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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 intersted 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.

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A simple example is a vibrating string (in the plane, oriented horizontally) whose ends move on vertical tracks while experiencing a restoring force whose strength is proportional to the displacement of the string. I first saw this example in section 1.4 of Strauss' book on PDE. –  Aaron Hoffman Apr 27 '12 at 15:42
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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.

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Thanks! I didnt know about this. Does this sort of say that high energy/frequency waves have to die out at infinity because we expect that Robin -> Dirichlet as $\omega \to \infty$? –  Otis Chodosh May 3 '12 at 1:29
    
I'm not really sure what this says about "why" there is dispersion, but I'm really the wrong person to ask. One nice thing about this approach is that it generalizes to Riemannian manifolds with a potential naturally, i.e. you can end up reducing certain dispersive statements about the wave equation to some geometric assumptions about the manifold (like behavior of trapped geodesics and asymptotic flastness) and/or spectral assumptions about $\Delta_g + V$. From what I can tell, there is a large literature on this. A more recent paper is arxiv.org/abs/1105.0873. See Proposition 1.38. –  Yakov Shlapentokh-Rothman May 4 '12 at 13:39
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There is a book by Daniel J. Hoppe, "Impedance Boundary Conditions In Electromagnetics", CRC, 1995. I give here its summary from Amazon.com which seems to answer the question:

Electromagnetic scattering from complex objects has been an area of in-depth research for many years. A variety of solution methodologies have been developed and utilised for the solution of ever increasingly complex problems. Among these methodologies, the subject of impedance boundary conditions has interested the authors for some time. In short, impedance boundary conditions allow one to replace a complex structure with an appropriate impedance relationship between the electric and magnetic fields on the surface of the object. This simplifies the solution of the problem considerably, allowing one to ignore the complexity of the internal structure beneath the surface. This book examines impedance boundary conditions in electromagnetics. The introductory chapter provides a presentation of the role of the impedance boundary conditions in solving practical electromagnetic problems and some historical background. One of the main objectives of this book is to present a unified and thorough discussion of this important subject. A method based on a spectral domain approach is presented to derive the Higher Order Impedance Boundary Conditions (HOIBC). The method includes all of the existing approximate boundary conditions, such as the Standard Impedance Boundary Condition, the Tensor Impedance Boundary Condition and the Generalised Impedance Boundary Conditions, as special cases. The special domain approach is applicable to complex coatings and surface treatments as well as simple dielectric coatings. The spectral domain approach is employed to determine the appropriate boundary conditions for planar dielectric coatings, chiral coatings and corregated conductors. The accuracy of the proposal boundary conditions is discussed. The approach is then extended to include the effects of curvature and is applied to curved dielectric and chiral coatings. Numerical data is presented to critically assess the accuracy of the results obtained using various forms of the impedance boundary conditions. A number of appendices that provide more detail on some of the topics addressed in the main body of the book and a selective list of references directly related to the topics addressed in this book are also included.

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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.

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