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MA40203: Theory of Partial Differential Equations

Miles Wheeler

Appendix H Hints for exercises

Here we collect the hints to all of the exercises in the lecture notes, including many exercises which will not be assigned. It is only really useful if you are reading a pdf version ‘with hints in appendix’.

1 Maximum principles for ordinary differential equations
1.1 Maximum principles

Exercises

1.1.1. (PS1) Some basic counterexamples.
1.1.1.a
Hint.
It is enough to consider solutions of \(Lu=0\text{,}\) which are explicit. (But not just any solution will do!)
1.1.1.b
Hint 1.
Same as for the previous part.
Hint 2.
While \(L\) satisfies the hypotheses of Theorem 1.4, this theorem also requires \(M \ge 0\text{.}\) So your counterexample will have \(M \lt 0\text{.}\)
1.1.2. (PS1) A concrete application.
1.1.2.b
Hint.
The constant function \(-1\) is a ‘particular solution’ which solves the ODE but not the boundary conditions. Conclude that \(u_1\) must have the form
\begin{equation*} u_1 = Ae^x + Be^{-x} - 1 \end{equation*}
for some constants \(A,B\text{,}\) and then solve for these constants using the boundary conditions.
1.1.3. (PS1) Basic lemma for \(h \le 0\).
Hint.
Follow the proof of Lemma 1.1.
1.1.6. (PS1) Alternative proof of Theorem 1.2.
Hint.
Assume for the sake of contradiction that \(u(c)=M\) for some \(c \in (a,b)\text{,}\) and then apply Theorem 1.3 to the restrictions of \(u\) to \((a,c)\) and \((c,b)\text{.}\)

2 Preliminaries
2.3 The chain rule

Exercises

2.3.2. (PS2) Derivatives of symmetric functions.
2.3.2.e
Hint.
Using the summation convention, you will likely end up with a term involving \(\delta_{ii}=N\text{.}\)
2.3.2.f
Hint.
As with the previous part you will likely encounter a term involving \(\delta_{ii}=N\text{.}\)
2.3.3. The Kelvin transform.
Hint.
This is a challenging calculation, and somewhat beyond the scope of what we will need in this unit. Before attempting the general case, it might be worthwhile to look at the special case \(N=2\) where the formula for \(v\) is simpler.

2.5 Regularity of domains

Exercises

2.5.1. (PS3) A domain with a corner.
2.5.1.a
Hint.
Look for an interior ball. Putting the centre along the \(y\)-axis will make things easier.
2.5.1.b
Hint 1.
Draw some pictures to convince yourself that this is true! Proving it requires a bit more ingenuity than the previous part.
Hint 2.
One relatively simple proof given by a student in previous years can be outlined as follows. Suppose that \(B_r(x_0,y_0)\) were an exterior ball at \(0 \in \partial\Omega\text{,}\) and assume without loss of generality that \(x_0 \ge 0\text{.}\) Then argue that for sufficiently small \(x \gt 0\text{,}\) the point \((x,x) \in \partial\Omega\) satisfies \(\abs{(x,x)-(x_0,y_0)}^2 \lt r^2\text{.}\)
2.5.2. (PS3) Locally constant functions.
2.5.2.a
Hint.
Fix \(z \in u(\Omega)\text{.}\) Supposing that \(u \not \equiv z\text{,}\) prove that the sets \(\{ x \in \Omega : u(x)=z \}\) and \(\{x \in \Omega : u(x) \ne z\}\) are both open and nonempty.

2.6 Symmetric matrices

Exercises

2.6.1. Lemma 2.31 for positive definite matrices.
2.6.1.b
Hint.
Following the proof of Lemma 2.31, we can write \(A_{ij} B_{ij} = \lambda_k \mu_\ell (U^\top V)_{k\ell}^2 \ge 0\text{.}\) What would have to happen in order for the double sum on the right hand side to be exactly zero?
2.6.2. (PS3) Invariance of the Laplacian.
Hint 1.
Since \(A\) is orthogonal, \(AA^\top\) is the identity matrix, i.e. \(A_{ik} A_{jk} = \delta_{ij}\text{.}\)
Hint 2.
Recall Notation 2.14.

