Undergraduate Course: Elliptic Partial Differential Equations (MATH10089)

Course Outline
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
SCQF Credits	10	ECTS Credits	5
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- Safonov theorem.
Course description	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. Krylov-Safonov's theorem.

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)	Co-requisites
Prohibited Combinations		Other requirements	None

Information for Visiting Students
Pre-requisites	None

Course Delivery Information

Academic year 2015/16, 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)	MATH10089 Elliptic Partial Differential Equations - Visiting Students		2:00

Academic year 2015/16, Part-year visiting students only (VV1)		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
No Exam Information

Academic year 2015/16, Part-year visiting students only (VV2)		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
No Exam Information

Learning Outcomes
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.

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