Undergraduate Course: Theory of Elliptic Partial Differential Equations (MATH11184)
|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
|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.
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).
Information for Visiting Students
|High Demand Course?
Course Delivery Information
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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.
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||Mr Martin Delaney
Tel: (0131 6)50 6427