Pseudodifferentialoperatorer

**Valid from:** Autumn 2021**Decided by:** Professor Thomas Johansson**Date of establishment:** 2021-04-15

**Division:** Mathematics**Course type:** Third-cycle course**Teaching language:** English

Throughout mathematics and physics, linear elliptic partial differential equations make their appearance. A natural class of operators arising in this context is that of pseudodifferential operators, which after the fact can be described in terms of the class of operators of the form A=f(D) for reasonable functions f and some elliptic partial differential operator D. In particular, the class of pseudodifferential operators contain parametrices and solution operators to elliptic partial differential equations. The purpose of the course is to introduce the details of pseudodifferential calculus, sometimes reducing deep questions about partial differential equations to first year calculus, and to study its consequence in the global analysis of elliptic partial differential equations.

*Knowledge and Understanding*

For a passing grade the doctoral student must

- be able to present a coherent picture of the fundamentals of pseudodifferential calculus, the notion of ellipticity and the construction of local as well as global parametrices of elliptic operators.
- be able to justify functional analytic and spectral properties of elliptic operators as operators on Sobolev spaces.
- be able to explain the relevant features of an elliptic operator making it Fredholm and how its index is determined by local data.

*Competences and Skills*

For a passing grade the doctoral student must

- be able to solve elliptic partial differential equations up to smooth errors.
- be able to perform symbolic computations and functional calculus with pseudodifferential operators starting both from a symbolic presentation and an integral kernel presentation.
- be able to prove Fredholm properties for simpler boundary value problems.

In this course we will study pseudodifferential operators with a view towards global results. Since the 1960s pseudodifferential operators have been used for the study of elliptic differential operators. The highlights from this include the Weyl law describing their spectral behaviour and Atiyah-Singer’s index theorem computing their index. After working through the basic methods of algebraic and analytic flavour, we study their operator theoretic consequences. We shall also consider applications in the study of boundary value problems and Hörmander’s proof of the Weyl law, asymptotically describing the eigenvalues of elliptic pseudodifferential operators.

- Shubin, M.A.: Pseudodifferential Operators and Spectral Theory. Springer Science & Business Media, 2001. ISBN 9783540411956.
- Hörmander, L.: The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators. Springer Science & Business Media, 2007. ISBN 9783540499374.

Chapters XVIII and XX in Shubin's book and Chapters XVIII and XX in Hörmander.

**Type of instruction:** Lectures

**Examination formats:** Oral exam, written assignments**Grading scale:** Failed, pass**Examiner:**

**Assumed prior knowledge:** Basic course on partial differential equations , e.g. FMAN55 Kontinuerliga system.
Fourier analysis, basic functional analysis, distribution theory and differential geometry.

**Course coordinators:**