Laplacian with Robin Boundary Conditions Question:
$\bigtriangledown^2u=0, r<a$
$\frac{\partial{u}}{\partial{r}}+\gamma u = h(\theta), r=a$
This one is giving me a hard time.
I guess $u(r,\theta) = R(r)\Theta(\theta)$
and get somewhere for the pde... but the Bonundary Conditions .. I have no idea what to do with.
Question states to solve and write the answer in terms of Fourier coefficients of $h$.
please help
 A: Step by step solution with details to be filled in:


*

*Apply separation of variables $u=R\Theta$ to the Laplacian in polar-coordinates: $\nabla^2u = u_{rr} + \frac{u_r}{r} + \frac{u_{\theta\theta}}{r^2} = 0$ to get the two equations 


$$R_{rr} + \frac{R_r}{r} - \frac{k^2R}{r^2} = 0$$
$$\Theta_{\theta\theta} + k^2\Theta = 0$$


*

*Solve for $\Theta$ and based on the solution give an argument for why $k$ has to be integer (and why we only need only consider $k\geq 0$).

*For $R(r)$ try a test solution $R = r^n$ to find the two solutions. Discard the solution that is not finite at $r=0$.

*The steps above gives us that the most general solution can be written $u(r,\theta) = \sum_{k=0}^\infty r^k\left(A_k\sin(k \theta) + B_k\cos(k\theta)\right)$

*Apply the boundary condition to the expression above to get $h(\theta) = \sum_{k=0}^\infty \left(ka^{k-1}  +\gamma a^k\right)\left(A_k\sin(k \theta) + B_k\cos(k\theta)\right)$. 

*Finally expand $h$ in a Fourier series $h(\theta) = \sum_{k=0}^\infty \alpha_k \sin(k\theta) + \beta_k \sin(k\theta)$ and compare with the formula above to get an expression for $A_k,B_k$ in terms of the Fourier coeficients ($\alpha_k,\beta_k$) of $h$. This will give you
$$u(r,\theta) = \sum_{k=0}^\infty \frac{(r/a)^k}{k/a + \gamma}\left(\alpha_k\sin(k \theta) + \beta_k\cos(k\theta)\right)$$
