In Week 2 Lecture 2, we started Section 2.3 with an example of calculating the partials of a composition \(f(u(s,t),v(s,t),s,t)\text{.}\) Later that same lecture we looked at another composition \(\phi(a\cdot x,x_1
x_2)\text{.}\)
Appendix C Problems classes
Often in problems classes, and indeed in some lectures, we will consider examples that don’t appear elsewhere in the lecture notes. For completeness, I will briefly summarise these here, and give links to the relevant recordings. I may also summarise examples which come up in office hours, if they seem particularly helpful.
Example 1. Chain rule examples.
Example 2. Differentiating using index notation.
At the start of Week 2 Lecture 3, we used index notation to calculate \(\partial_i (a\cdot x)\) and then calculated \(\partial_i[f(a\cdot x)]\) using the chain rule.
Example 3. Another chain rule example.
At the start of Week 3 Lecture 2, we came up with a definition of what \(\partial_\theta F\) ought to mean for a function \(F=F(x,y)\) on the plane, and then used the chain rule to show that \(\partial_\theta F = -y\partial_x F + x\partial_y F\text{.}\)
Example 4. Checking ellipticity.
At the end of Week 4 Lecture 1, we checked that, on \(\Omega = \R^2\text{,}\) the operator \((3+\sin \abs x)\partial_{11} + 4 \partial_{22}\) is uniformly elliptic (with ellipticity constant \(2\)).
Example 5. An example we keep coming back to.
In Week 4 Lecture 3, we set \(\Omega = B_1(0) \subset \R^2\) and considered a function \(u\) satisfying
\begin{align*}
\left\{
\begin{alignedat}{2}
-1 \le \Delta u \amp\le 0 \amp\quad\amp \ina \Omega,\\
u \amp= 0 \amp\quad\amp \ona \partial\Omega,
\end{alignedat}
\right.\text{.}
\end{align*}
We applied the weak maximum principle to \(-u\) to deduce that \(u \ge 0\) in \(\Omega\text{.}\) We gave an alternative argument for this fact using the comparison principle. Playing around with potential comparison functions and seeking an upper bound, we eventually settled on \(v = \tfrac 14 (\abs x^2 -
1)\) and showed that \(u \le v\) in \(\Omega\text{.}\)
In Week 5 Lecture 3, we came back to this example and used the strong maximum principle and Hopf lemma to find out even more about this function \(u\text{.}\) We were able to replace non-strict inequalities with strict inequalities or \(\equiv\text{,}\) and to obtain related inequalities for normal derivatives.
Example 6. A more challenging comparison principle argument.
At the end of Week 4 Lecture 3, we set \(\Omega = B_1(0) \subset \R^2\) and considered a function \(u\) solving the PDE
\begin{align*}
\left\{
\begin{alignedat}{2}
(\Delta - 1) u \amp= 0 \amp\quad\amp \ina \Omega,\\
u \amp= 1 \amp\quad\amp \ona \partial\Omega
\end{alignedat}
\right.\text{.}
\end{align*}
I suggested that we might be able to find good upper and lower bounds for \(u\) by considering comparison functions \(v\) of the form \(A\abs x^2
+ B\) for constant \(A,B\text{,}\) and briefly mentioned why choosing \(A,B\) appropriately was slightly more complicated than in the previous example.
At the end of Week 6 Lecture 3, we returned to this example and decided that comparison functions of the form \(A(\abs x^2 -1) + 1\) would do the trick, with \(A=1/4\) and \(A=1/5\text{,}\) concluding that
\begin{gather*}
\tfrac 14( \abs x^2 - 1) + 1 \le u \le \tfrac 15 (\abs x^2 - 1) + 1
\qquad \ina \Omega.
\end{gather*}
In particular \(0.75 \le u(0) \le 0.8\) which is a reasonably tight approximation!
Finally, I mentioned that this problem has an ‘explicit’ solution \(u(x)=I_0(\abs x)/I_0(1)\text{,}\) where \(I_0\) is a modified Bessel function of the first kind of order \(0\text{.}\) This gives \(u(0) \approx 0.78985\text{.}\) Bessel functions are of course not at all examinable in this unit! Even without knowing anything about Bessel functions, we could have guessed (and indeed proven) that \(u\) depends only on \(\abs x\) because of the rotational symmetry of the problem it solves; this is the topic of Chapter 5.
Example 7. 2024 Question 3(b).
At the start of Week 7 Lecture 3 we worked through Question 3(b) on The 2024 exam.
Example 8. Checking a function is subharmonic.
At the end of Week 7 Lecture 3, we let \(u \in C^2(\Omega) \cap C^0(\overline\Omega)\) be harmonic, and showed that the function \(v \in C^0(\overline \Omega)\) defined by
\begin{align*}
v(x) \amp=
\begin{cases}
1 \amp \abs{u(x)} = 1, \\
(u(x))^2 \abs{u(x)} \gt 1
\end{cases}
\end{align*}
was subharmonic. We did this by first checking that \(\Delta u^2 = \Delta(u^2) = 2
\abs{\nabla u}^2 \ge 0\) so that \(u^2\) is subharmonic, and then noticing that \(v = \max(u^2,1)\) is subharmonic by Lemma 4.16.
