Abstract

We consider the heat equation on a bounded $C^1$ domain in ${ℝ}^n$ with Dirichlet boundary conditions. Our primary aim is to prove that the heat equation is observable from any measurable set with a Hausdorff dimension strictly greater than $n-1$. The proof relies on a novel spectral estimate for linear combinations of Laplace eigenfunctions, achieved through the propagation of smallness for solutions to Cauchy–Riemann systems as established by Malinnikova, and uses the Lebeau–Robbiano method. While this observability result is sharp regarding the Hausdorff dimension scale, our secondary goal is to construct families of sets with dimensions less than $n-1$ from which the heat equation is still observable.

Keywords

Mathematical Subject Classification

Milestones

Authors