Elliptic boundary value problem
Differential equations describe a large class of natural phenomena, from the heat equation describing the evolution of heat in (for instance) a metal plate, to the Navier-Stokes equation describing the movement of fluids, including Einstein's equations describing the physical universe in a relativistic way.
Although all these equations are boundary value problems, they are further subdivided into categories.
Boundary value problems and partial differential equations specify relations between two or more quantities.
Boundary value problems can involve space, time and other quantities such as temperature, velocity, pressure, magnetic field, etc.
For instance, if one hangs a clothesline between the house and a tree, then in the absence of wind, the clothesline will not move and will adopt a gentle hanging curved shape known as the catenary.
[1] This curved shape can be computed as the solution of a differential equation relating position, tension, angle and gravity, but since the shape does not change over time, there is no time variable.
Elliptic boundary value problems are a class of problems which do not involve the time variable, and instead only depend on space variables.
The main example for boundary value problems is the Laplace operator, where
can be interpreted as the stationary or limit distribution of heat in a metal plate shaped like
represents the intensity of heat generation at each point in the plate.
After waiting for a long time, the temperature distribution in the metal plate will approach
In general, a boundary-value problem (BVP) consists of a partial differential equation (PDE) subject to a boundary condition.
For now, second-order PDEs subject to a Dirichlet boundary condition will be considered.
, any of the following equivalent conditions hold: If the second-order partial differential operator
is any trace operator, one can construct the boundary value problem In the rest of this article, we assume that
There is a subtlety here in that the partial derivatives must be defined "in the weak sense" (see the article on Sobolev spaces for details.)
is a Hilbert space, which accounts for much of the ease with which these problems are analyzed.
Unless otherwise noted, all derivatives in this article are to be interpreted in the weak, Sobolev sense.
The first step to cast the boundary value problem as in the language of Sobolev spaces is to rephrase it in its weak form.
and integrate by parts using Green's theorem to obtain We will be solving the Dirichlet problem, so that
is a general elliptic operator, the same reasoning leads to the bilinear form We do not discuss the Neumann problem but note that it is analyzed in a similar way.
There are many possible choices, but for the purpose of this article, we will assume that The reader may verify that the map
A regularity theorem for a linear elliptic boundary value problem of the second order takes the form Theorem If (some condition), then the solution
, the space of "twice differentiable" functions whose second derivatives are square integrable.
While in exceptional circumstances, it is possible to solve elliptic problems explicitly, in general it is an impossible task.
Because of the good properties we have enumerated (as well as many we have not), there are extremely efficient numerical solvers for linear elliptic boundary value problems (see finite element method, finite difference method and spectral method for examples.)
Equipped with the spectral theorem for compact linear operators, one obtains the following result.
such that If one has computed the eigenvalues and eigenvectors, then one may find the "explicit" solution of
Implemented on a computer using numerical approximations, this is known as the spectral method.
Consider the problem The reader may verify that the eigenvectors are exactly with eigenvalues The Fourier coefficients of
