So, I'm revisiting Complex Analysis mainly focused on the application of conformal maps in PDE's. Using Churchill's book (Complex Variables and Applications), the way to solve was using the fact that harmonic functions still harmonic after composed with (complex) conformal maps, the same for the boundary conditions. All of this together with the Riemann Mapping Theorem, I was hoping that will solve the dirichlet problem in any Jordan Region, by sending the problem to the unit circle, solving with the Poisson Integral, and returning the problem. That last part of the transformations of boundary conditions was the point where the proof did not feel quite right or rigorous enough to me. Does the Churchill approach really proves this relation?
To me I didn't feel satisfied, it was better to study a little more, so I get the Ahlfors one. In this book, after the Riemann Mapping Theorem, it is used the idea of approaching the boundary, and it is shown that this map can be extended to the boundary. So, again, I was hoping that the Riemann Mapping Theorem would be enough, but it actually goes for subharmonic functions and Perron's method (also, Conway's book does this too), and it actually proves (almost, the Conway's do the full proof) that the problem can be solved in any Jordan region.
In the end I couldn't see why the idea of sending the problem to the unit circle, solving and then returning was not used, that is also how is used to solve problems. Does this Ahlfors proof shows that this approach works? Because it seemed to me a completely different way to solve.
My goal was to prove that the approach of solving in a easier region works, to use it to solve some problems.