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The system I'm trying to solve is $$\nabla^2 C_{(r,\theta)} =0$$ $$C_{(\infty,\theta)}=C_0$$ $$ [ \frac{\partial C_{(r,\theta)}}{\partial r} \cos(\theta)+\frac{1}{r}\frac{\partial C_{(r,\theta)}}{\partial\theta} \sin(\theta) ]|_{r=a*\cos(\theta)}=\frac{k}{D}C_{(r,\theta)}|_{r=a*\cos(\theta)}$$ and I have no idea how to enter the bottom equation into Mathematica. The problem I have is the inclusion of the boundary condition. How can I indicate to Mathematica that I want the derivatives to be taken and then have everything evaluated on the sphere $r=a*\cos(\theta)$ ?

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    $\begingroup$ I talked about solving PDEs on non-rectangular domains here and here. Note v10 seems to have added some new functionality, at least for numerical solutions. $\endgroup$ – Timothy Wofford Jul 23 '14 at 7:46