Elliptic PDE Lecture Notes and Problems (LTCC course 2026)
Lecturer: Dr. Shengwen Wang (QMUL)
Email: shengwen.wang@qmul.ac.uk
This is the lecture notes for LTCC course "Elliptic Partial Differential Equations (Advanced)" taught in fall Spring 2026. The course contains five 2-hour lectures. We will focus primarily on divergence-form elliptic PDEs, which arise naturally from variational problems and the variational structure could simplified some of the technical proofs. The plan for the course is as follows:
- In lecture 1, we gather some key regularity estimates for harmonic functions, which are extendable to more general elliptic PDEs.
Notes for Lecture 1 (updated)
- In lecture 2, we will present the Schauder estimates, which gives higher order H\"older regularity of solutions.
Notes for Lecture 2 (updated)
- In lecture 3, we will give some approaches to the existence theory of elliptic PDEs. In modern PDE, the existence and regularity theory are usually treated separately. One first obtain existence of weak solutions and then prove regularity of the solutions.
Notes for Lecture 3 (updated)
- In lecture 4, we will go over the De Giorgi - Nash - Moser theory, which provides the initial $L^\infty$ and H\"older regularity of the solutions before applying Schauder estimates.
- In lecture 5, we will talk about some applications in the geometric PDE of harmonic maps, e.g. the $\epsilon$ - regularity theorem and partial regularity theory.
Take-home exam: Choose any 2 questions from Problem sets 1 - 4 and write down the solutions/proofs.