Boundary Conditions

Prerequisites: Differential Equations. Full path (2 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

Intuitive

Generically, if we can formulate the equations of motion for a theory, we have everything at our disposal to describe the solutions of the theory. However, in general we have to supplement the equations with the situation we want to actually describe with the theory. In the case of the planet, we have to add where the planet was and where it moved to at a certain instance of time. Otherwise the equation of motion would give us the solutions for all possible initial positions and velocities of the planet, and thus an infinite number of possible solutions to the theory. Such additional information are called boundary conditions. They select out of any possible kind of behavior described by a theory the particular one which is compatible with the state a system is in.http://axelmaas.blogspot.de/2012/03/equations-that-describe-world.html

Concrete

Important Spatial Boundary Conditions:

Important Boundary Conditions for Differential Equations:

Recommended Resources:

• What Are Boundary Conditions? Numerics Background | SimScale
• For boundary conditions for waves, see chapter 9 in Georgi's "THE PHYSICS OF WAVES"
• For boundary conditions in gauge field theory, see section 4.5 "Cauchy problem and gauge conditions" in Rubakov's "Classical Theory of Gauge Fields".

Abstract

Well-posedness. A boundary value problem is well posed in the sense of Hadamard if a solution exists, is unique, and depends continuously on the data. Which boundary data achieve this depends on the type of the operator. For the Poisson equation $\Delta u = f$ on a bounded connected domain $\Omega$ with smooth boundary, Dirichlet data $u|_{\partial\Omega} = g$ give a unique solution. Neumann data $\partial_n u|_{\partial\Omega} = g$ determine $u$ only up to a constant and require the compatibility condition

$$\int_\Omega f \, d^n x = \int_{\partial\Omega} g \, dS ,$$

which follows from the divergence theorem. For hyperbolic equations such as the wave equation, the natural data are Cauchy data (field and normal derivative) on a spacelike hypersurface. On a globally hyperbolic spacetime, Cauchy data for a linear wave equation on a Cauchy surface determine a unique solution on the whole spacetime, and data on a piece of the surface determine it in the domain of dependence of that piece.

Boundary conditions and self-adjointness. In quantum mechanics the boundary condition is part of the definition of an observable. A differential expression such as $-\frac{d^2}{dx^2}$ on $L^2([0,a])$ is symmetric on smooth functions supported away from the endpoints, but it is not self-adjoint there. Integrating by parts,

$$\langle \varphi, -\psi''\rangle - \langle -\varphi'', \psi\rangle = \Big[\overline{\varphi'}\,\psi - \overline{\varphi}\,\psi'\Big]_0^a ,$$

so a self-adjoint operator needs a domain on which this boundary form vanishes. Von Neumann's theory of deficiency indices classifies the possible choices. Here the deficiency indices are $(2,2)$, and the self-adjoint extensions form a family parametrized by $U(2)$. One way to write them, with a length scale set to $1$, is

$$\Psi - i\Psi' = U\,(\Psi + i\Psi'), \qquad \Psi = \begin{pmatrix}\psi(0)\\ \psi(a)\end{pmatrix}, \quad \Psi' = \begin{pmatrix}-\psi'(0)\\ \psi'(a)\end{pmatrix}, \quad U \in U(2),$$

where $\Psi'$ collects the outward normal derivatives. $U = -\mathbb 1$ gives Dirichlet, $U = \mathbb 1$ gives Neumann, $U = \sigma_x$ gives periodic conditions, and diagonal $U$ with other phases gives Robin conditions. Each choice is a different Hamiltonian with a different spectrum. Equivalently, the self-adjoint extensions correspond to Lagrangian subspaces of the space of boundary values with respect to the boundary form above.

Boundary conditions in the action principle. When we vary an action $S = \int_\Omega \mathcal L(\phi, \partial\phi)\, d^n x$, the boundary term $\int_{\partial\Omega} n_\mu \frac{\partial\mathcal L}{\partial(\partial_\mu\phi)}\,\delta\phi \, dS$ must vanish for the Euler-Lagrange equations to describe a stationary point. Fixing $\phi$ on the boundary (Dirichlet) does this. Leaving $\phi$ free on the boundary forces the natural boundary condition $n_\mu \,\partial\mathcal L/\partial(\partial_\mu\phi) = 0$, which for a free scalar field is the Neumann condition. Adding boundary terms to the action changes which boundary conditions are natural. For the Einstein-Hilbert action, the Gibbons-Hawking-York boundary term is needed so that the variational problem with the induced metric fixed on the boundary is well defined.

Why is it interesting?

The field equations and the boundary conditions are inextricably connected and the latter can in no way be considered less important than the formerV. Fock, The theory of space, time and gravitation

Now, Nature is described by fields, and this elegant and power

Path to this page

1. Calculus
2. Differential Equations