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Michela Egidi - Project


Title Multiscale Version of the Logvinenko-Sereda Theorem
DFG Code Ve 253/7-1
Principal Investigator Prof. Dr. Ivan Veselić
Researcher Dr. Michela Egidi


Summary

The aim of this project is to prove a multiscale version of the Logvinenko-Sereda Theorem. The classical Logvinenko-Sereda Theorem belongs to the realm of Harmonic or Fourier Analysis and asserts that the restriction of an L^p function on the real line to a thick subset has comparable L^p-norm to the function on the whole axis, provided that its Fourier transform is supported on an interval. Only the thickness and the size of the interval enter in the estimate, not its position. This result has been extended by Kovrijkine to the case where the Fourier transform has support in a union of a finite number of intervals of the same length. Again, only the number and the size of the intervals enters the estimate, not their position.
The validity of a multiscale version of the Logvinenko-Sereda Theorem is suggested by recent scale-free unique continuation estimates or uncertainty principles for eigenfunctions and spectral projections of Schrödinger operators. Here the functions are considered on intervals of length L, ranging over the positive reals. While the intended estimate appears at first sight simpler than in the case of the whole axis, one has the additional task to control effectively the dependence of the estimate on the additional size parameter L. Ideally, one would like to show that the estimates hold uniformly in L.
While the conjectured bound is a Harmonic Analysis result, it immediately triggers consequences in the theory of Inverse Problems, in particular under appropriate sparsity assumptions. Thus it can be seen as a continuum relative of compressed sensing and sparse recovery. Moreover, the conjectured estimates have applications in the spectral theory of Schrödinger operators and in the control theory of the heat equation. A multidimensional extension of this estimate will have even wider relevance.

Publications

Title: Sharp geometric condition for null-controllability of the heat equation on R d and consistent estimates on the control cost
Authors: Michela Egidi, Ivan Veselic
Year: 2017
Preprint https://arxiv.org/abs/1711.06088
Abstract: In this note we study the control problem for the heat equation on $R^d$, $d>= 1$, with control set a given subset of $R^d$. We provide a necessary and sufficient condition on the control set such that the heat equation is null-controllable in any positive time. We give an estimate of the control cost with explicit dependency on the characteristic geometric parameters of the control set. Finally, we derive a control cost estimate for the heat equation on cubes with periodic, Dirichlet, or Neumann boundary conditions, where the control sets are again assumed to be thick. We show that the control cost estimate is consistent with the $R^d$ case.

 

Title: Scale-free unique continuation estimates and Logvinenko-Sereda Theorems on the torus
Authors: Michela Egidi, Ivan Veselic
Year: 2016
Preprint https://arxiv.org/abs/1609.07020
Abstract: We study uncertainty principles or observability estimates for function classes on the torus. The classes are defined in terms of spectral subspaces of the energy or the momentum, respectively. In our main theorems, the support of the Fourier Transform of the functions is allowed to be supported in a (finite number of) parallelepipeds. The estimates we obtain do not depend on the size of the torus and the position of the parallelepipeds, but only on their size and number, and the size and scale of the observability set. Our results are on the one hand closely related to unique continuation and observability estimates which can be obtained by Carleman estimates and on the other hand to the Logvinenko and Sereda theorem. In fact, we rely on the methods used by Kovrijkine to refine and generalize the results of Logvinenko and Sereda.


Poster presentation

Title: Null-controllability of the heat equation on rectangular regions
Occasion: 5th Najman Conference on Spetral Theory and Differential Equations
Link coming soon
Date: September 2017


Talks

Title: Necessary and sufficient geometric condition for null-controllability of the heat equation on R^d
Speaker: Ivan Veselic
Occasion: Worskhop on Control theory of infinite-dimensional systems
Institution: FernUniversität Hagen
Date: 10th-12th January 2018

 

Title: Longvinenko-Sereda Theorems: from complex analysis to application in Control Theory
Speaker: Michela Egidi
Occasion: Worskhop on Control theory of infinite-dimensional systems
Institution: FernUniversität Hagen
Date: 10th-12th January 2018

 

Title: Logvinenko-Sereda Theorems and application to control theory of the heat equation
Speaker: Michela Egidi
Occasion: Group de travail
Institution: ENS Rennes
Date: 15th November 2017

 

Title: Unique continuation estimates and the Logvinenko Sereda Theorem
Speaker: Ivan Veselic
Occasion: Oberseminars Stochastik/Mathematische Physik
Institution: Fernuni Hagen
Date: 15th February 2017

 

Title: Logvinenko-Sereda Theorems for periodic functions
Speaker: Michela Egidi
Occasion: Summer school “Spectral Theory, Differential Equations and Probability“
Institution: Johannes Gutenberg Universität Mainz
Date: 4th-15th September 2016

 

Title: Uncertainty relations and applications to the Schrödinger and heat conduction equations
Speaker: Ivan Veselic
Occasion: Summer school “Spectral Theory, Differential Equations and Probability“
Institution: Johannes Gutenberg Universität Mainz
Date: 4th-15th September 2016

 

Title: Quantitative uncertainty principle on the torus
Speaker: Michela Egidi
Occasion: GGA Seminar
Institution: Durham University
Date: 23rd May 2016

Kontakt

TU Dortmund
Fakultät für Mathematik
Lehrstuhl IX
Vogelpothsweg 87
44227 Dortmund


Sie finden uns auf dem sechsten Stock des Mathetowers .


Sekretariat: Janine Textor
Raum M 620


E-mail:
janine.textor@tu-dortmund.de


Tel. (0231) 755-3063
Fax (0231) 755-5219