On the dimension of observable sets for the heat equation

Abstract

We consider the heat equation on a bounded $C^1$ domain in ${\mathbb{R}}^n$ with Dirichlet boundary conditions. The primary aim of this paper 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.

References in corpus (2)

• On the null-controllability of the heat equation in unbounded domains
• Propagation of smallness for solutions of elliptic equations in the plane