2019/12/16 by Nicolas Burq, Burq, Nicolas, Iván Moyano +1 · 6 citations
Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Stability and Controllability of Differential Equations
paper · doi:10.48550/arxiv.1912.07402
openalex publication_date 2019/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 from Logunov-Malinnikova [7, 6, 8] allows to extend the spectral projector type estimates from Jerison-Lebeau [3] from localisation on open set to localisation on arbitrary sets of non zero Lebesgue measure; we can actually go beyond and consider sets of non vanishing d -- δ (δ > 0 small enough) Hausdorf measure. We show that these new spectral projector estimates allow to extend the Logunov-Malinnikova's propagation of smallness results to solutions to heat equations. Finally we apply these results to the null controlability of heat equations with controls localised on sets of positive Lebesgue measure. A main novelty here with respect to previous results is that we can drop the constant coefficient assumptions (see [1, 2]) of the Laplace operator (or analyticity assumption, see [4]) and deal with Lipschitz coefficients. Another important novelty is that we get the first (non one dimensional) exact controlability results with controls supported on zero measure sets.