Partial Differential Equations

This course will cover elliptic partial differential equations, both linear and non-linear, with an emphasis on Sobolev functions and “a priori” estimates.


Unless there is sufficient student demand for in person lectures, the lectures will be in the form of recorded videoes, which can be found here.

The script is available as a pdf:

Chapter 1  – Chapter 2Chapter 3Chapter 4Chapter 5


The tutorials will take place on Tuesdays in Block 5 (15:30–17:00) in the seminar room on level 4 of Building B6 Section C (near Prof Schmidt and my offices). There will be no tutorial in the first week of semester.

Every week there will be an exercise sheet, which you should work on and submit for feedback. Exercise sheets are due Monday noon before the tutorial. So the first sheet is due Monday 20th Feb, before the first tutorial on Tuesday 21st Feb.

Sheet 1Solutions – Divergence Theorem and Revision

Sheet 2Solutions – Distributions

Sheet 3Solutions – Mean Value Property and Maximum Principle

Sheet 4Solutions – Weyl's Lemma, Poisson Representation Formula, Classical Maximum Principles – Graph of “Generalised Mollifier”

Sheet 5Solutions – Classical Maximum Principles, Banach Spaces

Sheet 6Solutions – Fixed Point theorems, Hölder Functions - Graph of Hölder constant

Sheet 7Solutions – Sobolev Functions

Sheet 8Solutions – More Sobolev Functions – Graph of Approximation of Absolute Value Function

Sheet 9Solutions – Sobolev Inequalities, Gårding Inequality

Sheet 10Solutions – Weak Solutions – Graph of Ex37

Sheet 11Solutions – Regularity Theorems

Sheet 12Solutions – Schauder Estimates

Final Week – PDE 5.2  – Review of key proofs