prof. alexandre kirilov

departamento de matemática

 

pseudodifferencial operators - 1st semester

 

place and time

description:

The class of pseudo-differential operators is a natural extension of the class differential operators, and contains most of the important operators. For example, if an elliptic differential operator has an inverse, this inverse is a pseudo-differential operator. Inside the class of pseudo-differential operators one can also take complex powers, and expression like (1-Δ)1/2 are well-defined pseudo-differential operators.

The lectures will start with an introduction of pseudo-differential operators on euclidean space. Here Fourier transformation is an important technique, and we will see how one can use the symbol calculus to invert many pseudo-differential operators, up to smoothing operators. Applications to partial differential equations will be developed. we also discuss adjoints and composition of operators, and regularity theory.

we will finish with a sequence of talks about problems of interest to the students, like: weak and strong solutions to Pseudo-Differential Equations; One-Parameter Semigroups Generated by Pseudo-Differential Operators, Local solvability of linear differential operators and Fredholm Pseudo-Differential Operators.

exercises

Seminars

In this series of talks, the goal of every speaker is talking about a topic of pseudo-differential operators directly related with their area of research:

Thiago formehl

Lucas siqueira

Wagner almeida

alexandre kirilov

 

references:

We closely follow the manuscript of Prof. Helmut Abels available in

http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/abels/PsDOSkriptWS0910.pdf

The following references are recommended

  1. Helmut Abels
    Pseudodifferential and singular integral operators. An introduction with applications
    (graduated lectures, de Gruyter, berlin, 2012)
    This is an improved version of the manuscript mentioned above. Here the author changed the order of presentation of some results, introduced a nice list of exercises and added new contents.
  2. X. Saint Raymond
    Elementary Introduction to the Theory of Pseudodifferential Operators
    (Studies in Advanced Mathematics, CRC Press, Boca Raton, 1991.)
    very nice introduction into the topic with some applications.
  3. M. w. Wong
    An Introduction to Pseudo-di?erential Operators
    (World Scientific, Singapore, 3rd ed., 2014)
    This text gives another careful introduction, very similar aim and scope as above.
  4. L. Hormander
    The Analysis of Linear Partial Differential Operators III, Pseudo-differential operators.
    (Springer, Berlin, 1994)
    A classic text that provides a tight introduction with precise but terse proofs.
  5. M. Shubin
    Pseudodifferential operators and spectral theory
    (Springer, Berlin, 2nd ed, 2001.)
    this book contains a short introduction to pseudodifferential operators and Fourier integral operators theory, with several applications to the spectral theory of linear elliptic equations. I have used this book extensively in recent years due to the accurate results about the asymptotic behavior of the spectrum of eliptic operators.
  6. Fabio Nicola, Luigi Rodino
    Global Pseudo-differential calculus on euclidean Spaces
    (Pseudo-Differential Operators. Theory and Applications, 4. Birkhäuser, Basel, 2010)
    This book is devoted to the theory of pseudo-differential operators defined globally on the Euclidean space, as well as applications in the realm of the general theory of PDEs, like global regularity, estimates in weighted Sobolev type spaces, the spectral theory and, in particular, decay and regularity properties of eigenfunctions etc.
  7. Hounie, J.
    Introdução aos Operadores Pseudo-diferenciais
    (16° CBM - impa - sbm, rio de janeiro, 1987)
    one of the few references in Portuguese on the topic.