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  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 LaxMilgram 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 nonlinear PDEs).

Information for Visiting Students
Prerequisites  None 
High Demand Course? 
Yes 
Course Delivery Information
Not being delivered 
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 LaxMilgram 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 halfspace. 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 Martin Dindos
Tel:
Email: M.Dindos@ed.ac.uk 
Course secretary  Mr Martin Delaney
Tel: (0131 6)50 6427
Email: Martin.Delaney@ed.ac.uk 

