Ross Ogilvie      14th October, 2024
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20.

The only constant is change

Let ${\lambda }_{𝜀}:{ℝ}^n\to ℝ$ be the standard mollifier. Let $F\in 𝒟(\mathrm{\Omega })$ be any distribution, not necessarily regular.

(a)

For any point $a\in \mathrm{\Omega }$, explain why $F({\lambda }_{𝜀}(x-a))$ is well-defined for $𝜀$ sufficiently small.
(1 point)

(b)

Expand the definitions to show $({\lambda }_{𝜀}\ast F)(a)=F({\lambda }_{𝜀}(x-a))$. (2 points)

(c)

Suppose that $F$ has the property that $F({\lambda }_{𝜀}(x-a))=0$ 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 $\phi \in 𝒟(\mathrm{\Omega })$ has total integral zero,

$${\int \; <!--\; nolimits\; -->}_{\mathrm{\Omega }}\phi (x)\mathit{𝑑𝑥}=0,$$

then $F(\phi )=0$. Prove that $F=F_c$ for $c\in ℝ$ the constant function. (3 points)
Hint. Define $c=({\lambda }_r\ast F)(a)$.

21.

Twirling towards freedom

Let $u\in C^2({ℝ}^n)$ be a harmonic function. Show that the following functions are also harmonic.

(a)

$v(x)=u(x+b)$ for $b\in {ℝ}^n$.

(b)

$v(x)=u(\mathit{𝑎𝑥})$ for $a\in ℝ$.

(c)

$v(x)=u(\mathit{𝑅𝑥})$ for $R(x_1,\dots ,x_n)=(-x_1,x_2,\dots ,x_n)$ the reflection operator.

(d)

$v(x)=u(\mathit{𝐴𝑥})$ for any orthogonal matrix $A\in O({ℝ}^n)$.

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 $u\in C^{\infty }({ℝ}^n)$ 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 $f_n:{ℝ}^2\to ℂ$ given by $f_n(x,y)={(x+\mathit{𝜄𝑦})}^n$ and let $u_n(x,y)$ and $v_n(x,y)$ be its real and imaginary parts respectively. Show that $u_n$ and $v_n$ are harmonic. (3 points)

(c)

A homogeneous polynomial of degree $n$ in two variables is a polynomial of the form $p=\sum <!--\; FUNCTION\; APPLICATION\; -->a_k{x}^k{y}^{n-k}$. Show that a homogeneous polynomial of degree $n$ is harmonic if and only if it is a linear combination of $u_n$ and $v_n$. (2 points + 2 bonus points)