Physical interpretation of Robin boundary conditions

In the case of the heat equation there is a very typical example.

If you have heat transfer by convection in one of the boundaries of your domain.

Imagine you have a solid where you are solving the heat equation, and in some of the boundaries, you have a liquid in contact at temperature $T_l$. If this liquid is moving (forced convection) or if you let it move by buoyancy of the hotter parts vs the colder ones, then the boundary condition you must impose on the solid on that boundary is:

$$\vec q\cdot \vec{n} = h\left(T-T_l\right)$$ where $h$ is the convection coefficient, and $\vec q$ is the heat loss by that boundary:

$$\vec q = -\kappa \boldsymbol{\nabla}T$$

so finally you have

$$\kappa\frac{\partial T}{\partial\vec{n}}+h\left(T-T_l\right)=0$$

and if you do a variable change like $\theta = T - T_l$

you get your homogeneous robin condition

$$\kappa\frac{\partial \theta}{\partial\vec{n}}+h\theta=0$$

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.

Check out Section II.1.7 of Tikhonov & Samarskii's text Equations of Mathematical Physics for a nice discussion of the physical interpretation of Dirichlet, Neumann and Robin boundary conditions for the simple example of the 1+1 wave equation ($u_{tt} = u_{xx}$) describing the transverse vibrations of a spring on the interval $x\in[0,l]$.

Here's a brief summary. The value of $u(t,0)$ is the transverse position of the spring at $x=0$. The value of $u_x(t,0)$ is the vertical component of the tension, so anything connected to the spring at this end will experience this vertical force, and by Newton's third law apply the same force to the end of the spring. Here are the interpretations. Dirichlet, $u(t,0)=0$: the end of the spring is transversally clamped or fixed. Neumann, $u_x(t,0)=0$: the end of the spring is undergoes free transverse motion, conversely no external transverse force acts on this end. Robin, $u_x(t,0) = k u(t,0)$: a linearly restorative transverse force is applied to the end of the spring, that is the end is transversally restrained, but elastically rather than rigidly. Actually, for the force to be restorative, one must pick a particular sign of $k$, which I don't feel like figuring out at the moment.