We're analyzing the equation $$\nabla^2\phi + \lambda\phi = 0$$ on a circular disk of radius R with boundary condition $\phi + a\nabla\phi \cdot n = 0$ where $a$ is an arbitrary constant and $n$ is the outward unit normal vector.
My question is, what does this boundary condition mean physically?
Furthermore, what can we say about the rayleigh quotient $$\lambda = \frac{\int_{dR}^{} \phi\nabla\phi\cdot n \cdot ds + \int_{R}^{} |\nabla\phi|^2 dx dy}{\int_{R}^{} \phi^2 dx dy}$$
if we can assign anything to $a$? When is it positive, and when would it be $= 0$?