Partial Differential Equations (PDEs) is a central part of mathematics, dealing with equations that involve partial derivatives, in contrast to ordinary differential equations which involve functions of a single variable.
Large parts of classical physics, and even much of modern physics, can be formulated in terms of PDEs. They also appear in many other areas, both within and outside mathematics.
In this second part of the course, we will introduce Sobolev spaces and use them to study weak solutions of elliptic and parabolic equations. We will also discuss some techniques that can be applied to show that these weak solutions enjoy additional regularity properties.
Sobolev spaces. Elliptic equations. Parabolic equations (if time permits). Regularity theory. Connection to Calculus of Variations
Tentative lecture plan
Lectures 1-3: Sobolev spaces, Chapter 5.1-5.7 in Evans
Lecture 4: Calculus of variation, Parts of Chapter 8 in Evans
Lectures 5-6: Elliptic PDEs, Chapter 6.1-6.5 in Evans, something about parabolic equations, Chapter 7.1 in Evans
Lecture 7: Regularity theory: Moser iteration and Widmans hole-filling technique, handwritten notes,
Lecture notes
Excercise problems (will be updated during the course) with some solutions.
Homework problems (now slightly revised with more hints)
Lecturer: Erik Lindgren, eriklin@kth.se
TA: Mina Farag
Lawrence C. Evans, Partial Differential Equations, second edition and handwritten lecture notes
Some material presented during the lectures are also taken from the following sources:
The use of artificial intelligence in this course is permitted only as a tool to support your learning.
The use of artificial intelligence is not permitted during any examination or assessment activities.
There will be four homework problems with deadline December 8, 2025 and a final oral exam is planned to take place December 15, 2025. The homework assignments will be posted in online, see assignments. Please mind the due date. We will not accept solutions handed in after the strict deadlines.
During the oral exam, you will be asked about the homework problems and you will be offered theoretical questions. For instance, you may be asked to derive a formula, prove a theorem from the lectures or complete a proof discussed during the course. The list of such possible exam questions can be found here.
Below is a schedule of possible slots for the oral exam. Please email me in order to reserve a slot.
Date 9:30 10:30 11:30 13:30
15 December free free taken taken
16 December free taken free taken
17 December taken taken taken taken