Solve Unperturbed Field on Rectangle w/ Small Disk

[Hootenanny]

Homework Statement

The overall question is to construct an asymptotic approximation for a harmonic field on a rectangle with a small disk. However, I'm having difficulty finding the unperturbed field. Perhaps I've been staring at it for too long, but I can't seem to find a solution.

The unperturbed field satisfies the following Neumann problem:

$$\Delta u =0 \;\;\;\;\;\;\text{in}\;\;\;\;\;\;\Omega = \left\{\left(x_1,x_2\right)\; : \; \left|x_1\right| < 2 \;,\; \left|x_2\right| < 1 \right\}\;,$$

$$\left.\frac{\partial u}{\partial x_2}\right|_{x_1=\pm2} = 1 \;,$$

$$\left.\frac{\partial u}{\partial x_1}\right|_{x_2=\pm1} = 2 \;,$$​

Homework Equations

N/A

The Attempt at a Solution

The usual method to solving such problems is to make an educated guess, however, I'm having some problems guessing the solution. Clearly u(x) = Const. is not a solution. My first thought was that u(x) must either be a linear combination of x₁ and x₂, or a linear combination of x₁.x₂. However, as far as I can see, none of these functions can satisfy either the top and bottom or left and right boundary conditions simultaneously.

A nudge towards the correct 'guess', or any other help would be very much appreciated.

Edit: I just thought that I'd add that a solution does exist since the existence condition is clearly satisfied,

$$\oint_{\partial\Omega}\frac{\partial u}{\partial n}dS = 0$$

 

[Dick]

u(x1,x2)=2*x1+x2?? Am I missing something?

 

[Hootenanny]

u(x1,x2)=2*x1+x2?? Am I missing something?

Gahh! I knew that it was going to be simple and I was just being dense!

#Bangs head against desk#

Many thanks Dick!