What's the solution to Laplace's equation $\nabla^2V=0 $ in the annulus with centre 0, inner radius 1, and outer radius 2, with boundary conditions $V=0$ on the inner boundary and $V=1$ on the outer boundary?
By separating variables, I got the general solution $V(r,\theta)=\Sigma[(C_n r^n + D_n r^{-n})(A_n \cos n\theta + B_n \sin n\theta)]$. The inner boundary condition gives $D_n = -C_n \forall n$. The outer boundary condition gives $\Sigma[C_n(2^n - 2^{-n})(A_n \cos n\theta + B_n \sin n\theta)] = 1.$ But this gives a Fourier series with zero constant term being identically equal to 1. By uniqueness of Fourier series, isn't this a contradiction?
Or is there a better method than using separation of variables, which gives a different general form for the solution?
Many thanks for any help with this!