Is it possible to find the following 2D Laplace equation's analytical solution on the shadowed region when having such Dirichlet boundar condition:
$$\dfrac{\partial^2 u(x,y)}{\partial x^2}+\dfrac{\partial^2 u(x,y)}{\partial y^2}=0$$
I found there were already similar analytical solution results here to Laplace equation on a rectangular region, but that solution does not handle rectanglular regions with a rectangle hole in it.
With $a=50, u_0=0, u_1=100$ specified, for the rectangle region with a rectangle hole on it, it is also easy to find the numerical solution to such a PDE, which can be visualized as:
How can I obtain its analytical or explicit solution? or how to prove such an analytical/ explicit solution cannot be obtained?