On the dimension of observable sets for the heat equation

2024/07/30 by Green, A. Walton, Balc'h, Kévin Le, Martin, Jérémy +1 · 3 citations
#35A02 #35K05 #35Q93 #58J35 #93B05 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2407.20954

Abstract

We consider the heat equation on a bounded C¹ domain in ℝⁿ 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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