We are concerned with the short-time observability constant of the heat equation from a subdomain of a bounded domain . The constant is of the form , where depends only on the geometry of and . Luc Miller (J. Differential Equations 204:1 (2004), 202–226) conjectured that is (universally) proportional to the square of the maximal distance from to a point of . We show in particular geometries that may blow up like when is a ball of radius , hence disproving the conjecture. We then prove in the general case the associated upper bound on this blowup. We also show that the conjecture is true for positive solutions of the heat equation.
The proofs rely on the study of the maximal vanishing rate of (sums of)eigenfunctions. They also yield lower and upper bounds for other geometric constants appearing as tunneling constants or approximate control costs.
As an intermediate step in the proofs, we provide a uniform Carleman estimate for Lipschitz metrics. The latter also implies uniform spectral inequalities and observability estimates for the heat equation in a bounded class of Lipschitz metrics, which are of independent interest.
"Observability of the heat equation, geometric constants in control theory, and a conjecture of Luc Miller." Anal. PDE 14 (2) 355 - 423, 2021. https://doi.org/10.2140/apde.2021.14.355