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.

Lectures

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 2 – Chapter 3 – Chapter 4 – Chapter 5

Tutorials

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 1 – Solutions – Divergence Theorem and Revision

Sheet 2 – Solutions – Distributions

Sheet 3 – Solutions – Mean Value Property and Maximum Principle

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

Sheet 5 – Solutions – Classical Maximum Principles, Banach Spaces

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

Sheet 7 – Solutions – Sobolev Functions

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

Sheet 9 – Solutions – Sobolev Inequalities, Gårding Inequality

Sheet 10 – Solutions – Weak Solutions – Graph of Ex37

Sheet 11 – Solutions – Regularity Theorems

Sheet 12 – Solutions – Schauder Estimates

Final Week – PDE 5.2  – Review of key proofs