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I have three boundary value problems.

  1. Find common solution of boundary value problem for Laplace's equation in a ball in $\mathbb{R}^3$. $ \begin{cases} \Delta u = 0 & \text{ if r $<$ 1} \\ u - \partial_n u = 3 \cos(\varphi)^2 & \text{ if r $=$ 1} \end{cases} $

I got the answer $u = P_0 (\cos(\theta)) - 2r^2 P_2 (\cos(\theta))$, where P is Legendre's polynomial.

But the right answer is $u = P_0 (\cos(\theta)) - 2r^2 P_2 (\cos(\theta)) + r Y_1 (\theta, \phi)$ where $Y_1$ is spherical function of order 1. How can I get the spherical function?

  1. Find common solution of boundary value problem for Laplace's equation in a disk in $\mathbb{R}^2$. $ \begin{cases} \Delta u = 0 & \text{ if r $<$ 1} \\ u - \partial_n u = 2 \cos(\varphi)^2 & \text{ if r $=$ 1} \end{cases} $

My answer is $u = 1 - r^2 \cos(2\varphi)$,

but the right answer is $u = 1 - r^2 \cos(2\varphi) + r (a\cos\varphi + b\sin\varphi)$ where $a$ and $b$ arbitrary constants.

Similarly, I do not understand how to get the last term.

  1. At what ratio between $A$ and $B$ boundary problem for Laplace's equation in $\mathbb{R}^3$ has solutions? Find all of them.

$ \begin{cases} \Delta u = 0 & \text{ if 1 $<$ r $<$ 2} \\ \partial_n u |_{r=1} = A + 2 \cos\theta \\ \partial_n u |_{r=2} = B - 2 \cos\theta \end{cases} $

Can you give me advice how to apply solvability condition to find this ratio or maybe I need to do something different here?

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