Undergraduate Course: Elliptic Partial Differential Equations (MATH10089)
|School||School of Mathematics
||College||College of Science and Engineering
|Credit level (Normal year taken)||SCQF Level 10 (Year 4 Undergraduate)
||Availability||Available to all students
|Summary||The aim of the course is to give an introduction to elliptic partial differential equations and their applications in geometry, material sciences, biology and finance. The student will learn about important concepts such as the notion of a weak solution, the maximum principle as well as important theoretical results, for example the Krylov-
Examples of elliptic PDEs and various types of weak solutions.
Weak differentiability, Sobolev spaces and classical solutions.
Harmonic functions: Mean value theorem, Greens function Poisson kernel.
Maximum principle for general linear equations.
Divergence form equations, Caccioppoli's inequality, Morrey's theorem in two dimensions.
Non-divergence form equations, strong and viscosity solutions.
Aleksandrov's maximum principle.
Information for Visiting Students
Course Delivery Information
|Not being delivered|
| 1. Reformulate equations of divergence form through integral identities using partial integration.
2. Evaluate or estimate the maximum and minimum of solutions of elliptic equation using the maximum principle.
3. Infer regularity of solutions from that of given data.
4. Explicitly compute the (radially) symmetric solutions and Green/Poisson kernels for the Laplace operator.
5. Estimate first and second order derivatives of solutions via integral norms of solution itself.
6. Compare qualitative properties of weak and viscosity solutions.
E. Landis, Second Order Equations of Elliptic and Parabolic Type, AMS 1998
Qing Han, Fanghua Lin, Courant Institute Lecture notes, NYU, 2011
D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2011
|Graduate Attributes and Skills
|Course organiser||Dr Martin Dindos
|Course secretary||Mrs Alison Fairgrieve
Tel: (0131 6)50 5045