Multiscale version of the Logvinenko-Sereda Theorem

• Applicant: Professor Dr. Ivan Veselic
• Subject Area: Mathematics
• Term: from 2015 to 2019
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 280969390

• Final Report Year: 2019

Final Report Abstract

The research project obtained two classes of results in mathematical analysis. The first part of the project was devoted to Fourier analysis, more precisely to uncertainty relations. The core result is an generalisation of Kovrijkine’s lower bounds for certain types of bandlimited functions on Lp seminorms associated to so called thick subsets. While the original results of Kovrijkine apply to (holomorphic) functions on the whole euclidean space, the estimates obtained in the project consider functions (Fourier series) defined on boxes. The same holds also for certain mixed geometries like strips, which are bounded in certain coordinate directions and unbounded in others. The second part of the project is devoted to an application of uncertainty relations to the null-control problem of the heat equation on the whole euclidean space and on large finite boxes. The control is allowed to act on subsets which are thick, as studied in the first part of the project, and which are equidistributed, as studied in the earlier DFG funded project. It was shown that thickness of the observation set is a necessary and sufficient criterion for the null controllability of the heat equation on the whole space to hold. Furthermore, explicit control cost estimates in term of geometric parameters were obtained. An important intermediate step was a operator theoretic result on control cost estimates which improved previously known statements.

Publications

• Scale-free unique continuation estimate and Logvinenko-Sereda theorem on the torus
  M. Egidi and I. Veselić
  (See online at https://doi.org/10.1007/s00023-020-00957-7)
• Exhaustion approximation for the control problem of the heat or schr\”odinger semigroup on unbounded domains
  A. Seelmann and I. Veselić
  
• On null-controllability of the heat equation on infinite strips and control cost estimate
  M. Egidi
  
• Sharp estimates and homogenization of the control cost of the heat equation on large domains
  I. Nakić, M. Täufer, M. Tautenhahn, and I. Veselić
  (See online at https://doi.org/10.1051/cocv/2019058)
• Sharp geometric condition for null-controllability of the heat equation on R^d and consistent estimates on the control cost. Arch. Math. (Basel), 111(1):85–99, 2018
  M. Egidi and I. Veselić
  (See online at https://doi.org/10.1007/s00013-018-1185-x)
• The reflection principle in the control problem of the heat equation
  M. Egidi and A. Seelmann
  
• Null controllability and control cost estimates for the heat equation on unbounded and large bounded domains
  M. Egidi, I. Nakić, A. Seelmann, M. Täufer, M. Tautenhahn, and I. Veselić
  (See online at https://doi.org/10.1007/978-3-030-35898-3_5)