This course is run as part of the Mathematics Taught Course
Centre organised in collaboration with Bristol, Imperial, Oxford and Warwick.
This is an introductory course for research students in pure or applied
mathematics who have some basic familiarity with Lebesgue
integration and Functional Analysis. (Follow the links for a summary of
background results that will be used in this course.)

Course assessment.

The assessment for this course consists of submitting solutions to the problems sheets (these should all have been submitted by Friday 26th January 2018) and completing a take home test over the weekend of 28th-29th January 2018. The outcome of the assessment is "pass" or "fail".

Week 1

**Lecturer**: Prof Jey Sivaloganathan,
University of Bath

**Topics**

- Distributions and their partial
derivatives
- Distributional
derivatives of functions
- Weak Solutions of
partial differential equations
- Sobolev Spaces
- Convolution and
mollification
- Density of smooth
functions in Sobolev Spaces
- The Sobolev
Inequality
- The Poincare Inequality
- Morrey's Inequality
- Rellich-Kondrachov compact embedding
theorem
- Simple examples on
existence of weak solutions
- Boundary values of Sobolev functions
- Use of weak compactness
methods

**Texts:**

R.A. Adams and J.J.F. Fournier, *Sobolev** Spaces* (2nd edition. Elsevier
2003).

L.C. Evans,* Partial Differential Equations *(2nd
edition American Math. Soc. 2010)

A comprehensive work is:

V.G. Maz'ya, *Sobolev**
Spaces*, Springer.