2.7 Maxima and minima

Exercises

2.7.1. (PS3) Minimising a quadratic polynomial.
2.7.1.d
Hint.
Use Corollary 2.29 to estimate \(\tfrac 12 \partial_{ij} f(y) h_i h_j\) from below. Plugging this estimate into the identity gives \(f(y+h) \le f(y)\) for all \(h \in \R^2\text{.}\)
2.7.1.e Optional.
Hint.
Look at the restrictions of \(f\) to lines spanned by eigenvectors of \(D^2 f\text{.}\)

2.8 Compactness and diagonal subsequences

Exercises

2.8.1. (PS3) Suprema, maxima, and closures.
Hint.
Clearly \(\sup_{\overline \Omega} f \ge \sup_\Omega f\text{.}\) To see the reverse inequality, recall that for any \(x \in \overline\Omega\) there exists a sequence \(x_n\) in \(\Omega\) which converges to \(x\text{,}\) and use the sequential continuity of \(f\text{.}\)
2.8.2. (PS4) Sufficient condition for equicontinuity.
Hint.
Let \(x \in B\) and consider the single-variable function \(g(t) = f_n(tx_0+(1-t)x)\text{.}\) Express \(g(1)-g(0)\) using either the mean value theorem or the fundamental theorem of calculus, and then estimate the result in terms of \(M\) and \(\abs{x-x_0}\text{.}\)
2.8.5. Sequences of functions on \(\R\).
2.8.5.b
Hint.
One option is to consider sequences of the form \(f_n(x)=g(x-n)\) for a fixed function \(g\) which vanishes outside of a fixed interval.

2.9 Integration and vector calculus

Exercises

2.9.1. Divergence theorem and the Laplacian.
Hint.
First apply the divergence theorem to \(\nabla u\text{.}\) Then assume for the sake of contradiction that there exists a point \(x \in \R^N\) where \(\Delta u - f\text{,}\) say. Now consider \(B=B_r(x)\) for small \(r \gt 0\text{,}\) and use the continuity of \(\Delta u - f\text{.}\)
2.9.2. Dirichlet principle.
2.9.2.a
Hint.
Apply the divergence theorem to \(v\nabla u\text{.}\)
2.9.2.b
Hint.
Expanding things out \(I(u+\varepsilon v)\) is a quadratic polynomial in \(\varepsilon\text{.}\)

3 Maximum principles for elliptic equations
3.1 Ellipticity and uniform ellipticity

Exercises

3.1.1. (PS4) Checking uniform ellipticity.
3.1.1.b
Hint.
One option is to work with (3.2) directly. Another option is to calculate the eigenvalues and get (3.2) from Corollary 2.29.
3.1.1.c
Hint.
Same as for previous part.
3.1.2. (PS4) Ellipticity after shifting and scaling.

3.2 The weak maximum principle

Exercises

3.2.1. Non-uniqueness.
3.2.1.a
Hint.
If you have any doubts about what the boundary of \(\Omega\) is, try drawing a picture.
3.2.1.b
Hint.
There are many ways to do this. One option which has been popular in previous years is to look for monomials \(x^\alpha\) which satisfy \(\Delta x^\alpha = 0\text{,}\) and then to define \(\Omega\) by thinking about where these monomials vanish.
3.2.3. (PS4) A two-sided estimate.
3.2.3.a
Hint.
Use the comparison principle. Note that while the operator \(\Delta\) is elliptic, the operator \(-\Delta\) is not.
3.2.4. (PS5) Domains contained in a slab.
3.2.4.a
Hint.
This is similar to an argument made in the proof of Theorem 3.2. Since \(c \le 0\) and \(z \ge 0\) in \(\Omega\text{,}\) the \(cz\) term is harmless.
3.2.4.b
Hint.
Note that \(-\sup_\Omega \abs f \le f \le \sup_\Omega \abs f\) in \(\Omega\text{,}\) and similarly for \(u\) on \(\partial\Omega\text{.}\)

