Question about the conformal map in the dirchlet problem(Stein and Shakarchi) The direchlet problem on a strip is given as:
.
For a more general Dirichlet problem, we have the following paragraph from Stein and Shakarchi:


My questions are:

*

*We solve the problem by taking the strip to a disc with a conformal map. Shouldn't the function F, go from $\Omega \to D$?  Instead Shakarchi says $D \to \Omega$.

*Also why is the symbol $\partial \Omega$ used to signify the boundary.

*What is the Poisson integral formula give - Does it give the solution to the boundary value problem?

I would appreciate some answers. I am self-studying stein and shakarchi.
 A: $1.\,\,$ $F$ goes from the disc to $\Omega$ and $F^{-1}$ goes from $\Omega$ to the disc.
$2.\,\,$  We need some symbol to signify the boundary.  Is your complaint that this is overloaded because the symbol is also used for partial derivative?  The meaning is clear from the context.
$3.\,\,$  Yes, the Poisson integral formula gives the solution on the disc to the Dirichlet problem.  See Stein and Shacharchi, Princeton Lectures in Analysis Volume II (Complex Analysis), Chapter 2, Problem 12, pp. 66 and 67.
$\underline{\texttt{UPDATE}} \text{ (further discussion on 1.)}$
If $F:\mathbb{D}\to\Omega$
then $\tilde{f}=f\circ F$ is defined on the circle (the boundary of the disc).  We can then solve for $\tilde{u}$ inside the disc, and then the solution in the strip is given by $u=\tilde{u} \circ F^{-1}.$
It turns out we need both $F$ and $F^{-1}$, so if you want to switch their roles, I guess it is okay, just notation, but since you are following the book, why not stick to the notation that is there.
