Section B.10 Problem Sheet 10
Due 10 May at 4pm, either in class or in the pigeon hole in 4W.
The last two problems are, at least in my opinion, significantly easier than the first two.
The relationships between different maximum and comparison principles is a constant thread throughout this unit, and Exercise 6.2.1 and Exercise 6.2.3 give you some practice with this for parabolic problems. The suggested trick in Exercise 6.2.3 highlights an important difference between the parabolic and elliptic cases which is easy to overlook, and so in making you work it out yourself I am hoping that it will stick in your head a bit better. Exercise 6.3.2 gives you some basic practice applying comparison results to a concrete examples. Such questions appear frequently appear on exams, and come in a wide range of difficulties. For a somewhat more challenging example see Exercise 6.2.2, which I didn’t assign this year but which I have released solutions for. Finally, Exercise 6.3.1 is an easy generalisation of Corollary 6.5. Compared to what we can prove in the elliptic setting, it’s surprisingly strong!
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.
