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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 \;,
\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
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