Sobolev Spaces

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".

****Please let me know by email by 15th December 2017 if you intend to take the assessment for this course****

****There will be an extra session for the course on Tuesday 12th December starting at 10.15am ****

 

 

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.