= Solution
Set $u=\cos\theta$. The boundary data simplify to $u+2u^2=\frac23P_0(u)+P_1(u)+\frac43P_2(u)$. The <axisymmetric harmonic extension from a sphere> keeps only decaying radial powers outside and regular powers inside. Hence
$$
\boxed{\Psi_{\rm out}=\frac{2a}{3r}+\frac{a^2}{r^2}\cos\theta+\frac{4a^3}{3r^3}P_2(\cos\theta),\quad r\ge a,}
$$
and
$$
\boxed{\Psi_{\rm in}=\frac23+\frac ra\cos\theta+\frac{4r^2}{3a^2}P_2(\cos\theta),\quad r\le a.}
$$
Each <Legendre polynomial> coefficient agrees at $r=a$. The exterior solution tends to zero, and the interior solution is finite at $r=0$. The <maximum principle for harmonic functions> proves <uniqueness> in these respective classes.
Back to article page
