1. Observability of the heat equation from very small sets
- Author
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Green, A. Walton, Balc'h, Kévin Le, Martin, Jérémy, and Orsoni, Marcu-Antone
- Subjects
Mathematics - Analysis of PDEs ,Mathematics - Classical Analysis and ODEs ,Mathematics - Complex Variables ,35A02, 35Q93, 35K05, 58J35, 93B05 - 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.
- Published
- 2024