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.

Exam

The exam has been scheduled to take place on 15 June in the morning. Please contact Ross for further information if have not yet received an appointment.

Lectures

The lectures will take the form of recorded videos. A playlist of these videos can be found here.

The first chapter and the sections 2.1-2.3 coincide with the second chapter and parts of the sections 3.1-3.4 of the lecture Introduction to PDE respectively. We are re-using those corresponding videos recorded for Intro PDE for this course. Because the Intro PDE lecture videos are in English and use a translated version of the script, we have included that below also, to make it easier to follow along. The remainding videos are in German.

It is expected that students watch the videos consistently throughout the semester in order to be able to participate in the tutorials. A suggested schedule of which videos to watch in which week will be published alongside the exercises.

The script is available as a pdf:

Chapter 1  – Chapter 2Chapter 3Chapter 4Chapter 5

Chapter 2 intro pdeChapter 3 intro pde

Since Prof Schmidt has a sabbatical this semester, the “question hour” will only be by explicit request. It is scheduled for Thursday Block 1 (8:30-11:00). Please email Prof Schmidt before lunchtime Wednesday if you want to have a question hour. It is of course possible to also ask questions in the tutorials.

Tutorials

The tutorials will take place on Tuesdays in Block 5 (15:30-17:00). We have moved the tutorial to the seminar room on level 4 of Building B6 Section C (near Prof Schmidt and my offices). The tutor is Ross Ogilvie [r.ogilvie uni-mannheim.de]. Please send him an email if you have not already done so, so that he can easily contact all students.

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.

Sheet 1Solutions – Intro PDE sections 2.1 and 2.2 – Divergence Theorem and Revision

Sheet 2Solutions – Intro PDE sections 2.3 and 2.4 – Distributions

Sheet 3Solutions – Intro PDE sections 3.1, 3.2 part 1, 3.3 – Mean Value Property and Maximum Principle

Sheet 4Solutions – Intro PDE sections 3.2 part 2, 3.4, PDE 2.4 part 1 – Weyl's Lemma, Poisson Representation Formula, Classical Maximum Principles – Graph of “Generalised Mollifier”

Sheet 5Solutions – PDE 2.4 part 2 and 3, PDE 3.1 part 1 – Classical Maximum Principles, Banach Spaces

Sheet 6Solutions – PDE 3.1 part 2 and 3, PDE 3.2 – Fixed Point theorems, Hölder Functions – Graph of Hölder constant

Sheet 7Solutions – PDE 3.3 part 1,2 and 3 – Sobolev Functions

Sheet 8Solutions – PDE 3.3 part 4, PDE 3.4 part 1 and 2 – More Sobolev Functions – Graph of Approximation of Absolute Value Function

Sheet 9Solutions – PDE 3.4 part 3, PDE 4.1 part 1 – Sobolev Inequalities, Gårding Inequality

Sheet 10Solutions – PDE 4.1 part 2 and 3 – Weak Solutions – Graph of Ex36, Cross section of that graph and derivatives

Sheet 11Solutions – PDE 4.2 part 1, 2 and 3 – Regularity Theorems

Sheet 12 – Solutions – PDE 4.2 part 4, PDE 5.1 – Schauder Estimates

Final Week – PDE 5.2  – Review of key proofs