Observability inequalities for the heat equation exhibit an exponential cost at small times. Is this exponential behavior an artefact of the usual Carleman and spectral methods, or is it forced by the underlying geometry?
We show that it is genuinely geometric: any weighted observability inequality must satisfy an exponential decay rate controlled by the maximal distance to the observation set. As a consequence, we answer an open question raised by Ervedoza and Zuazua concerning geometric lower bounds on the exponential rate appearing in infinite-time integrated observability inequalities for heat equations. Our results are established in a general geometric metric-measure setting and apply, in particular, to Riemannian manifolds, Schrödinger operators, sub-Riemannian manifolds, and Laplacians on metric graphs.
Joint work with Amaury Hayat and Emmanuel Trélat: arXiv:2607.13279.
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