# TMA4305 Partial Differential Equations 2018

## Exercises

Exercises appear here in *inverse cronological order* (i.e., newest on top).

### For week 46

From Borthwick: Problems 12.3, 5, 5.

Although there is no exercise class on Friday 16 November, I suggest these problems. I intend to post solutions (at least solution *sketches*) to these problems. **This is the final problem set.**

### For week 45 (Fri 9 November)

From Borthwick: Problems 11.4, 5, 6.

### For week 44 (Fri 2 November)

From Borthwick: Problems 10.5, 10.7.

Also Problem 5 from the December 2016 exam. (Yes, I know, the solutions are available, but try not to look.)

### For week 43 (Fri 26 October)

From Borthwick: Problems 10.1, 10.4, 10.6.

*Additionally* if \((u_k)\) is a sequence in \(L^1_{\text{loc}}(\Omega)\) and \(u\in L^1_{\text{loc}}(\Omega)\),
we say that \(u_k\to u\) *strongly* if \(\int_K|u_k-u|\,d^n\mathbf{x}\to0\) as \(k\to\infty\) for all compact sets \(K\subset\Omega\). We say that \(u_k\to u\) *in the sense of distributions* if \(\int_K u_k\psi\,d^n\mathbf{x}\to\int_K u\psi\,d^n\mathbf{x}\) for all \(\psi\in C^\infty_c(\Omega)\). Show that strong convergence implies convergence in the sense of distributions, with the same limit. (Yes, this is almost trivial. The meaning of “in the sense of distributions” will soon become clearer.)

If all the members of a sequence \((u_k)\) in \(L^1_{\text{loc}}(\Omega)\) have weak derivatives \(\partial_{x_j}u_k\) and \(u_k\to u\), \(\partial_{x_j}\to v\in L^1_{\text{loc}}(\Omega)\), both in the sense of distributions, show that \(u\) has a weak derivative \(\partial_{x_j}u\), and that \(\partial_{x_j}u=v\).

### For week 42 (Fri 19 October)

From Borthwick: 7.2, 3, 7

Also, let \(\Omega\subseteq\mathbb{R}\) be an open interval, and assume \(a\in\Omega\)
and \(g\in L^1_\text{loc}(\Omega)\) are given. Put
\[ f(x) = \int_a^x g(t) \,dt \]
Show that \(g\) is the weak derivative of \(f\).
*Hint*: Let \(\psi\in C^\infty_{\text{c}}(\Omega)\) be a test function, and look at
\[ \int_\Omega f(x)\psi'(x)\,dx. \]
The calculation may be simpler if you assume that \(a \le x\) for all \(x \in \operatorname{supp}(\psi)\).
There is no loss of generality, since moving \(a\) to a different location merely adds a constant to \(f\).

### For week 41 (Fri 12 October)

**No exercises this week.**

### For week 40 (Fri 5 October)

From Borthwick: Exercise 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.

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.)

### For week 39 (Fri 28 September)

From my notes on the weak maximum principle for the heat equation: Exercises 1 and 2.

From Borthwick: Exercises 9.2 and 9.3.

**For 9.2,** 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. Exercise 9.2 *is* that proof, with the details filled in.

**Added after the week's exercise class:** The image below (which I struggled with on the blackboard) tells much of the story for Bortwick 9.2. (Also available in pdf format.) The point is that the dark shaded area is a subset of the total (dark and light) shaded area, *and* that the small ball (red in the figure) is contained in each of the larger balls (black in the figure). Indeed, the latter fact is easier to prove, and the former fact follows from this.

### For week 38 (Fri 21 September)

From Borthwick: Exercise 6.3.

Work through the detail of the proof of Theorem 2 in the posted note on the energy method for the wave equation, and then do the following problem:

Applying ideas from exercise 6.4 and the proof mentioned above, prove uniqueness for the problem \(u_t-\Delta u=f(t,\mathbf{x},u)\) in \((0,T)\times\Omega\) with given initial values and boundary values of the bounded region \(\Omega\subset\mathbb{R}^n\). What requirements do you need to impose on \(f\) for the proof to work?

### For week 37 (Fri 14 September)

From Bortwick: Exercises 4.7, 4.8, 4.9.

In exercise 4.8, show that, more generally, \(u(t,\mathbf{x})=v(\mathbf{a}\cdot\mathbf{x}-t)\) defines a solution, for arbitrary \(C^2\) functions \(v\) and some vectors \(\mathbf{a}\).

### For week 36 (Fri 7 September)

Since this is posted late, and last week's exercise class was cancelled, I include only two problems this week.

From Borthwick: Exercises 4.5, 4.6.

For exercise 4.6, I suggest the following shortcut: Motivated by the factorisation \(\partial_t^2-c^2\partial_x^2 =(\partial_t-c\partial_x)(\partial_t+c\partial_x) =(\partial_t+c\partial_x)(\partial_t-c\partial_x) \), put \(w_1=(\partial_t+c\partial_x)u\) and \(w_2=(\partial_t-c\partial_x)u\), then move straight to proving (4.50). Then do (c) and (d).

### For week 35 (Fri 31 August)

From Borthwick: Exercise 3.6 (but the book misplaced the apostrophe in Burgers' equation!). What happens to the solution in (a) when \(a\to0\)? (The resulting solution is known as a *rarefaction wave*.)

From Borthwick: Exercise 3.7. Additionally, note that \(w=u_x\) satisfies Burgers' equation!

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 \(x\in\mathbb{R}\), \(t\in[0,T)\)?

Solve the IVP (initial value problem) \[uu_x+y^2u_y=yu,\quad u(x,1)=x.\] What is the largest domain on which a classical solution exists?

The last two problems are not easily solved using material from the book. Consult the notes on quasi-linear equations (see the Messages page).