Spectral estimates, heat observability, and thickness on Riemannian manifolds.

par Alix Deleporte (LMO - Université Paris Saclay)

lundi 29 sept. 2025, 14:00 → 15:00 Europe/Paris

Amphi Schwartz

Eigenfunctions of the Laplacian cannot vanish on a set of positive measure. Quantitative versions of this
unique continuation are well-known on fixed Riemannian manifolds : the L2 norm of an eigenfunction is
bounded by its L2 norm on a set of positive measure times a constant which grows exponentially with the frequency. This growing rate is sharp and reflects in observability properties for the heat equation.

In this talk, I will present recent results, in collaboration with M. Rouveyrol (Uni. Bielefeld) about these questions in a non-compact setting, and/or uniformly with respect to the metric. Quantitative unique continuation, and observability of the heat equation, hold under a necessary and sufficient condition of thickness of the observed set : it must intersect every large enough metric ball with a mass bounded from below, proportionally to the mass of the ball itself. 

I will talk about the case of non-compact hyperbolic surfaces, then about much more general Riemannian manifolds (in progress!). The proof crucially uses the Logunov-Mallinikova estimates.