TMA4305 Partial Differential Equations 2019

Exercises

There are no mandatory exercises in the course. There will, however, be exercises designed to help in the learning of the course material. They will be posted here.

Exercises are posted in inverse chronological order, meaning the newest on top.

B refers to an exercise in Borthwick's book.
X is for exercises stated here

The exercise set for the coming Friday is marked with a filled diamond (◆). Exercise sets for the future are marked with an open diamond (◇).

Week Problems and solutions
46 Friday 15 November
B 12.3, 12.4
Solution
45 Friday 8 November
B 11.5, 11.6
Solution
44 Friday 1 November
X8 below
B 11.1, 11.2
Solution
43 Friday 25 October
X7 below
B 10.1, 10.5, 10.7
Solution (a bit sketchy, but hopefully enough for you to get the point)
42 Friday 18 October
B 7.1, 7.2, 7.3, 7.6, 7.7.
Solution
41 Friday 11 October
Exam 2018-12: Problems 6, 2 (in that order – think of problem 2 as a review problem)
You can find the solution on the “Old exams” page in the menu.
To be honest, I forgot that we haven't introduced the wave kernel yet. Think of problem 2 as an introduction to the wave kernel, then.
40 Friday 4 October
B 9.5.
The assumptions in the exercise are not sufficient (as far as we know, anyhow) to guarantee the existence of $\partial_r u(\mathbf x_0)$. Just assume that the derivative does exist. For example, $u \in C^2(B)\cap C^1(\overline{B})$ should be sufficient. (It may not be very clear, but $B$ is any ball, which may be assumed without loss of generality to be centred at the origin.)
X6 below .
Solution
39 Friday 27 September
From my notes on the parabolic maximum principle: Exercise 1 and 3 (with $A=1$).
B 9.2. But first, take a look at Edward Nelson's marvellous 1961 proof of Liouville's theorem! It is only nine lines of text, without a single formula. Available at DOI: 10.2307/2034412 (you need to be within the NTNU network). Exercise 9.2 is that proof, with the details filled in.
X4 and X5 below
38 Friday 20 September
B 6.1, 6.3, 6.4
Solution
37 Friday 13 September
Note that the exercise class on 13 September was cancelled.
X3 below: An alternative derivation of d'Alembert's solution
B 4.7, 4.9
36 Friday 6 September
X2 below
B 4.1, 4.5.
Solution
35 Friday 30 August
B 3.6. What happens to the solution in (a) when $a\to0$? (The resulting solution is known as a rarefaction wave.)
B 3.7 (you need to assume that $u \in C^2$). Additionally, note that $w=u_x$ satisfies Burgers' equation!
X1 below
Solution

X1

Consider the PDE with initial data: $u_t+u^2u_x=0,\quad u(0,x)=\frac1{1+x^2}.$ What is the largest $T$ so that the problem has a classical solution for $x\in\mathbb{R}$, $t\in[0,T)$?

X2

Consider the PDE with initial data: \begin{aligned} u_t+xuu_x&=0&&\text{for } x>0, t>0 u(0,x)&=g(x)&&\text{for } x>0 \end{aligned} Find the solution expressed by $g$ (on implicit form). Why is there no need to specify a boundary value at $x=0$? Under what conditions does a classical solution exist for all $t\in[0,T]$ and $x\in[0,\infty)$?

X3

For week 37: An alternative approach to the one-dimensional wave equation: Fill in the details of the following outline.

Start with the equation $u_{tt}-c^2u_{xx}=0$. Assuming that $u$ is a solution, consider the two functions $u_t \pm c u_x$. These satisfy (first order) transport equations, so each is a traveling wave with speed $\pm c$. That is, there are functions $w_\pm$ so that \begin{aligned} u_t-cu_x &= -2c w_+'(x-ct)&&\text{(a right-traveling wave),}\\ u_t+cu_x &= 2c w_-'(x+ct)&&\text{(a left-traveling wave).} \end{aligned} The factors $\pm 2c$ and the derivative on the right hand side are just for convenience in what follows. They are not essential. Add the two equations and integrate with respect to $t$, then subtract them and integrate with respect to $x$. Note that the first requires an integration “constant” $C_1(x)$, whereas the second an integration “constant” $C_2(t)$. Conclude that the two must be equal, hence an actual constant $C$, and conclude that [\ u(t,x) = w_+(x-ct)+w_-(x+ct)+C. \] (But we might as well let one of the functions $w_{\pm}$ absorb $C$, so we can throw it away.)

Finally, substitute this into the initial data $u(0,x) = g(x), \quad u_t(0,x) = h(x)$ and derive the d'Alembert solution.

X4

(a) Consider the cube $Q=(0,\pi)\subset\mathbb{R}^n$ and the function $w(t,\mathbf{x})=e^{-nt}\prod_{i=1}^{n} \sin x_i\qquad(t\ge0,\mathbf{x}\in\overline{Q}).$ Verify that $w$ satisfies the standard heat equation.

(b) Let $\Omega$ be a region with $\overline\Omega\subset Q$, Assume that $u \in C\bigl([0,\infty)\times\overline{\Omega}\bigr) \cap C^2\bigl((0,\infty)\times \Omega\bigr)$ satisfies $u_t-\Delta u=0$ for $(t,\mathbf{x}) \in (0,\infty)\times \Omega$ and $u(t,\mathbf{x})=0$ for $(t,\mathbf{x}) \in [0,\infty)\times\partial \Omega$. Show that $\lim_{t\to\infty} u(t,\mathbf{x})=0$ uniformly in $\mathbf{x}$.

Hint: Apply the maximum principle to $mw \pm u$ where $m$ is a suitably large constant.

