prof. alexandre kirilov
departamento de matemática
pseudodifferencial operators - 1st semester
place and time
- monday and wednesday at 10:00, pc05/pc03
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
- First list (deadline 21/03) - Fourier transform
- second list (deadline 14/04) - tempered distributions
- third list (deadline 28/04) - symbol classes
- fourth list (deadline 18/05) - composition and formal adjoint
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
- Weak Solutions of Pseudo-Differential Equations
- Gårding's Inequality
Lucas siqueira
- One-Parameter Semigroups Generated by Pseudo-Differential Operators
Wagner almeida
- Local solvability of linear differential operators
- Wave front sets of solutions of partial differential equations
alexandre kirilov
- Fredholm Pseudo-Differential Operators
- Symmetrically Global Pseudo-Differential Operators
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
- 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. - 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. - 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. - 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. - 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. - 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.
- 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.