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Open PDE & Analysis Seminar

Cyril Letrouit (Ecole Normale Supérieure)

Subelliptic wave equations are never observable

Friday 24th April 2020, 3pm (Paris time)

Abstract : In this talk, we explain a result we obtained recently, concerning the wave equation with a sub-Riemannian (i.e. subelliptic) Laplacian. Given a manifold $M$, a measurable subset $\omega\subset M$, a time $T_0$ and a subelliptic Laplacian $\Delta$ on $M$, we say that the wave equation with Laplacian $\Delta$ is observable on $\omega$ in time $T_0$ if any solution $u$ of $\partial_{tt}^2u-\Delta u=0$ with fixed initial energy satisfies $\int_0^{T_0}\int_\omega |u|^2dxdt\geq C$ for some constant $C>0$ independent on $u$. It is known since the work of Bardos-Lebeau-Rauch that the observability of the elliptic wave equation, i.e. with a Riemannian Laplacian, in time $T_0$ is almost equivalent to the geometric control condition (GCC), which stipulates that any geodesic ray meets $\omega$ within time $T_0$. We show that in the subelliptic case, as soon as $M\backslash \omega$ has non-empty interior and $\Delta$ is ``subelliptic but not elliptic", GCC is never verified, which implies that subelliptic wave equations are never observable. The proof is based on the construction of sequences of solutions of the wave equation concentrating on geodesics (for the distance on $M$ associated to $\Delta$) spending a long time in $M\setminus \omega$.

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Thomas Alazard (ENS Paris-Saclay)         Nicolas Burq (Orsay)         Iván Moyano (Nice)