Introduction to Partial Differential Equations
Exercise Sheet 9

Ross Ogilvie      28th October, 2024
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27.
To be or not to be

Consider the Dirichlet problem for the Laplace equation Δu = 0 on Ω with u = g on Ω, where Ω n is an open and bounded subset and g is a continuous function. We know from the weak maximum principle that there is at most one solution. In this question we see that for some domains, existence is not guaranteed.

(a)
Consider Ω = B(0,1) {0}, so that the boundary Ω = ∂𝐵(0,1) {0} consists of two components. We write g(x) = g1(x) for x ∂𝐵(0,1) and g(0) = g2. Show that there does not exist a solution for g1(x) = 0 and g2 = 1.

Hint. Use Lemma 3.23, even if we haven’t reached it in lectures yet. (3 points)

(b)
Generalise this: What are the necessary and sufficient conditions on g for the Dirichlet problem to have a solution on this domain? (3 points)
(c)
Generalise again: What can you say about the Dirichlet problem for bounded domains whose boundaries have isolated points? (1 point)
28.
Do nothing by halves

Let H1+ = {x = (x1,,xn) nx1 > 0} be the upper half-space and H10 = {x = (x1,,xn) nx1 = 0} the dividing hyperplane. We call R1(x) = (x1,x2,,xn) reflection in the plane H0.

(a)
A reflection principle for harmonic functions Let u C2(H1+¯) be a harmonic function that vanishes on H10. Show that the function v : n defined through reflection
v(x) = { u(x)  for x1 0 u(R1(x)) for x1 < 0

is harmonic. (3 points)

(b)
Green’s function for the half-space Show that Green’s function for H1+ is
G(x,y) = Φ(x y) Φ(R1(x) y).

(2 points)

(c)
Green’s function for the half-ball Compute the Green’s function for B+. (3 points)
Hint. Make use of both the Green’s function for the ball and part (b).
29.
Teach a man to fish
(a)
Using the Green’s function of H1+ from the previous question, derive the following formal integral representation for a solution of the Dirichlet problem Δu = 0 in H1+,u|H10 = g
u(x) = 2x1 nωnH10 g(z) |x z|ndσ(z)

Here, ‘formal’ means that you do not need to prove that the integrals are finite/well-defined.
(3 points)

(b)
Show that if g is periodic (that is, there is some vector L n1 with g(x + L) = g(x) for all x n1) then so is the solution. (2 points)
(c)
Now consider the plane n = 2 with g function with compact support. Approximate the value of u(x) for large |x|. Feel free to modify this question as you see fit, what interesting things can you say about the growth of u? (Bonus Points as deserved)