MA40203: Theory of Partial Differential Equations

Due 20 March at 4pm, either in class or in the pigeon hole in 4W.

Exercise 4.1.1 is a classic and important exercise, and I believe that a version of it was on an old exam. It is also a more accessible version of the calculation in the proof of Lemma 4.2.

Exercise 4.1.2 proves a result we need in the lectures, and also gives you practice deriving new estimates from old ones. Like deriving new maximum principles from old ones, this is extremely useful both to get a conceptual understanding of how these estimates work, and as a labour saving device to avoid writing unnecessarily repetitive proofs. I have not assigned Exercise 4.1.3, which is just a fancier version of Exercise 4.1.2 where you use induction. Hopefully after doing Exercise 4.1.2 you will feel confident that you could do this exercise if you really had to, even if you wouldn’t be looking forward to writing it up carefully. In any case these exercises naturally lead to Exercise 4.2.2, which guides you through a proof that \(C^2\) harmonic functions are in fact smooth.

Exercise 4.2.1 is mostly here as light entertainment. If you’ve been dying to use a tedious mechanical procedure to explicitly solve a PDE with your bare hands in this unit, it may be your only chance. Do not worry about having to use this particular technique on the exam. At the same time, the general idea that PDEs involving polynomials might have solutions (or sub- and supersolutions) which are also polynomials is perhaps a useful takeaway.

Please feel free to email me, or drop by office hours (Wednesdays 1:15–2:05 in 4W 1.12). I am also more than happy to meet one-on-one or in a small group.