(Corrected 2019-09-27) Most of the occurrences of $\Omega$ in question (b) were unfortunately written $Q$ instead, which unfortunately ruins the point.

X5

If $u$ is a harmonic function on the upper half space $\mathbb{R}\times (0,\infty)$, and it is continuous on $\mathbb{R}\times [0,\infty)$ with $u(x,0)=0$ for all $x\in\mathbb{R}$, extend $u$ by setting $u(x,-y)=-u(x,y)$ and show that the extension is harmonic on $\mathbb{R}$.

Hint: The mean value property.

(2019-09-27) Note: More regularity is needed than I had realised at first. I'll get back to this.

X6

Let $\Omega$ be a bounded region with a piecewise $C^1$ boundary, and assume that $\Omega$ admits a Green's function.

1. For any $\mathbf x\in\Omega$, show that $G_{\mathbf x}(\mathbf y)\to\infty$ when $\mathbf y \to \mathbf x$.

2. Show that $G_{\mathbf x}(\mathbf y)>0$ for any $\mathbf x,\mathbf y\in\Omega$ with $\mathbf y\ne\mathbf x$. (Hint: Use the strong maximum principle on a suitable subregion of $\Omega$.)

3. Conclude that $-\partial_n G_{\mathbf x}\ge0$ at $C^1$ boundary points, $-\partial_n G_{\mathbf x}>0$ at $C^2$ boundary points. At a $C^2$ boundary point you can always place a ball inside $\Omega$ whose boundary is tangent to $\partial\Omega$ at this point.

4. Show that $-\int_{\partial\Omega} \partial_n G_{\mathbf x}\,dS = 1$.

5. (A challenge) Assuming that the Dirichlet problem is always solvable on $\Omega$, show that $-\partial_n G_{\mathbf x}$ behaves in a manner analogous to what we found for the Poisson kernel on the two-dimensional disk, in particular that it is an “approximate delta” as $\mathbf x$ approaches the boundary of $\Omega$. (Part of the challenge is to make this somewhat precise. Hint: Look at boundary values that are one at a given point on the boundary, then drop off rapidly to zero.)

X7

Let $\Omega\subseteq\mathbb{R}^n$ be a region, and $\psi \in C_c(\Omega)$. Let $\rho$ be a standard mollifier, and $\rho_\delta(x)=\delta^{-n}\rho(x/\delta)$ as usual. Then $\psi*\rho_\delta\to\psi$ uniformly. This follows from the uniform continuity of $\psi$ and the usual properties of $\rho$. You are not asked to prove this.

1. Assume instead that $\psi\in C^k(\Omega)$. Conclude from the above that $D^\alpha(\psi*\rho_\delta)\to D^\alpha\psi$ uniformly for every multi-index with $\lvert\alpha\rvert\le k$.
Conclude further that $C_c^\infty(\Omega)$ is dense in $C_c^k(\Omega)$ in the following sense: For any $\psi \in C_c^k(\Omega)$ there is a sequence of functions $\psi_j \in C_c^\infty(\Omega)$ so that $D^\alpha \psi_j \to D^\alpha\psi$ uniformly for every multi-index $\lvert\alpha\rvert \le k$, and there exists a compact set $K\subset\Omega$ with $\operatorname{supp}(\psi_j)\subseteq K$ for all $j$.
(This defines convergence in $C_c^k(\Omega)$.)

2. Conclude from the above that the partial integration formula $\int_\Omega (D^\alpha f) \psi \,d^nx = (-1)^{\lvert\alpha\rvert}\int_\Omega f D^\alpha \psi \,d^nx$ holds for any $\psi\in C_c^k(\Omega)$, where $f \in L^1_{\text{loc}}(\Omega)$ has a weak derivative $D^\alpha f$.

3. Show the product rule $\nabla(fg)=g\nabla f+f\nabla g$ where $f \in W^{1,1}_{\text{loc}}(\Omega)$, $g\in C^1(\Omega)$, and $\nabla(fg)$ and $\nabla f$ are weak gradients.

4. If $f \in L^1_{\text{loc}}(\mathbb{R}^n)$ has a weak derivative $D^\alpha f$, show that $D^\alpha(f*\psi)=(D^\alpha f)*\psi$ for any $\psi\in C_c^k(\mathbb{R}^n)$, provided $\lvert\alpha\rvert\le k$.

X8

If $f \in C(\mathbb{T}^n)$ and $u \in C^2(\mathbb{T^n})$ is a classical solution of $-\Delta u=f$, show that $\int_{\mathbb{T}^n}f\,d^x=0$.

Write $H^m_{\perp}(\mathbb{T}^n)=\{f \in H^m(\mathbb{T}^n) | \int_{\mathbb{T}^n} f\,d^nx=0\}$. (Non-standard notation. I was tempted to write $H^m_0$, but that would be too confusing. This space is the orthogonal complement of the constant functions.)

If $f \in H^0_{\perp}(\mathbb{T}^n)$ write $\mathcal{D}_f[u]=\int_{\mathbb{T}^n}(\tfrac12\lvert\nabla u\rvert^2+uf)\,d^nx$ for $u \in H^1_{\perp}(\mathbb{T}^n)$. State and show an analogue of Dirichlet's principle for weak solutions of $-\nabla u=f$ in $H^1_{\perp}(\mathbb{T}^n)$, and show that this problem has a unique solution.

You “only” need to repeat the steps of the proof for the usual Dirichlet principles, adjusting to the new setting and checking that it all still works. There is only one major obstacle: There is no Poincaré inequality in $H^1(\mathbb{T}^n)$. (The constant functions violate it.) However, there is such an inequality in $H^1_{\perp}(\mathbb{T}^n)$, and that is all you need. You can prove it using Fourier analysis. It's easier than it looks at first – try it!