Download this article
 Download this article For screen
For printing
Recent Issues
Volume 7, Issue 3
Volume 7, Issue 2
Volume 7, Issue 1
Volume 6, Issue 4
Volume 6, Issue 3
Volume 6, Issue 2
Volume 6, Issue 1
Volume 5, Issue 4
Volume 5, Issue 3
Volume 5, Issue 2
Volume 5, Issue 1
Volume 4, Issue 4
Volume 4, Issue 3
Volume 4, Issue 2
Volume 4, Issue 1
Volume 3, Issue 4
Volume 3, Issue 3
Volume 3, Issue 2
Volume 3, Issue 1
Volume 2, Issue 4
Volume 2, Issue 3
Volume 2, Issue 2
Volume 2, Issue 1
Volume 1, Issue 4
Volume 1, Issue 3
Volume 1, Issue 2
Volume 1, Issue 1
The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
 
Subscriptions
 
ISSN 2578-5885 (online)
ISSN 2578-5893 (print)
Author Index
To Appear
 
Other MSP Journals
On the dimension of observable sets for the heat equation

A. Walton Green, Kévin Le Balc’h, Jérémy Martin and Marcu-Antone Orsoni

Vol. 7 (2025), No. 3, 639–660
Abstract

We consider the heat equation on a bounded C1 domain in n with Dirichlet boundary conditions. Our primary aim 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.

Keywords
observability, heat equation, spectral estimate, propagation of smallness
Mathematical Subject Classification
Primary: 35A02
Secondary: 35K05, 35Q93, 58J35, 93B05
Milestones
Received: 12 November 2024
Revised: 3 April 2025
Accepted: 30 May 2025
Published: 21 July 2025
Authors
A. Walton Green
Washington University
St. Louis, MO
United States
Kévin Le Balc’h
Inria, Laboratoire Jacques-Louis Lions
Sorbonne Université, Université de Paris, CNRS
Paris
France
Jérémy Martin
Inria, Laboratoire Jacques-Louis Lions
Sorbonne Université, Université de Paris, CNRS
Paris
France
Marcu-Antone Orsoni
Université Laval
Quebec, QC
Canada