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Preprints, Working Papers, ... Year : 2024

Observability of the heat equation from very small sets

Abstract

We consider the heat equation set on a bounded $C^1$ domain of $\mathbb R^n$ with Dirichlet boundary conditions. The first purpose of this paper is to prove that the heat equation is observable from any measurable set $\omega$ with positive $(n-1+\delta)$-Hausdorff content, for $\delta >0$ arbitrary small. The proof relies on a new spectral estimate for linear combinations of Laplace eigenfunctions, obtained via a Remez type inequality, and the use of the so-called Lebeau-Robbiano's method. Even if this observability result is sharp with respect to the scale of Hausdorff dimension, our second goal is to construct families of sets $\omega$ which have codimension greater than or equal to $1$ for which the heat equation remains observable.
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hal-04666012 , version 1 (01-08-2024)

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Andrew Walton Green, Kévin Le Balc'H, Jérémy Martin, Marcu-Antone Orsoni. Observability of the heat equation from very small sets. 2024. ⟨hal-04666012⟩
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