Problems to Chapter 8

$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$

### Problems to Chapter 8

Problem 1.

1. Write Laplace equation in spherical coordinates.
2. Look for a solution in the form $u=P(\rho)\Phi(\phi)\Theta(\theta)$.
3. Write equations and their solutions for $P$ and $\Theta$.
4. Write equation (with parameters $l$ and $m$) for $\Phi$; what values can take $(l,m)$?

Problem 2.

For equation to $\Phi$ find solutions in the form $L(\cos(\phi))$ where $L$ is a polynomial of degree $l$ with respect to $\cos(\phi),\sin(\phi)$ in the following cases:

1. $l=0$ and all possible values $m$;
2. $l=1$ and all possible values $m$;
3. $l=2$ and all possible values $m$;
4. $l=3$ and $m=0$.

Problem 3.

1. Solve \begin{align} &\Delta u=0 && x^2+y^2+z^2< a^2,\label{8.P.1}\\ &u=g(x,y,z) && x^2+y^2+z^2=a^2 \label{8.P.2} \end{align} with $g(x,y,z)$ defined below and $a=1$.
Hint. If $g$ is a polynomial of degree $m$ look for $$u=g - P(x,y,z)(x^2+y^2+z^2-a^2) \label{8.P.3}$$ with $P$ a polynomial of degree $(m-2)$. Here $a$ is the radius of the ball. If $g$ has some rotational or reflectional symmetry, so $P$ has.
2. Represent $u$ as a sum of homogeneous harmonic polynomials.
3. Plug $x=\rho \sin(\phi)\cos(\theta)$, $y=\rho \sin(\phi)\sin(\theta)$, $z=\rho \cos(\phi)$.

4. $g=x^2+y^2-z^2$.
5. $g=z(x^2+y^2)$.
6. $g=xyz$.
7. $g=x^4+y^4+z^4$.
8. $g=x^2yz$.
9. $g=xyz^2$.
10. $g=x^2y^2$.
11. $g=x^2z^2$.
12. $g=xz^3$.
13. $g=x^3z$.

Problem 4. Find function $u$, harmonic in $\{x^2+y^2+z^2\ge a^2\}$, decaying at infinity, and coinciding with $g(x,y,z)$ as $x^2+y^2+z^2=a^2$ with $g(x,y,z)$, defined in Problem 3 (a)-(j), and $a=1$.
Hint. Combine Problem 4 and Problem 7.P.6.