université Paris-Sud)

Abstract: We consider coefficient determination problems for hyperbolic partial differential equations. A typical problem of this type is to determine a spatially varying speed of sound in an acoustic wave equation given the Neumann-to-Dirichlet map associated to the equation. These problems are motivated by applications, for example, in geophysical imaging. There are two traditional approaches to solve coefficient determination problems for wave equations: the Boundary Control method originating from [1] and a method using geometric optics. The former applies only to equations with time-independent coefficients, but gives very strong results in this case, whereas the latter allows for time-dependent coefficients [2], but requires certain convexity from the geometric setup. We give an introduction to both the approaches in simple settings and then discuss recent developments of coefficient determination problems.
References
[1] Belishev, M. I. (1987). An approach to multidimensional inverse problems for the wave equation. Doklady Akademii Nauk SSSR, 297(3), 524–527.
[2] Stefanov, P. D. (1989). Uniqueness of the multi-dimensional inverse scattering problem for time dependent potentials. Math Z, 201(4), 541–559.

Lauri Oksanen (University of Helsinki)