We consider the heat equation on a bounded
domain
in
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
. 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
from
which the heat equation is still observable.
Keywords
observability, heat equation, spectral estimate,
propagation of smallness