B4.3 Distribution Theory - Material for the year 2020-2021

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Prof. Jan Kristensen
General Prerequisites: 

Part A Integration is essential. A good working knowledge of Part A core Analysis is expected. Part A Integral Transforms and Introduction to Manifolds are desirable but not essential.

Course Term: 
Course Weight: 
1.00 unit(s)
Course Level: 

Assessment type:

Course Overview: 

Distribution theory can be thought of as the completion of differential calculus, just as Lebesgue integration theory can be thought of as the completion of integral calculus. It was created by Laurent Schwartz in the 20th century, as was Lebesgue's integration theory.

In this course we give an introduction to distributions. It builds on core material in analysis and integration studied in Part A. One of the main areas of applications of distributions is to the theory of partial differential equations, and a brief treatment, mainly through examples, is included. A more systematic study is deferred to Fourier Analysis and PDEs.

Learning Outcomes: 

Students will become acquainted with the basic techniques that in many situations form the starting point for the modern treatment of PDEs.

Course Synopsis: 

Test functions and distributions on $\mathbb{R}^n$: definitions and examples, Dirac $\delta $- function, approximate identities and constructions using convolution of functions. Density of test functions in Lebesgue spaces. Smooth partitions of unity. [4 lectures]

The calculus of distributions on $\mathbb{R}^n$ : functions as distributions, operations on distributions, adjoint identities, consistency of derivatives, convolution of test functions and distributions. The Fundamental Theorem of Calculus for distributions. Support and singular support of a distribution. [6 lectures]

Examples of distributions de fined by principal value integrals and fi nite parts. Examples of distributional boundary values of holomorphic functions defi ned in a half plane. [2 lectures]

Distributional and weak solutions of PDEs, absolutely continuous functions, Sobolev functions. Examples of fundamental solutions. Weyl's Lemma for distributions. [4 lectures]

Reading List: 

The main recommended book is

1) R.S. Strichartz, A Guide to Distribution Theory and Fourier Transforms
(World Scientific, 1994. Reprinted: 2008, 2015)
In particular, Chapters 1, 2 and 6.

Further Reading: 

2) L.C. Evans, Partial Diļ¬€erential Equations (Amer. Math. Soc. 1998)
3) E.H. Lieb and M. Loss, Analysis (Amer. Math. Soc. 1997)
4) E.M. Stein and R. Shakarchi, Fourier analysis. An introduction (Princeton Univ. Press 2003)