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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 $= 0$?

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    $\begingroup$ These conditions are sometimes called Robin boundary conditions. $\endgroup$ – Delta-u Apr 16 '18 at 14:51
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    $\begingroup$ There are a few examples at mathoverflow.net/questions/95316/… $\endgroup$ – Anthony Carapetis Apr 16 '18 at 14:52
  • $\begingroup$ Generally speaking, the Laplacian operator computes the “average” change in a function on a small circle around a point. $\endgroup$ – amd Apr 16 '18 at 19:13
  • $\begingroup$ In that case, what can you say about the first term in the Rayleigh quotient: ∫dR ϕ∇ϕ⋅n⋅ds? When would it be ≥ 0? $\endgroup$ – colinrob Apr 18 '18 at 0:51

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