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Selected topics in partial differential equations
Pseudodifferential Operators and Microlocal Analysis
Class number: 250503
Class type: VO (lecture course)
Semester hours: 3
Time and place: Mon 13:15--14:55 2A310 (UZA 2);
Thu 13:15--14:55 D1.07 (UZA 4)
Start: 2.10.2006
General introduction: The theory of pseudodifferential operators
emerged in the mid 1960-ies through the work of Kohn and Nirenberg and
has earlier roots in Fourier analysis and singular integrals. It was
subsequently refined by a number of mathematicians, most notably by Lars Hörmander, and turned into one of the most
esstential tools in PDE. More precisely, it allows a flexible way of
applying Fourier techniques to the study of variable-coefficient operators
and singularities of distributions. By the latter we mean the wave front set, a
refinement of the notion of singular support, in the sense that also the
the "bad frequency directions'' of the distribution are taken into account---the
study of which is called microlocal analysis.
Applications of pseudodifferential operators are multitude. The most
prominent and classical is the propagation of the wave front set for
solutions of partial differential equations. However, over the years,
in particular pseudodifferential calculus has turned into a general method that
has successfully been applied in diverse fields of analysis, recently also
in time frequency analysis.
For a (slightly) more technical introduction and an outline
of the main aims of the course click here.
Contents: At leat for the first three quarters of the course we
will rather closely follow the book Elementary Introduction to the Theory
of Pseudodifferential Operators by Xavier Saint Raymond (Studies in
Mathematics, CRC-Press, Boca Raton, 1991). The main chapters will be:
- 0. Motivation and Introduction
- 1. Prerequisites: A short collection of the basic facts on Fourier transform and Sobolev spaces.
- 2. Pseudodifferential Symbols: Symbols and their asymptotic expansions, oscillatory integrals, operations on symbols.
- 3. Pseudodifferential operators: definition and action in S, S', and H^s, change of variables.
- 4. Applications: local solvability of linear PDEs, microlocal analysis, propagation of singularities, ...
Literature: A short commented list of books may be found here.
What you already should know: A solid background in analysis is indispensable. Further you need to be aquainted with distibution theory (approximately the contents of a 3-4 hours course, in particular tempered distributions, Fourier transform on S' and L^2, Sobolev spaces based on L^2--- there will be, however, a short and terse account on these topics (Ch. 1) to refresh your memory). You will also appreciate some basic background from topology, functional analysis (Lebesgue spaces) and PDE (classical equations).
Target audience: This course is designed for the doctoral college (IK) Mathematical Analysis and Applications: Time-frequency Analysis and Microlocal Analysis and can be seen as a continuation of my course on distribution theory from summer term 2006. Hence it serves both as an element of the PhD curriculum in the framework of the IK as well as a preparation for a dimploma thesis in analysis, in particular with the research group DiANA.
Remark: The course will normally be held 4 hours a week but suspended some weeks on (duly) announcement.
Position within curriculum: Doctoral college (IK) Mathematical Analysis and Applications: Time-frequency Analysis and Microlocal Analysis and Diplomstudium Mathematik, 2. Abschnitt, Studienschwerpunkt Analysis.
Exams: oral; by arrangement.