I'm trying to understand what the essential differences between Dirichlet/Neumann and Robin boundary conditions are. Therefore, let $\omega \in \left(0, 2\pi\right)$ and let \begin{equation*} \Omega = \left\{ \left(r,\varphi\right) : r > 0, \varphi \in \left(0, \omega\right) \right\} \end{equation*} be the wedge with angle $\omega$ in the origin.

Consider for $\alpha \in \mathbb{C} \setminus \left\{ 0 \right\}$ the Laplace equation with Robin boundary conditions: \begin{align*} \Delta u &= 0 \quad \text{in } \Omega, \\ \partial_n u + \alpha u &= 0 \quad \text{on } \partial \Omega. \end{align*}

Because of the simple geometry we have $\partial_n = \pm \frac{1}{r} \partial_\varphi$. Therefore, the above Robin boundary condition can be written as

\begin{align*} -\frac{1}{r}\partial_\varphi u + \alpha u &= 0 \quad \text{for } \varphi = 0, \\ \frac{1}{r}\partial_\varphi u + \alpha u &= 0 \quad \text{for } \varphi = \omega. \end{align*}

In the Dirichlet or Neumann case simple separation of variables $u = R\left(r\right) \Phi\left(\varphi\right)$ leads to explicitly solvable eigenvalue problems. However in the Robin case the boundary conditions don't allow for separation of variables. So far I've tried the following:

- Consider the Robin boundary condition as a heterogeneous Dirichlet/Neumann condition. But that doesn't really solve the problem...
Mellin transforming the equation, but again the Robin boundary messes everything up...

Finally starting from a Dirichlet boundary problem for $\hat{u}$:

\begin{align*} \Delta \hat{u} &= 0 \quad \text{in } \Omega, \\ \hat{u} &= 0 \quad \text{on } \partial \Omega. \end{align*}

I tried looking at the solution $\tilde{u}$ of the ODE

\begin{equation*} \frac{f\left(\varphi\right)}{r} \tilde{u}_\varphi + \alpha \tilde{u} = \hat{u}. \end{equation*}

For some function $f: \left[0, \omega\right] \to \mathbb{C}$ such that $f\left(0\right) = -1$ and $f\left(\omega\right) = 1$. By construction $\tilde{u}$ now satisfies the Robin boundary condition. But I can't derive a useful PDE then.

I'm aware that Robin boundary conditions are kind of tricky. Already in the second order ODE case the eigenvalues are only known as solutions to some exponential-algebraic equations.

I was hoping that due to the simple geometry I could at least get some analytically tractable results. So my question is, is there any known result concerning problems like this? Or might it even be impossible to write something down in terms of standard functions? I would appreciate any help or hints. Thanks in advance.