Section B.5 Problem Sheet 5
Due 13 March at 4pm, either in class or in the pigeon hole in 4W.
Exercise 3.2.4 is another problem on the comparison principle. The end result is a generalisation of the weak maximum principle without any sign conditions on \(Lu\text{,}\) but with an additional term on the right hand side. The last part of the problem hints at how such results can be used to study sequences of functions, and is related to arguments we will make repeatedly in Chapter 4.
Next we move on to Section 3.3 with Exercise 3.3.1, which gives you some practice working with normal derivatives. If you made it through Problem Sheet 2, this is hopefully pretty easy. Exercise 3.3.4 is an interesting fact in its own right, and also gives you practice applying the results from Section 3.3. Getting the pieces to fit together is perhaps harder than you might initially assume — certainly much harder than one would assume by just skimming through a solution without having worked on the problem beforehand.
Finally, Exercise 3.3.6 is a nice ‘capstone’ question for Chapter 3 in that it ties together a great many of its results. Even if you run out of time to finish this exercise for this sheet, I highly recommend coming back to it later as a way to help keep all of these different variants of the maximum principle straight in your head.
In general, the end of Chapter 3 is a good time to take stock and perhaps do a bit of a mini-revision. The results from this chapter will be our main tools in both Chapter 4 and Chapter 5, and the parabolic results in Chapter 6 can be thought of as generalisations of Chapter 3 to a different class of PDE.
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.