3.3 The Hopf lemma and the strong maximum principle

Exercises

3.3.1. (PS5) Normal derivatives of radial functions.
Hint.
Think back to the first part of Exercise 2.3.2.
3.3.2. Minimum principles.
3.3.2.a
Hint.
Consider the function \(-u\text{,}\) and use the fact that \(\inf_\Omega u = -\sup_\Omega(-u)\text{.}\)
3.3.2.b
Hint.
Same as for the previous part.
3.3.3. A strong comparison principle.
Hint.
First apply Proposition 3.4. Then, apply Theorem 3.8 to the difference \(w = v-u\text{.}\) If \(c \equiv 0\text{,}\) then the argument is simpler; you may want to try that case first.
3.3.4. (PS5) Uniqueness for other boundary conditions.
3.3.4.a
Hint.
Consider the difference \(w=u-v\text{,}\) and apply the weak or strong maximum principle followed by Theorem 3.9. If \(c \equiv 0\text{,}\) then the argument is simpler; you may want to try that case first.
3.3.6. (PS5) Maximum principles in the reverse order.
3.3.6.a
Hint.
The function \(\tilde u = u - \sup_\Omega u\) satisfies \(\sup_\Omega \tilde u = 0\text{.}\)

4 The Dirichlet problem
4.1 Interior estimates

Exercises

4.1.1. (PS6) A maximum principle for the gradient.
Hint.
Write \(v = \partial_j u \partial_j u\) using the summation notation, and then differentiate twice using the product rule. As in the proof of Lemma 4.2, \(u\) being harmonic means that \(\partial_j \Delta u = \partial_{iij} u = 0\text{.}\)
As in the proof of Lemma 4.2, \(\abs{D^2 u}^2 = \partial_{ij} u\, \partial_{ij} u\) is the matrix norm from Definition 2.5.
4.1.2. (PS6) Interior estimate for second derivatives.
Hint.
The idea is to apply Lemma 4.2 to the \(C^3\) harmonic function \(v = \partial_i u\text{,}\) and then to \(u\) itself. Perhaps the trickiest part of the exercise is to get the various balls to nest correctly. One way to do this is to first fix a point \(x \in B_{r/2}(x_0)\text{,}\) and then consider the three balls
\begin{equation*} B_{r/8}(x) \subset B_{r/4}(x) \subset B_{r/2}(x) \subset B_r(x_0)\text{.} \end{equation*}

4.2 The Dirichlet problem in a ball

Exercises

4.2.1. (PS6) Explicitly solving a Dirichlet problem.
Hint.
Use (4.9).

4.3 The mean value property

Exercises

4.3.1. (PS7) Reflecting harmonic functions.
Hint.
By Theorem 4.11, it is enough to show that, for every \(x \in \Omega\text{,}\) \(u\) satisfies the mean-value property on \(B_r(x)\) for \(r\gt0\) sufficiently small. If \(x_N \gt 0\) or \(x_N \lt 0\text{,}\) pick \(r\) small enough that you can apply the mean-value property for \(u\) (or its reflection). If \(x_N = 0\text{,}\) use symmetry to argue that the average has to be \(0=u(x)\text{.}\)

4.4 Subharmonic functions

Exercises

4.4.1. (PS7) Directly checking \(\abs x\) is subharmonic.
4.4.1.c
Hint.
Note that an arbitrary point \(x \in (a,b)\) may be expressed as \(x=ta+(1-t)b\) for some \(t \in (0,1)\text{.}\)
4.4.3. (PS7) Harmonic liftings.
Hint 1.
This is much more subtle than the second part of Lemma 4.16. I highly recommend that you draw yourself a picture of the various sets involved. In particular, this is useful when you need to think about the boundary of an intersection.
Hint 2.
First show that \(u \le U\) in \(\Omega\text{.}\) Then, consider an arbitrary ball \(B'\) with \(\overline{B'} \subseteq \Omega\) and a harmonic function \(h\) on \(B'\) with \(U \le h\) on \(\partial B'\text{.}\) Certainly \(U \le h\) on \(B' \without B\text{,}\) and so it remains to show \(U \le h\) on the set \(B' \cap B\text{.}\) Since \(U\) and \(h\) are both harmonic on this set, there is a chance you can apply the comparison principle.

