{
  "id": "1912.07402",
  "version": "v1",
  "published": "2019-12-16T14:28:25.000Z",
  "updated": "2019-12-16T14:28:25.000Z",
  "title": "Propagation of smallness and control for heat equations",
  "authors": [
    "Nicolas Burq",
    "Iván Moyano"
  ],
  "categories": [
    "math.AP"
  ],
  "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 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 -- $\\delta$ ($\\delta$ > 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-16T14:28:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "heat equations",
      "propagation",
      "spectral projector estimates",
      "non zero lebesgue measure",
      "laplace operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}