Introductory description

Partial differential equations have always been fundamental to applied mathematics, and arise throughout the sciences, particularly in physics. More recently they have become fundamental to pure mathematics and have been at the core of many of the biggest breakthroughs in geometry and topology in particular. This course covers some of the main material behind the most common 'elliptic' PDE. In particular, we'll understand how analysis techniques help find solutions to second order PDE of this type, and determine their behaviour. Along the way we will develop a detailed understanding of Sobolev spaces.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Partial differential equations have always been fundamental to applied mathematics, and arise throughout the sciences, particularly in physics. More recently they have become fundamental to pure mathematics and have been at the core of many of the biggest breakthroughs in geometry and topology in particular. This course covers some of the main material behind the most common 'elliptic' PDE. In particular, we'll understand how analysis techniques help find solutions to second order PDE of this type, and determine their behaviour. Along the way we will develop a detailed understanding of Sobolev spaces.

Learning outcomes

By the end of the module, students should be able to:

• This course is most suitable for people who have liked the analysis courses in earlier years. It will be useful for many who intend to do a PhD, and essential for others. There are not too many prerequisites, although you will need some functional analysis, and some facts from Measure Theory will be recalled and used (particularly the theory of Lp spaces, maybe Fubini's theorem and the Dominated Convergence theorem etc.). It would make sense to combine with "MA3G1 Theory of PDEs", in particular the parts about Laplace's equation, in order to see the relevant context for this course, although this is not essential.

Subject specific skills

Students who have successfully taken the module will be able to investigate and solve second order partial differential equations using functional analysis techniques and also understand the regularity properties of the solutions obtained. They will obtain a solid grounding in Sobolev and Hölder spaces and understand the inequalities that relate them.

Transferable skills

They will be left with an excellent foundation for further study at graduate level, and will be able to apply the techniques they have learned in pure mathematics, the applied sciences and in many activities across engineering and finance.