Lecture 19/08: We derived a first order, scalar conservation law from first principles (i.e., some physical considerations along with the divergence theorem). We introduced some notation, most importantly multiindex notation (p. 701). We defined what we mean by a PDE (or a system of PDEs), and defined the terms linear, semilinear, quasilinear and nonlinear. We also looked at the main mathematical questions in PDE theory: Solution formulas, existence, uniqueness and stability. Relevant material is Evans Chapter 1 and Appendix A.
Lecture 21/08: We solved linear transport equations with constant coefficients using the method of characteristics, and used the same idea to compute the solution of the corresponding non-homogeneous problem (this was a particular case of Duhamel's principle). We started on the Laplace and Poisson equation: Derived the equation from first principles and started computing a rotation invariant solution. Relevant material: Evans, Section 2.1-2.2.1.
Problem sheet (updated 20/08). Solutions to problems
Lecture notes from week 1

Lecture 26/08: We derived the fundamental solution of Laplace's equation (Evans, 2.2.1).
Lecture 28/08: We will prove that the solution of Poisson's equation can be written as a convolution with the fundamental solution of Laplace's equation (Evans, 2.2.1).
Problem sheet (updated 4/09). Solutions to problems

Lecture 02/09: Cancelled!
Lecture 04/09: We proved the mean value property of harmonic functions and its converse (2.2.2). We also stated and proved the strong maximum principle (2.2.3).
Problem sheet . Solutions to problems

Lecture 09/09: We finished the proof of the maximum principle, from which properties on uniqueness of solutions of Laplace's and Poisson's equations follow (2.2.3). We also reviewed the theory of mollifiers (Appendix C.5) and we showed that any continuous function satisfying the Mean Value Property is actually in $LaTeX: C^{\infty}(U)$ . In particular, any harmonic function $LaTeX: u\in C^2(U)$ actually lies in $LaTeX: C^\infty(U)$ .
Lecture 11/09: We proved some estimates on the derivatives of harmonic functions (theorem 2.3.7).

Lecture 25/09: We stated the symmetry of Green's function without proof and started on Green's function in the half-plane (up to eq. (33) on p. 37).
Problem sheet. Solutions to problems, proof of Theorem 14

Lecture 07/10: We derived the heat equation ("diffusion equation") and derived the fundamental solution of the heat equation.
Lecture 09/10: We found and proved a formula for the solution of the Cauchy problem for the (homogeneous and inhomogeneous) heat equation in $LaTeX: \mathbb{R}^n$ .
Problem sheet, solutions

Lecture 14/10: We finished the proof of the formula for the solution of the Cauchy problem. We stated (without proof) the mean value property and maximum principles for the heat equation. We also saw how these imply uniqueness of the initial-boundary value problem on bounded domains. Last, we started on the proof of the maximum principle on $LaTeX: \mathbb{R}^n$ .
Lecture 16/10: We finished the proof of the maximum principle on $LaTeX: \mathbb{R}^n$ , and we saw how this implies uniqueness of solutions of the Cauchy problem. We stated the fact that solutions of the (homogeneous) heat equation are $LaTeX: C^\infty$ (both in space and in time). We also gave an alternative proof of uniqueness of the (inhomogeneous) heat equation on bounded domains using an energy estimate, and proved backward-in-time uniqueness.

Lecture 21/10: We solved some of the problems from week 8, then moved on to the wave equation and solved the Cauchy problem in dimensions (d'Alembert's formula).
Lecture 23/10: We solved the wave equation on the half-line $LaTeX: x\in\mathbb{R}_+$ using the reflection method. We then used this to solve the wave equation in dimensions, and finally applied the method of descent to solve the problem in 2 dimensions using the solution in 3 dimensions.

Lecture 28/10: We finished section 2.4 (energy methods) and moved on to section 3.2 (the method of characteristics).