Undergraduate Course: Theory of Elliptic Partial Differential Equations (MATH11184)
Course Outline
| School | School of Mathematics |
College | College of Science and Engineering |
| Credit level (Normal year taken) | SCQF Level 11 (Year 5 Undergraduate) |
Availability | Available to all students |
| SCQF Credits | 10 |
ECTS Credits | 5 |
| Summary | NB. This course is delivered *biennially* with the next instance being in 2025-26. It is anticipated that it would then be delivered every other session thereafter.
The partial differential equations (PDEs) plays a central role in many areas of modern science. This course focuses on elliptic PDEs and introduces the basics of modern theory of such PDEs. This course generalises many key ideas from complex analysis - such as harmonic functions, the maximum principle, and boundary value problems - though prior knowledge of Complex Analysis is not required. |
| Course description |
- How questions in physics and mechanics give rise to different types of elliptic PDEs.
- Harmonic functions: Mean value theorem, gradient estimates, the Fundamental solution and the Green's function.
- Maximum principle for general linear elliptic equations.
- Weak differentiability, Sobolev spaces, and Sobolev inequalities.
- The concept of weak solutions, the Lax-Milgram theorem, well-posedness of the Dirichlet and Neumann boundary value problems.
- Regularity theory of weak solutions.
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Information for Visiting Students
| Pre-requisites | This is a Year 4/5 advanced, Honours level course. Visiting students are expected to have an academic profile equivalent to at least the first three years of the BSc (Hons) Mathematics programme (UTMATHB), and should be confident that their background has equipped them to undertake a course at this level. Students should have passed courses equivalent to Honours Analysis (MATH10068); or Algebra and Calculus (PHYS08041); or Linear Algebra and Several Variable Calculus (PHYS08042). |
| High Demand Course? |
Yes |
Course Delivery Information
|
| Academic year 2025/26, Available to all students (SV1)
|
Quota: None |
| Course Start |
Semester 1 |
Timetable |
Timetable |
| Learning and Teaching activities (Further Info) |
Total Hours:
100
(
Lecture Hours 22,
Seminar/Tutorial Hours 5,
Summative Assessment Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
69 )
|
| Assessment (Further Info) |
Written Exam
80 %,
Coursework
20 %,
Practical Exam
0 %
|
| Additional Information (Assessment) |
Coursework 20%, Examination 80% |
| Feedback |
Not entered |
| Exam Information |
| Exam Diet |
Paper Name |
Minutes |
|
| Main Exam Diet S1 (December) | MATH11184 Theory of Elliptic Partial Differential Equations | 120 | |
Learning Outcomes
On completion of this course, the student will be able to:
- Demonstrate understanding of Sobolev spaces and their relations to other spaces of functions.
- Reformulate equations of divergence form through integral identities using partial integration so that the Lax-Milgram theorem can be applied.
- Evaluate or estimate the maximum and minimum of solutions of elliptic equation using the maximum principle.
- Infer regularity of solutions from that of given data.
- Explicitly compute the Green/Poisson kernels for the Laplace operator in radially symmetric case and the upper half-space. Estimate first and second order derivatives of solutions via integral norms of solution itself.
|
Reading List
Recommended :
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 |
Additional Information
| Graduate Attributes and Skills |
Not entered |
| Keywords | TEPDE |
Contacts
| Course organiser | Dr Linhan Li
Tel:
Email: linhan.li@ed.ac.uk |
Course secretary | Mr Martin Delaney
Tel: (0131 6)50 6427
Email: Martin.Delaney@ed.ac.uk |
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