We study stationary solutions of the damped wave equation on a compact and smooth Riemannian manifold without boundary. In the high frequency limit, we prove that a sequence of $\beta$-damped stationary solutions cannot be completely concentrated in small neighborhoods of a small fixed hyperbolic subset made of $\beta$-damped trajectories of the geodesic flow.The article also includes an appendix (by S. Nonnenmacher and the author) where we establish the existence of an inverse logarithmic strip without eigenvalues...

We study asymptotic properties of eigenfunctions of the Laplacian on compact Riemannian surfaces of Anosov type (for instance negatively curved surfaces). More precisely, we give an answer to a question of Anantharaman and Nonnenmacher [4] by proving that the Kolmogorov-Sinai entropy of a semiclassical measure $\mu$ for the geodesic flow $g^t$ is bounded from below by half of the Ruelle upper bound. (This text has been written for the proceedings of the ${37}^{\text{èmes}}$ Journées EDP (Port d’Albret-June, 7-11 2010))

We extend a result of the second author [27, Theorem 1.1] to dimensions $d\ge 3$ which relates the size of $L^p$-norms of eigenfunctions for $2<p<2(d+1)/d-1$ to the amount of $L^2$-mass in shrinking tubes about unit-length geodesics. The proof uses bilinear oscillatory integral estimates of Lee [22] and a variable coefficient variant of an "$ϵ$ removal lemma" of Tao and Vargas [35]. We also use Hörmander’s [20] $L^2$ oscillatory integral theorem and the Cartan–Hadamard theorem to show that, under the assumption of nonpositive curvature,...

This paper is a proceedings version of the ongoing work [20], and has been the object of the talk of the second author at Journées EDP in 2012.In this work we investigate optimal observability properties for wave and Schrödinger equations considered in a bounded open set $\Omega \subset {ℝ}^n$, with Dirichlet boundary conditions. The observation is done on a subset $\omega$ of Lebesgue measure $\left|\omega \right|=L\left|\Omega \right|$, where $L\in (0,1)$ is fixed. We denote by ${𝒰}_L$ the class of all possible such subsets. Let $T\&gt;0$. We consider first the benchmark problem of maximizing...

We consider the wave and Schrödinger equations on a bounded open connected subset $\Omega$ of a Riemannian manifold, with Dirichlet, Neumann or Robin boundary conditions whenever its boundary is nonempty. We observe the restriction of the solutions to a measurable subset $\omega$ of $\Omega$ during a time interval $[0,T]$ with $T>0$. It is well known that, if the pair $(\omega ,T)$ satisfies the Geometric Control Condition ($\omega$ being an open set), then an observability inequality holds guaranteeing that the total energy of solutions can be...

Following joint work with Dyatlov [DyGu], we describe the semi-classical measures associated with generalized plane waves for metric perturbation of ${ℝ}^d$, under the condition that the geodesic flow has trapped set $K$ of Liouville measure $0$.