Elliptic boundary value problem explained

In mathematics, an elliptic boundary value problem is a special kind of boundary value problem which can be thought of as the steady state of an evolution problem. For example, the Dirichlet problem for the Laplacian gives the eventual distribution of heat in a room several hours after the heating is turned on.

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. This is necessary because each category must be analyzed using different techniques. The present article deals with the category of boundary value problems known as linear elliptic problems.

Boundary value problems and partial differential equations specify relations between two or more quantities. For instance, in the heat equation, the rate of change of temperature at a point is related to the difference of temperature between that point and the nearby points so that, over time, the heat flows from hotter points to cooler points. Boundary value problems can involve space, time and other quantities such as temperature, velocity, pressure, magnetic field, etc.

Some problems do not involve time. 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

In two dimensions, let

x,y be the coordinates. We will use the subscript notation u _{x,} u _{xx} for the first and second partial derivatives of u with respect to x , and a similar notation for y . We define the gradient \nabla u = (u _{x,u} _{y)} , the Laplace operator \Delta u = u _{xx} + u _{yy} and the divergence \nabla ⋅ (u,v) = u _{x} + v _{y} . Note from the definitions that \Delta u = \nabla ⋅ (\nabla u) .

The main example for boundary value problems is the Laplace operator,

\Delta u = f in \Omega, u = 0 on \partial \Omega;

where

\Omega is a region in the plane and \partial \Omega is the boundary of that region. The function f is known data and the solution u is what must be computed.

The solution

u can be interpreted as the stationary or limit distribution of heat in a metal plate shaped like \Omega with its boundary \partial \Omega kept at zero degrees. The function f 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 u .

Second-order linear problems

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.

Let

U be an open, bounded subset of R ^{n} , denote its boundary as \partial U and the variables as x = (x _{1,...,x} _{n)} . Representing the PDE as a partial differential operator L acting on an unknown function u = u(x) of x\in U results in a BVP of the form$$\backslash left\backslash$$

Notes and References

1. Swetz, Faauvel, Bekken, "Learn from the Masters", 1997, MAA, pp.128-9