Section B.4 Problem Sheet 4
Due 6 March at 4pm, either in class or in the pigeon hole in 4W.
Exercise 2.8.2 is one of the simplest ways to verify equicontinuity in practice, and plays an important role in Chapter 4. The style of argument involving the mean value theorem or fundamental theorem of calculus will also appear when we try to work with nonlinear PDEs. Exercise 2.8.5, which we’re skipping this year, is a follow-up to Exercise 2.8.2 giving you a taste of how we will use it in practice in Chapter 4.
Almost all of the results in this unit have uniform ellipticity (or its close cousin uniform parabolicity in Chapter 6) as one of their assumptions. Exercise 3.1.1 ensures that you’ve verified this by hand at least once, and also seen some examples where it can fail. Exercise 3.1.2 is a sequel to Exercise 2.3.1 from the previous problem sheet, and asks you to think about how the ellipticity of an operator behaves under simple changes of variables.
Exercise 3.2.3 gives us a chance to get our hands dirty and use the comparison principle to obtain explicit bounds. This is similar in spirit to the first few parts of Exercise 1.1.2.
There are a fair number of unassigned exercises that we are skipping over. If you are looking for more things to attempt, my first recommendation is the (relatively easy) Exercise 3.2.1, which asks you to think about what can happen when the hypotheses of Corollary 3.5 (and by extension of the weak maximum principle and comparison principle) do not hold.
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.
