MA40203: Theory of Partial Differential Equations

Yes and no. This particular calculation is quite involved, and would eat up far too much time on an exam. On the other hand, this general strategy of applying elliptic operators to well-chosen functions and then fussing with inequalities is one of the central themes of the unit, and certainly will appear on the exam in various guises. As an example, take a look at Exercise 4.1.1. This calculation is thematically related to the one in the proof of Lemma 4.2, but is much simpler. I put it on a problem sheet this year, and it is also quite similar to an old exam question (not written by me).

No. Of course a rough knowledge of how these proofs go will likely help you deal with related problems, which could indeed appear on an exam.

No. On the other hand you could be asked to think about, for instance, how a PDE simplifies when we assume that the solution only depends on \(x_1\text{,}\) say, or on \(\abs x\) (see Chapter 5!).

On the one hand, thinking creatively about comparison functions (and other ‘auxiliary functions’) is one of our main techniques in the unit. So you should definitely be prepared to play around with simple functions (e.g. linear functions, constants, functions depending only on \(t\) in the parabolic case). Some questions on past exams have also strongly suggested which comparison functions to consider — be prepared to take such hints when they are offered.

On the other hand, if you are considering a complicated comparison function with five different parameters all to be determined, you are almost definitely doing something wrong. (Although such things do appear in research papers!)

If a question asks you to prove something, and you can prove it using the tools from the unit, then you should be good. If, for some reason, I did want to forbid you from using a certain result in your arguments, I would have to make this explicit in the statement of the question.

You were asked to check Definition 3.1 for a variety of examples in Exercise 3.1.1. One way to do this is to diagonalise the relevant matrix. Do not expect anything dramatically harder than Exercise 3.1.1 to appear on the exam. For instance, it’s highly unlikely that you’ll have to find the eigenvalues of \(4 \times 4\) matrix all of whose entries are nonzero.

We needed the divergence theorem, more specifically Theorem 2.49, several times in Section 4.3 on the Mean value property. You are expected to be able to do similar manipulations, and perhaps to do the occasional simple integral of an explicit function (e.g. a constant function). You would certainly not be expected to do something much more complicated than Example 2.50, and even that I would treat as something which would likely eat up a fair bit of time.

On the other hand, a problem which has appeared on some past exams is to calculate a quite nasty looking integral over a ball or sphere, but where the nastiest terms can be grouped together into a harmonic function and hence calculated using Theorem 4.10 by simply evaluating said harmonic function at the centre of the ball. I’d say this is fair game.

This is a difficult question to answer. On the one hand, this unit has a lot of analysis prerequisites, and we often use techniques from these earlier analysis units in our proofs. On the other hand, for us most of these techniques are a means to an end — not the sort of thing you’d expect to see an entire exam question on.

We made a lot of arguments involving sequences of functions in Chapter 4, and so certainly these would be fair game on an exam.

Since Theorem 6.7 is our only result which explicitly treats (a class of) nonlinear parabolic problems, it should certainly be the first thing you think to use. On the other hand, there are other approaches that you can take, most of which revolve around reinterpreting the nonlinear information you’ve been given as a linear equation or inequality to which you can apply the tools from Section 6.2. Indeed, Question 4(b) on The 2021 exam exam was carefully engineered so that Theorem 6.7 did not directly apply, and you were forced to think more creatively.

We need uniform ellipticity/parabolicity to do much of anything, and so it’s not unreasonable to assume in a given problem that this probably holds and you just need to either make a note of it in the case of an operator you’ve seen before or else prove it. On the other hand, it’s certainly possible to imagine questions where this story is complicated somewhat. In Question 4(b) on The 2021 exam, for instance, there is a linear operator \(L\) whose parabolicity is wrapped up in the properties of an unknown function \(u\text{.}\) In this particular case the resolution is not so complicated, but one can imagine more complicated variations.

No, it is not. Moreover, I would say that going over the arguments not covered in class would not be particularly good preparation for the exam.

No, and you will notice that nothing this open-ended appears on past exams

¹

Technically, The 2020 exam did have an extra ‘Question 5’ which was open-ended, but this was the case for all Level 3–5 Maths units that semester. As far as I can tell, the Library has scrubbed all of these questions from their records.

either.

The result in Exercise 2.8.2 is certainly worth knowing. For instance, it is easy to imagine exam questions based on some of the results and techniques in Section 4.1, where Exercise 2.8.2 is used to show equicontinuity. Ideas from problem sheets have frequently appeared on past exams, and indeed there have sometimes even been opportunities to save time by quoting the conclusion of a problem sheet directly (e.g. a formula from Problem Sheet 2). On the other hand, there is probably not much point in memorising, for instance, the precise PDE that appears in Exercise 6.3.2.

While this isn’t the best style, it would be fine on an exam. We used the notation in (3.1) (and (6.1)) for almost all of the unit, and so if you write \(a_{ij}\text{,}\) \(b_j\) or \(c\text{,}\) I will by default assume that this is what you mean.

On the other hand, if you are also using the letter \(c\) for some other purpose, e.g. the coefficients in a linear function \(x \mapsto ax_1 + bx_2 + cx_3\text{,}\) then this could get confusing in a hurry, and you could very well lose marks for being unclear.

For what it’s worth, the coefficient \(c\) in (3.1) is called the ‘zeroth-order coefficient’ or ‘zeroth-order term’. So instead of “because \(L\) has \(c=0\)”, one can write “because \(L\) has no zeroth-order term”. Similarly the \(b_i\) are called ‘first-order coefficients’ and \(b_i\partial_i\) are the ‘first-order terms’.