In a (bounded) domain $\Omega \subset \mathbb{R}^n$, if we're studying the Laplace equation or heat equation or such PDE's we can impose the Dirichlet
$u|_{\partial\Omega} \equiv 0$,
Neumann
$D_{\nu} u|_{\partial\Omega}\equiv 0$
or Robin (for $\alpha \in \mathbb{R}$)
$(D_{\nu} u + \alpha u)|_{\partial \Omega} \equiv 0$.
I know that, for example for the heat equation, Dirichlet eigenvalues correspond physically to the boundary being in contact with a (large) heat bath at $T=0$. Or, in the Laplace equation, if we're interested in the modes supported by $\Omega$ (as a drum), Dirichlet boundary conditions can be thought of keeping the boundary from moving.
Neumann boundary conditions, for the heat flow, correspond to a perfectly insulated boundary. For the Laplace equation and drum modes, I think this corresponds to allowing the boundary to flap up and down, but not move otherwise.
My question is: what sort of physical interpretations are there for the Robin boundary conditions? Wikipedia says that they are related to electromagnetic problems, but gives no details. I'd be happy with answers that are not necessarily physics-related, for example, if there was somewhere that Robin boundary conditions naturally arise in a mathematical context, I'd be interested to know about that as well.
Best Answer
Here is an example where $\Omega = \mathbb{R}^3$. One way to establish dispersion for the wave equation involves taking a temporal Fourier transform. In order to do this one has to multiply by a cutoff function supported in $t \in [0,\infty)$. You then get the equation
$(\Delta+\omega^2)\psi = F$
where $\psi$ is the temporal Fourier transform of the product of the original solution with the cutoff, $\omega$ is the Fourier variable, and $F$ is a function controllable by initial data via a finite time energy inequality. If this plan of attack is going to work, we need to make sure that $\psi$ is uniquely determined by $F$. This of course requires appropriate boundary conditions at $\infty$. These turn out to be
1) $\psi = O\left(|x|^{-1}\right)$
2) $\frac{\partial\psi}{\partial r} - i\omega\psi = O\left(|x|^{-2}\right)$
This is a sort of Robin condition at infinity. See http://terrytao.wordpress.com/2011/04/21/the-limiting-absorption-principle/ for more details.