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Victor Ivrii:
Consider problem:
\begin{align}
& \Delta u=0 &&\text{in   }x^2+y^2<1, \ \ y>0,
\label{1}\\
& u|_{y=0}=x^2 &&\text{as   }|x|<1,\label{2}\\
& u|_{x^2+y^2=1}=1,&& \text{as   } y>0.
\label{3}
\end{align}
We want to separate variable $r$ and $\theta$ but the conditions as $\theta=0,\pi$ are inhomogeneous.

So we want to make them homogeneous. Find $v$, so that $u:=v$ satisfies (\ref{1}) and (\ref{2}) but not necessarily (\ref{3}), so $v$ is not unique. Can you suggest a candidate?

Then $w=u-v$ will satisfy (\ref{1}), homogeneous condition (\ref{2}), modified (\ref{3}). Find $w$ by separation, and then $u=v+w$.

Andrew Hardy:
Only considering (1) and (2),
My candidate is
$$ v = x^2 -y^2 $$
which is harmonic, (has zero laplacian) and $$ v(x,0) = x^2 $$
By homogeneous (2) do you mean?
$$ w|_{y=0}=0  $$
How do I modify (3) ?

Victor Ivrii:

--- Quote ---How do I modify (3) ?

--- End quote ---
Calculate $w|_{x^2+y^2=1}$ as $y>0$, if $u$ satisfies (\ref{3}) and $v=x^2-y^2$ you know.

Andrew Hardy:
 To calculate $w|_{x^2+y^2=1} $ I make a substitution that if  $$u|_{x^2+y^2 =1}=1   $$ then  $$ v|_{x^2+y^2 =1}= 1  - y^2 - y^2 = 1-2y^2 $$
$$ v|_{x^2+y^2 =1}= x^2 - (1-x^2) = 2x^2 - 1 $$
 so then  $$w|_{x^2+y^2=1}=2y^2 = 2- 2x^2$$ 

Not sure if I'm on the right track, I'll return to this later today.

Sheng Gao:
By Laplace equation
$u_{rr} +\frac{1}{r}u_r +\frac{1}{r^2}u_{\theta\theta}=0,  r<1, 0<\theta<\pi$
$u(r,0)=r^2$
$u(r,\pi)=r^2$
$u|_{r=1}=1$

let $v=x^2-y^2$, then
$\Delta v=v_{xx}+v_{yy}=2-2=0$
$v|_{y=0}=x^2$  $v|_{x^2+y^2=1}=1-2y^2=1-2r^2sin\theta=1-2sin\theta, since$  $r=1$

And
Let $w=u-v, then    \Delta w=0$ $w|_{y=0}=0$, $w|_{x^2+y^2=1}=2sin^2\theta$
                                         

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