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DRPS : Course Catalogue : School of Mathematics : Mathematics

Undergraduate Course: Theory of Elliptic Partial Differential Equations (MATH11184)

Course Outline
SchoolSchool of Mathematics CollegeCollege of Science and Engineering
Credit level (Normal year taken)SCQF Level 11 (Year 5 Undergraduate) AvailabilityAvailable to all students
SCQF Credits10 ECTS Credits5
SummaryNB. This course is delivered *biennially* with the next instance being in 2023-24. 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 will introduce the fundamental concepts used in the PDE theory such as the notion of a weak solution and Sobolev spaces. The course will then focus on elliptic PDEs and will introduce the basics of modern theory of such PDEs.
Course description Types of weak solutions for elliptic PDEs.
Questions in physics and mechanics giving rise to elliptic PDEs.
Weak differentiability, Sobolev spaces and classical solutions.
Divergence form equations, the Lax-Milgram theorem, solvability of the Dirichlet and Neumann boundary value problems.
Harmonic functions: Mean value theorem, gradient estimates, the Fundamental solution and the Green's function.
Maximum principle for general linear equations, Aleksandrov's maximum principle (with some extensions to non-linear PDEs).
Entry Requirements (not applicable to Visiting Students)
Pre-requisites Students MUST have passed: Honours Analysis (MATH10068) OR Algebra and Calculus (PHYS08041) OR Linear Algebra and Several Variable Calculus (PHYS08042)
Prohibited Combinations Other requirements None
Information for Visiting Students
High Demand Course? Yes
Course Delivery Information
Academic year 2023/24, 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 95 %, Coursework 5 %, Practical Exam 0 %
Additional Information (Assessment) Coursework 5%, Examination 95%
Feedback Not entered
Exam Information
Exam Diet Paper Name Hours & Minutes
Main Exam Diet S1 (December)MATH11184 Theory of Elliptic Partial Differential Equations2:00
Learning Outcomes
On completion of this course, the student will be able to:
  1. Demonstrate understanding of Sobolev spaces and their relations to other spaces of functions.
  2. Reformulate equations of divergence form through integral identities using partial integration so that the Lax-Milgram theorem can be applied.
  3. Evaluate or estimate the maximum and minimum of solutions of elliptic equation using the maximum principle.
  4. Infer regularity of solutions from that of given data.
  5. 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
Course organiserDr Linhan Li
Course secretaryMr Martin Delaney
Tel: (0131 6)50 6427
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