$\newcommand{\const}{\mathrm{const}}$ $\newcommand{\erf}{\operatorname{erf}}$

### APM 346 (2012) Home Assignment 9

#### Problem 1

Find function $u$ harmonic in $\{x^2+y^2+z^2\le 1\}$ and coinciding with $g=z^3$ as $x^2+y^2+z^2=1$.

Hint. According to Subsection 30.1 solution must be a harmonic polynomial of degree $3$ and it should depend only on $x^2+y^2+z^2$ and $z$ (Explain why). The only way to achive it (and still coincide with $g$ on $\{x^2+y^2+z^2=1\}$) is to find \begin{equation*} u= z^3 + az(1-x^2-y^2-z^2) \end{equation*} with unknown coefficient $A$.

#### Problem 2

Apply method of descent described in Subsection 28.4 but to Laplace equation in $\mathbb{R}^2$ and starting from Coulomb potential in $3D$ $$U_3(x,y,z)=-\frac{1}{4\pi} \bigl(x^2+y^2+z^2\bigr)^{-\frac{1}{2}}, \label{eq-1}$$ derive logarithmic potential in $2D$ $$U_2(x,y,z)=\frac{1}{2\pi}\log \bigl(x^2+y^2\bigr)^{\frac{1}{2}}, \label{eq-2}$$ Hint. You will need to calculate diverging integral $\int_0^\infty U_3 (x,y,z)$. Instead consider $\int_0^N U_3 (x,y,z)$, subtract constant (f.e. $\int_0^N U_3 (1,0,z)$) and then tend $N\to \infty$.

#### Problem 3

Using method of reflection (studied earlier for different equations) construct Green function for

1. Dirichlet problem
2. Neumann problem

for Laplace equation in

1. half-plane
2. half-space

as we know that in the whole plane and space they are just potentials \begin{gather} \frac{1}{2\pi}\log \bigl((x_1-y_1)^2+(x_2-y_2)^2\bigr)^{\frac{1}{2}},\label{eq-3}\\ -\frac{1}{4\pi} \bigl((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2\bigr)^{-\frac{1}{2}} \label{eq-4} \end{gather} respectively.

#### Problem 4

Apply method of descent but now looking for stationary solution of $-\Delta u=f(x_1,x_2,x_3)$ instead of non-stationary solution of \begin{align*} & u_{tt}-\Delta u=f(x_1,x_2,x_3),\\ & u|_{t=0}=g(x_1,x_2,x_3),\\ & u_t|_{t=0}=h(x_1,x_2,x_3) \end{align*} start from Kirchhoff formula (28.12) and derive for $n=3$ (26.10) with $G(x,y)$ equal to (\ref{eq-4}) here.