Propagation of smallness and control for heat equations

Abstract

In this note we investigate propagation of smallness properties for solutions to heat equations. We consider spectral projector estimates for the Laplace operator with Dirichlet or Neumann boundary conditions on a Riemanian manifold with or without boundary. We show that using the new approach for the propagation of smallness of Logunov and Malinnikova (2018) allows one to extend the spectral projector type estimates of Jerison and Lebeau (1999) from localisation on open sets to localization on arbitrary sets of non-zero Lebesgue measure; we can actually go beyond and consider sets of non-vanishing $d-\delta$ ($\delta >0$ small enough) Hausdorff measure. We show that these new spectral projector estimates allow one to extend Logunov–Malinnikova’s propagation of smallness results to solutions to heat equations. Finally, we apply these results to the null controllability of heat equations with controls localized on sets of positive Lebesgue measure. The main novelty here is that we can drop the constant coefficient assumptions of Apraiz et al. (2013, 2014) on the Laplace operator (or the analyticity assumption of Escauriaza et al. (2017) and Lebeau and Moyano (2019)) and deal with Lipschitz coefficients. Another important novelty is that we get the first (non-one-dimensional) exact controllability results with controls supported on measure zero sets.