$\newcommand{\const}{\mathrm{const}}$ $\newcommand{\erf}{\operatorname{erf}}$
Deadline Wednesday, December 5.
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 $A$.
Apply method of descent described in Subsection 28.4 but to Laplace equation in $\mathbb{R}^2$ and starting from Coulomb potential in $3D$ \begin{equation} U_3(x,y,z)=-\frac{1}{4\pi} \bigl(x^2+y^2+z^2\bigr)^{-\frac{1}{2}}, \label{eq-1} \end{equation} derive logarithmic potential in $2D$ \begin{equation} U_2(x,y,z)=\frac{1}{2\pi}\log \bigl(x^2+y^2\bigr)^{\frac{1}{2}}, \label{eq-2} \end{equation} 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 $N\to \infty$.
Using method of reflection (studied earlier for different equations) construct Green function for
for Laplace equation in
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.
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.