NMAK24002U Partial Differential Equations 2 (PDE2)

Volume 2024/2025

MSc Programme in Mathematics
MSc Programme in Mathematics with a minor subject


This advanced course on partial differential equations will be centered around methods for obtaining a priori estimates for elliptic equations, with applications to existence and uniqueness for such equations and proving regularity (such as differentiability) of their solutions.

We will in the main part of the course be dealing mostly with linear equations with variable coefficients, but always with a view towards applications to nonlinear equations (also with a few examples of this), giving proofs of all statements in high detail.

In the final part of the course, we will branch out with a lighter treatment of some modern advanced broader topics in PDEs.

Main Part (first 6 weeks):

  • Quick review of "PDE" material: Laplace equation, harmonic functions, second order elliptic equations in divergence form, Green’s functions, Poincaré-Sobolev Inequality.
  • Hopf's maximum principle, Serrin's comparison principle.
  • Gradient estimates via Bernstein's method (cut-offs and maximum principles).
  • Morrey’s and Campanato’s lemmas.
  • De Giorgi-Nash-Moser's iteration argument.
  • Moser's Harnack Inequality.
  • Regularity of weak solutions of second order elliptic equations in divergence form.
  • Hilbert's 19th Problem vs. De Giorgi's & Nash's solutions to it.
  • Nonlinear elliptic equations.
  • Alexandrov's maximum principle.
  • Alexandrov–Bakelman–Pucci estimates.
  • Method of moving planes, Gidas-Ni-Nirenberg theory.
  • Krylov–Safonov Hölder estimates and Harnack inequality.

Tapas of Topics in PDEs (last 2 weeks):

  • Nonlinear Schrödinger equations.
  • Quasi-linear heat equations.
  • Blow-up behavior at singular times for semilinear evolution equations: heat, wave and Schrödinger equations.
  • Examples from active research areas at MATH within PDEs, e.g. problems from quantum theory, mean curvature flow and variational problems in geometry such as minimal surfaces.
Learning Outcome
  • Knowledge: To display knowledge of the course topics and content.
  • Skills: To be able to use the acquired knowledge to read and understand current research papers.
  • Competences: The student should be able to apply the theory to solve problems of moderate difficulty within the topics of the course.

See Absalon for course literature. 

Literature may include:

  • Q. Han, F. Lin: Elliptic partial differential equations, Second Edition, Courant Lecture Notes, AMS (2011), 147 pages.
    ISBN 978-0-8218-5313-9.
  • Last 2 weeks' "Tapas of Topics": Own lecture notes and/or excerpts from various other sources.
MSc students, who have knowledge corresponding to the courses PDE or DifFun, as well as preferably AdVec and/or FunkAn.

Having academic qualifications equivalent to a BSc degree is recommended.
4 hours of lectures and 2 hours of exercises each week for 8 weeks.

Please note that to prepare a lecture is a substantial time commitment.
So participants should plan their time accordingly.
  • Category
  • Hours
  • Lectures
  • 32
  • Preparation
  • 158
  • Theory exercises
  • 16
  • Total
  • 206
Continuous feedback during the course of the semester
7,5 ECTS
Type of assessment
Continuous assessment
Type of assessment details
Three written assignments combined with two in-class oral presentations:
- One lecture (45 minutes) about a relevant topic (to be decided together with the lecturer or teaching assistant).
- One short presentation of a written assignment solution (to be decided together with the lecturer or teaching assistant).
For the final grade, the student's performance in the course is assessed as a whole, based on both the handed in written assignments and the in-class oral presentations.
All aids allowed
Marking scale
7-point grading scale
Censorship form
No external censorship
One internal examiner

30 minutes oral exam without preparation time and without aids.

Criteria for exam assesment

The student should convincingly and accurately demonstrate the knowledge, skills and competences described under Intended learning outcome.