5 Radial symmetry
5.1 Statement of the theorem

Exercises

5.1.1. (PS8) Necessity of \(u\gt0\) in Theorem 5.1.
Hint.
Take \(N=1\) and think back to Exercise 1.1.1. If you are a big fan of Bessel functions and the Helmholtz equation, you may enjoy coming up with similar examples in dimensions \(N=2\) and \(N=3\text{.}\)
5.1.3. (PS8) Very old exam question.
Hint.
There are many possible techniques for tackling this problem
  1. Consider the restriction of \(u\) to the set \(\{x \in \Omega : u \gt 1\}\text{.}\)
  2. Evaluate the PDE at a maximum \(\gt 1\) or a minimum \(\lt -1\text{,}\) and think about the meaning of the sign of \(\Delta u\)
  3. Use Lemma 5.4 (or the idea behind its proof) to compare \(u\) to an appropriate constant function.

5.3 A semilinear comparison principle

Exercises

5.3.1. (PS9) Necessity of an assumption in Lemma 5.4.
Hint 1.
Take \(N=1\) and think back to Exercise 1.1.1.
Hint 2.
Consider solutions of \(u''+u=v''+v=0\text{.}\)

5.4 Proof of the theorem

Exercises

5.4.1. A calculation needed in Lemma 5.6.
5.4.1.a
Hint.
The summation convention is not particularly useful here. You will need to calculate \(\partial_1 \phi_1\) and \(\partial_1 \phi_2\) at \(x_1=0\text{.}\)
5.4.1.b
Hint.
The summation convention is not particularly useful here. You will need to calculate \(\partial_{11} \phi_1\) and \(\partial_{11} \phi_2\) at \(x_1=0\text{.}\)
5.4.1.d
Hint.
This is not particularly easy; it’s optional for a reason.
5.4.2. (PS9) Monotonicity in an annulus.
Hint.
There is no need to rewrite the entire proof of the theorem and all of the lemmas! This question is primarily here to get you to go back and work through all of the steps in the proof, seeing where the modifications need to be made for this different geometry.
First, convince yourself that a version of Lemma 5.6 still applies on the outer boundary \(\abs x = R_2\text{.}\) By drawing some pictures, convince yourself that the proofs of Lemma 5.7 and Lemma 5.8 still work provided we restrict our attention to \(\lambda \ge (R_1+R_2)/2\text{.}\)
5.4.3. (PS9) Gidas–Ni–Nirenberg for an ellipse.
Hint.
As with Exercise 5.4.2, do not worry too much about writing things out in complete detail. This exercise is mostly meant to get you thinking.

6 Maximum principles for parabolic equations
6.2 The weak maximum principle

Exercises

6.2.2. Asymptotics for the heat equation.
6.2.2.a
Hint.
Be careful with all of the minus signs when calculating \(\partial_t v\text{!}\)
6.2.3. (PS10) Comparison principle.
Hint.
Show that change of variable \(\bar u = e^{\gamma t} u\) and \(\bar v = e^{\gamma t}v\) transforms the inequality \(L u \le Lv\) into an inequality \(\bar L \bar u \le \bar L \bar v\) where \(\bar L\) is a uniformly parabolic operator. Then choose \(\gamma\) appropriately so that the zeroth order coefficient \(\bar c\) of \(\bar L\) is \(\le 0\text{.}\)

6.3 Semilinear comparison principles

Exercises

6.3.1. (PS10) Uniqueness for semilinear problems.
Hint.
Use the (semilinear) comparison principle.
6.3.2. (PS10) Bounding a solution to a semilinear problem.
Hint.
The constant functions \(0\) and \(\pi\) can be used as comparison functions.
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