Introduction to Partial Differential Equations
Exercise Sheet 7

Ross Ogilvie      14th October, 2024
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20.
The only constant is change

Let λ𝜀 : n be the standard mollifier. Let F 𝒟(Ω) be any distribution, not necessarily regular.

(a)
For any point a Ω, explain why F(λ𝜀(x a)) is well-defined for 𝜀 sufficiently small.
(1 point)
(b)
Expand the definitions to show (λ𝜀 F)(a) = F(λ𝜀(x a)). (2 points)
(c)
Suppose that F has the property that F(λ𝜀(x a)) = 0 for all a,𝜀 (for which it is defined). Argue using Exercise 19 that F = 0. (2 points)
(d)
Suppose that F has the following property: if a test function φ 𝒟(Ω) has total integral zero,
Ωφ(x)𝑑𝑥 = 0,

then F(φ) = 0. Prove that F = Fc for c the constant function. (3 points)
Hint. Define c = (λr F)(a).

21.
Twirling towards freedom

Let u C2(n) be a harmonic function. Show that the following functions are also harmonic.

(a)
v(x) = u(x + b) for b n.
(b)
v(x) = u(𝑎𝑥) for a .
(c)
v(x) = u(𝑅𝑥) for R(x1,,xn) = (x1,x2,,xn) the reflection operator.
(d)
v(x) = u(𝐴𝑥) for any orthogonal matrix A O(n).

Together these show that the Laplacian is invariant under similarities (Euclidean motions, reflection and rescaling). (6 points)

22.
Harmonic Polynomials in Two Variables
(a)
Let u C(n) be a smooth harmonic function. Prove that any derivative of u is also harmonic. (1 point)
(b)
Choose any positive degree n. Consider the complex valued function fn : 2 given by fn(x,y) = (x + 𝜄𝑦)n and let un(x,y) and vn(x,y) be its real and imaginary parts respectively. Show that un and vn are harmonic. (3 points)
(c)
A homogeneous polynomial of degree n in two variables is a polynomial of the form p = akxkynk. Show that a homogeneous polynomial of degree n is harmonic if and only if it is a linear combination of un and vn. (2 points + 2 bonus points)