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arXiv · 2407.20954

On the dimension of observable sets for the heat equation

A. Walton Green·Kévin Le Balc'h·Jérémy Martin·Marcu-Antone Orsoni
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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.

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A. Walton Green, Kévin Le Balc'h, Jérémy Martin, Marcu-Antone Orsoni. 2024-07-30. On the dimension of observable sets for the heat equation. https://doi.org/10.2140/paa.2025.7.639

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