On the dimension of observable sets for the heat equation

TL;DR

This paper proves that the heat equation on a bounded domain can be observed from any measurable set with Hausdorff dimension greater than n-1, using spectral estimates and propagation of smallness techniques.

Contribution

It establishes a sharp dimension threshold for observability of the heat equation and constructs sets with lower dimensions where observability still holds.

Findings

Observability holds for sets with Hausdorff dimension > n-1.

Spectral estimates for Laplace eigenfunctions are developed.

Sets with dimension less than n-1 can also be observable.

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.