$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$
Problem 1.
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:
Problem 3.
a. Solve
\begin{align}
&\Delta u=0 && x^2+y^2+z^2< 1,\label{8.P.1}\\
&u=g(x,y,z) && x^2+y^2+z^2=1 \label{8.P.2}
\end{align}
with $g(x,y,z)$ defined below.
Hint. If $g$ is a polynomial of degree $m$ look for
\begin{equation}
u=g - P(x,y,z)(x^2+y^2+z^2-R^2)
\label{8.P.3}
\end{equation}
with $P$ a polynomial of degree $(m-2)$. Here $R$ is the radius of the ball. If $g$ has some rotational symmetry, so $P$ has.
b. Represent $u$ as a sum of homogeneous harmonic polynomials.
c. Plug $x=\rho \sin(\phi)\cos(\theta)$, $x=\rho \sin(\phi)\sin(\theta)$, $z=\rho \cos(\phi)$.