TMA4305 Partial Differential Equations 2017

Exercises have been posted here in inverse chronological order.

Solution sketch.

From Borthwick: Problems 12.3, 4.

For 12.3, refer also to the “harmonicfunctionology” note, esp. the fundamental solution (top of p. 7) and Theorem 12 (p. 11; see also Theorem 12.10 in Borthwick). The upper half plane \(\mathbb{H}\) is certainly not bounded, so we can't strictly speaking apply that theorem. Go ahead anyway and do the formal calculation, and afterwards try to justify your calculations by imposing suitable restrictions on \(g\).

Solution sketch.

From Borthwick: Problems 11.2, 4, 5.

For the latter two you only need the definition of the Rayleigh quotient (just below the middle of page 218) and the statement of Theorem 11.9. No deeper theory from Ch 11 is required, so it doesn't matter that we haven't covered it in the lectures yet!.

While doing the problems myself, I found that 11.2 is wrong: It is not the case that \(\log r \in H^1_0(\mathbb{D})\). The best you can get, is \(\log r \in W^{1,p}_0(\mathbb{D})\) for \(1\le p<2\). This is close to what we want, but not quite there. One could, however, replace \(\log r\) by \( (-\log(r/2))^a-(\log 2)^a\) where \(0<a<\frac12\).

Solution sketch.

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.)

Solution sketch.

From Borthwick: Problems 10.1, 10.3, 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 \(\partial_{x_j}u=v\).

Solution sketch.

From Borthwick: Problems 7.1, 2, 3, 5, 6, 7. (It looks like a lot, but several of them are very simple and quick to do.)

Also prove the polarisation identity \[ \langle u,v \rangle = \frac14 \sum_{k=0}^3 i^k \lVert u+i^kv \rVert ^2 \] for vectors in an inner product space.

Solution sketch.

1. In a single space dimension, the Laplacian of \(u\) is given by \(u''\), and the fundamental solution for \(-u''\) can be written \(\Phi(x)=-\frac12|x|\). Explain why, and verify that \(u=\Phi*f\) solves \(-u''=f\) for any continuous function \(f\) with compact support. What is Green's function for the “unit ball” \([-1,1]\)?
2. In \(n\) space dimensions, show that a \(C^2\) function \(u\) is subharmonic if, and only if, the spherical averages \(\tilde u_{\mathbf{x}}(r)\) are (not necessarily strictly) increasing functions of \(r\) for all \(\mathbf{x}\) and \(r\)
3. In light of the previous question, extend the definition of subharmonic to continuous, not necessarily differentiable, functions by requiring spherical averages to increase with the radius. Show that the weak and strong maximum principles hold for subharmonic functions in this generalized sense.
4. From Borthwick: Problem 9.5.

Solution sketch

1. Verify that convolution is commutative and associative: \(f*g=g*f\) and \((f*g)*h=f*(g*h)\). Just do it as a formal calculation without worrying too much about mathematical rigour. (But if you must know, assuming the absolute value of each function has a finite integral over \(\mathbb{R}^n\) is sufficient.)
2. Show by direct calculation that \(H_s*H_t=H_{s+t}\) for \(s,t\ge0\), where \(H_t\) is the heat kernel on \(\mathbb{R}^n\). With reference to solutions of the heat equation, why should this identity hold?
3. The (now corrected) “additional” problem from week 39 (see below).
4. From Borthwick: Problems 9.3 and 9.4. (The hint for 9.4 mentions the maximum principle. I believe the mean value property is intended.)

Solution sketch including the corrected extra problem.

All three exercises from my note on the parabolic maximum principle (make sure to get the latest version, dated 2017-09-25).

In addition, use the energy method with energy \(u^2/2\) to show uniqueness of solutions to \[ u_t-\nabla \cdot (\mathbf{A} \nabla u) + \mathbf{b} \cdot \nabla u =0 \] on \(\Omega_T=(0,T)\times\Omega\), where \(\Omega \subset \mathbb{R}^n\) is a bounded region with piecewise \(C^1\) boundary, \(\mathbf{A}\) is a constant positive definite symmetric \(n\times n\) matrix, \(\mathbb{b}\) is a constant vector, and initial/boundary values are given on the parabolic boundary of \(\Omega_T\).

Ooops! (2017-09-28 12:00) The \(\nabla\) in \(\mathbf{b} \cdot \nabla u\) was missing. The exercise makes no sense without it. This was corrected much too late, so we moved this problem to week 40.

Solution sketch.

From Borthwick: Exercises 6.3 and 6.4.

Applying ideas from exercise 6.4 and the note on the energy method for the wave equation, prove uniqueness for the problem \(u_t-\bigtriangleup u=f(t,\mathbf{x},u)\) 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?

Solution sketch.

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 \(\mathbf{a}\).

Solution sketch.

From Borthwick: Exercises 4.1, 4.2, 4.5, 4.6.

For exercise 4.6, I offer 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).

Solution sketch (corrected/updated 2018-07-23).

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).