An Introduction to Partial Differential Equations Spring 2008 Course Document

Lecturer: David Rule Office: JCMB 5315 Phone: 0131 650 5055 Email: david.rule@ed.ac.uk Lectures: Thursdays 1-3pm, JCMB 6309, The King’s Buildings Website: www.maths.ed.ac.uk/˜ rule/pde/ Textbooks: G.B. Folland, Introduction to Partial Differential Equations, Chapters 0, 2 & 6; L.C. Evans, Partial Differential Equations, Chapter 5 Course Summary: The aim of this course is to study some classical methods in partial differential equations which are particularly useful in modern research. The principal goal is to understand boundary value problems for the Laplacian via layer potentials, while acquiring useful tools along the way. Syllabus: The following is the approximate content of each lecture.

1. Convolutions and the Fourier transform

2. Distributions and Compact operators

3. The Laplace operator; Green’s identities, symmetry and mean value properties

4. The maximum principle and Louisville’s theorem

5. The fundamental solution and applications

6. Dirichlet and Neumann problems; Green’s function

7. Dirichlet’s principle and the Dirichlet problem for a half-space

8. The reflection principle and the Kelvin transform

9. Solving the Dirichlet and Neumann problems via layer potentials; definitions and motivation

10. Properties of integral operators

11. Properties of the double layer potential

12. Properties of the single layer potential

13. Solutions to the boundary value problems

14. Sobolev spaces; extensions and traces

15. Sobolev inequalities; Poincaré’s inequality David Rule