Example 9. An extended problem.
In Week 8 Lecture 3, we worked through the end of Exercises 3.4 and most of Exercises 4.6, but changed the right hand side to be \(x_2^3\) instead of \(x_2\text{.}\)
Example 10. Simple parabolic comparison.
In Week 10 Lecture 3, we considered \(\Omega = (-1,1) \subset \R\) and applied the (linear, parabolic) comparison principle find upper and lower bounds for a function \(u\) satisfying
\begin{align*}
\left\{
\begin{alignedat}{2}
-1 \le (-\partial_t +\partial_{xx}) u \amp\le 0 \amp\quad\amp \ina D,\\
u \amp= 0 \amp\quad\amp \ona \Sigma,
\end{alignedat}
\right.
\end{align*}
ultimately finding \(u \ge 0\) as well as the two upper bounds \(u \le t\) and \(u \le \frac 12 (1-x^2)\text{.}\) As in the elliptic case (e.g. Example 5), the trickiest part was coming up with comparison functions. Here the functions we considered were either constant, depended only on \(t\text{,}\) or depended only on \(x\text{.}\)
Example 11. Simple semilinear parabolic comparison.
Later in Week 10 Lecture 3, we considered a more general bounded \(\Omega \subset \R^N\) and a function \(u \in
C^2(\overline D)\) satisfying
\begin{align*}
\left\{
\begin{alignedat}{2}
-\partial_t u + \Delta u - u\cos(\partial_1 u) = 0 \amp\quad\amp \ina D,\\
u \amp = 0 \amp\quad\amp \ona \partial\Omega \times [0,T], \\
\abs u \amp \le 1 \amp\quad\amp \ona \Omega \times \{0\}.
\end{alignedat}
\right.
\end{align*}
Applying the semilinear comparison principle, we first considered constant comparison functions and showed that \(\abs u \le 1\) in \(D\text{.}\) Then we considered comparison functions depending on \(t\) and showed that \(\abs
u \le e^{-t}\) in \(D\text{.}\)
Example 12. 2024 Q4(b).
At the end of Week 11 Lecture 1, we looked at Q4(b) on the The 2024 exam. The first part turned out to be a relatively straightforward application of the semilinear comparison. We then speculated a bit about the second part, which has to do with reflection symmetry.
Example 13. Recap of Chapter 3.
In Week 11 Lecture 2, we did an overview of many of the results from Chapter 3, and talked about some common errors with applying them. In previous years this has probably been the most popular revision session with students, could definitely be worth watching the recording if you weren’t there.
Example 14. Tricks for nonlinear problems.
In Week 11 Lecture 3, we started with a quick recap of our three main nonlinear results (Theorem 5.1, Lemma 5.4, and Theorem 6.7). The we mostly talked about various tricks for applying linear results (e.g. from Chapter 3) to nonlinear problems. For instance, we talked about how the nonlinear PDE \(\Delta u - u^3 = 0\) could be thought of as a linear problem \(Lu=f\) where \(L=\Delta\) and \(f=u^3\text{,}\) or alternatively the linear problem \(Lu=0\) where \(L=\Delta - u^2\text{.}\)
If you are at all unsure about what we mean by a differential operator like \(\Delta - u^2\) please get in touch and we can go through some additional examples — confusion about this is a common cause of lost marks on exams.
Example 15. 2019 Q1(c).
At the end of Week 11 Lecture 3, we let \(\Omega\) be bounded and tried to prove that solutions \(u \in C^2(\Omega) \cap
C^0(\overline \Omega)\) of
\begin{align*}
\left\{
\begin{alignedat}{2}
\Delta u - u^3 \amp = f \amp\quad\amp \ina \Omega,\\
u \amp = 0 \amp\quad\amp \ona \partial\Omega
\end{alignedat}
\right.
\end{align*}
are unique. The trick was to find a linear PDE satisfied by the difference \(w=u-v\) of two solutions, and here the cubic nonlinearity meant that we could get away with purely algebraic calculations. Alternatively, we could have mimicked the argument in the proof of Lemma 5.4 (or indeed Theorem 6.7). In either case, we obtained a linear problem of the form
\begin{align*}
\left\{
\begin{alignedat}{2}
\Delta w + c w \amp = 0 \amp\quad\amp \ina \Omega,\\
w \amp = 0 \amp\quad\amp \ona \partial\Omega
\end{alignedat}
\right.
\end{align*}
where the (bounded) zeroth-order coefficient satisfied \(c \le 0\text{.}\) The result then follows at once from Corollary 3.5.
