Using conformal mapping to solve a boundary value problem, Use conformal mapping to solve the following boundary value problem for $u=u(x,y)$ in the planar region $R=\{(x,y) \in \mathbb{R}^2: x^2 + y^2 > 1 \text{ and } y>0\}$:


*

*u solves $U_{xx}+U_{yy}=0$ in R 

*$\frac{d}{d_n}=0$ on the boundary of R, where $\frac{d}{d_n}$ denotes the normal derivative.

*$U_x(x,y)\to 1$  , as $x^2+y^2\to\infty$ in R.


I have no experience in solving these types of problems, nor do I have any experience in using conformal mappings / linear fractional transformations.  So, I don't know how to even get started.  Any help will be greatly appreciated. 
Overall, I do not even know what the question is really trying to ask, so if someone can start off by helping me understand what the question is, that'd be awesome.  (FYI, this is not homework, just old problems that have been written for our math department.)
I know that the first bulletpoint means we need to find a function u that is harmonic in R, which is the region in the upper half plane, and outside of the unit disk.  I don't know what the second and third bulletpoints are even asking for, let alone proceed to write up a possible solution.
(I'll start with finding out what a normal derivative is...)
 A: The question asks you to solve the Laplace equation in an indented domain. The domain is the upper half-plane less the upper half of the unit disk $x^2+y^2 < 1$. This is an uncomfortable geometry to work with, so they want you to use a conformal mapping $w=F(z)$ to transform the geometry into something easier to handle - I'm guessing into another half-plane - where the solution to Laplace's equation can be written in terms of boundary integrals.
The normal derivative indicates the rate of change of $u$ along the outward normal direction on the boundary of your region at any point on the boundary. Specifying $\partial_n u =0$ is a typical "insulated" boundary condition: If you imagined $u$ as the temperature in a steady-state heat conduction problem, it suggests that no heat flow occurs across the boundary. This is one example of a Neumann boundary condition, where you impose conditions on the derivatives of your unknown quantity.
The third condition is a boundary condition at infinity. Typically you require a bc at infinity to get a unique solution to the bvp. The bc at infinity also has physical significance - in this case (interpreted physically), it is like a constant heat-flux at infinity.  
