MATP16 Partial Differential Equations is an optional course for a Master of Science degree in mathematics.

It covers first order quasi-linear equations, classification into types of second-order equations, the Cauchy-Kovalevskaya theorem, Laplace's equation, the wave equation and the heat equation, Sobolev spaces, the Lax-Milgram theorem and elliptic equations, maximum principles for elliptic equations, evolution equations.

Teaching
The teaching consists of lectures, seminars. Lecturer is Erik Wahlén.

Assessment
The course is assessed through a written and an oral examination.

Due to Covid-19, the written examination is replaced by a take-home exam from 8.00-14.00 on May 29. The exam will be given as an assignment in Canvas. Oral exams will take place through Zoom. Please see the following page for more information.

The resit exam on August 21 will be a regular written exam in physical form. The exam is from 8.00-13.00 in rooms 309 A, B and C.

Course literature
Lawrence C. Evans, Partial Differential Equations, 2nd edition, 2010, ISBN: 978-0821849743.

Erik Wahlén, Fourier methods and distributions, 2020 (updated April 10)

Schedule
From March 24 we have lectures online, Tuesdays, 15.15-17.00 in Zoom and seminars Thursdays, 10.15-12.00, every second week (April 2, April 16, ...).

You can access the Zoom lectures here.

You can access the Zoom seminar here.

You will find a module for each lecture and seminar on the left.

Official Course Description

Course program
Course program 1 for the first weeks (updated Feb 24) .

Course program 2 for March (updated April 1).
Course program 3 for April

Course program 4 for May

The course will cover selected part of chapters 1-7 in Evans. In addition, we will use the lecture notes on Fourier methods and distribution theory above.

Exercises

Additional exercises

Summary of the course
The following summary can be useful when studying for the written and oral exams.

Course evaluation

Evaluation report from this year.

Evaluation report from the last time the course was offered (2018).

Links
Errata for Evans' book

Tychonoff's counterexample to uniqueness for the heat equation can be found here (in French). In my example I should have written $LaTeX: g\left(t\right)=e^{-t^{-\alpha}},\:t>0$ and $LaTeX: g\left(t\right)=0,\:t\le0$ with $LaTeX: \alpha>1$ (this is from Fritz John's book)

Uniqueness of non-negative solutions of the heat equation on the real line was originally shown by Widder (available through Lund University)

A nice article about modelling vibrating strings (available through Lund University)

The article by G. Aronsson mentioned by Evans in Exercise 1 of Chapter 4.7