On the dimension of observable sets for the heat equation
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866913953560395776 |
|---|---|
| author | Green, A. Walton Balc'h, Kévin Le Martin, Jérémy Orsoni, Marcu-Antone |
| author_facet | Green, A. Walton Balc'h, Kévin Le Martin, Jérémy Orsoni, Marcu-Antone |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20954 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the dimension of observable sets for the heat equation Green, A. Walton Balc'h, Kévin Le Martin, Jérémy Orsoni, Marcu-Antone Analysis of PDEs Classical Analysis and ODEs Complex Variables 35A02, 35Q93, 35K05, 58J35, 93B05 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. |
| title | On the dimension of observable sets for the heat equation |
| topic | Analysis of PDEs Classical Analysis and ODEs Complex Variables 35A02, 35Q93, 35K05, 58J35, 93B05 |
| url | https://arxiv.org/abs/2407.20954 |