We will follow the book Partial Differential Equations, second edition, by L. C. Evans. You can find the exact reference in Leganto. The book should be available at Akademika.
The final syllabus is as follows:
- Chapter 1: Everything.
- Section 2.1 and 3.2 (first-order ODEs). You need to be able to solve simple first-order equations using the method of characteristics. You need to be able to recognise when requirements for the method of characteristics break down, and argue why a solution might not exist.
- Section 2.2 (Laplace/Poisson). You should know all of the results and understand most of the proofs here. We did not focus so much on Green's functions.
- Section 2.3 (heat equation). You should have an idea of most of the proofs here. We did not go in detail about the mean value formula and its corollaries, so understanding the ideas here is sufficient.
- Section 2.4 (wave equation). You need to know the derivation of d'Alembert's formula, and know roughly the idea behind the solution formulae in multiple dimensions.
- For all time-dependent equations, you need to know Duhamel's principle and be able to apply it in given problems.
- You need to know how to apply the energy method to obtain stability and uniqueness. You need to be able to prove maximum principles, either using mean value formulas or directly (i.e., assume there is an extremum, deduce a contradiction), and know how to use these to get uniqueness.
- Section 3.3 (Hamilton–Jacobi equations). You need to know the Hopf–Lax formula, what conditions it relies on, and have an idea of why it gives a solution to the Hamilton–Jacobi equation – and in what sense it solves the PDE. You need to be aware that the solution is not necessarily unique, but that the Hopf–Lax formula gives the "correct" solution (the viscosity solution, although that is outside the scope of this course).
- Section 3.4 (nonlinear conservation laws). You need to be able to solve simple Cauchy problems using the method of characteristics. You need to know that discontinuities might appear, that we work with weak solutions, and that these might not be unique. You need to know the Rankine–Hugoniot condition and be able to use it to check whether a given function is a weak solution. You also need to know what entropy conditions are, and to be able to check that a given function is an entropy solution. (Evans does not treat this subject in full, so you might want to consult these lecture notes – see Chapter 3.)