General Prerequisites:
Part A Integration is essential. A good working knowledge of Part A core Analysis is expected. Part A Integral Transforms, and ASO: Multidimensional Analysis and Geometry (previously called: Introduction to Manifolds) are desirable but not essential.
Course Term: Michaelmas
Course Lecture Information: 16 lectures
Course Weight: 1
Course Level: H
Assessment Type: Written Examination
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.
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 \(Rn\): definitions and examples, Dirac δ- 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 \(Rn\) : 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. Convolution with a compactly supported distribution.
Examples of distributions defined by principal value integrals and finite parts. Examples of distributional boundary values of holomorphic functions defined in a half plane. [8 lectures]
Distributional and weak solutions of PDEs, absolutely continuous functions, Sobolev functions. Examples of fundamental solutions. Weyl's Lemma for distributions. Convolution rules for support and singular support. [4 lectures]