Laplace equation on semidisk

Question

I am interested in the solution of the following boundary value problem on the semidisk $D=\{(r,\theta): 0<r<1, 0<\theta<\pi\}$: $$u_{xx}+u_{yy}=0 \mbox{ in } D, $$ $$u(1,\theta)=0 \mbox{ for } 0<\theta<\pi$$ and $$u(x,0)=f(x) \mbox{ for } -1<x<1$$ where $f(x)$ is a given function. I don't see if separation of variables could work, or maybe I need more advanced method.

Answer 1

Firstly you need to use polar coordinates, so you seek the solution:

$u(x,y)=u(r(x,y),\theta(x,y))=R(r)\Theta (\theta)$ (here we use separation of variables)

If you follow the calculations through, you will be lead to a cauchy euler equation.

Can you take it from here?

Also $r^2=x^2+y^2$ and $\tan\theta=\frac { y } { x}$.

Answer 2

Following Hans Lundmark's suggestion, we compose 3 maps: $\frac{z+1}{1-z}$, $z^2$, $\frac{-i+z}{i+z}$, to get the map $$h(z)=\frac{(1+z)^2-i(1-z)^2}{(1+z)^2+i(1-z)^2}$$ which maps the upper boundary of the semidisk to the upper boundary of the unit disk, and the lower line segment of the semidisk to the lower boundary of the unit disk.

Now we solve the problem on the unit disk, with $g(\theta)=0$ on the upper semicircle, and $$g(\theta)=f\left(\frac{1-\sqrt{-\tan(\theta/2)}}{1+\sqrt{-\tan(\theta/2)}}\right)$$ on the lower semicircle. See this post. But finding Fourier coefficients of $g$ seems to be formidable even if $f$ is simple. So I am not sure if this approach is the best way to solve this problem.