Let $D=\{z\in\mathbb{C}|z\le1\}\setminus{-1,1}$.Find an explicit function $u$ on $D$ such that u is continuous on $D$ and harmonic in the interior $D$ and $u(z)=1$ on $|z|=1,Im\ z>0$ and $u(z)=-1$ on $|z|=1,Im\ z<0$.
My first guess is to use the Poisson Kernel to represent such a harmonic function,but Poisson Kernel need the harmonic function continuous up to the whole boundary,there are two jumping points here is the formula still valid?
Even though we use the Poisson Kernel,it seems that the integral cannot produce an explicit function.