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. 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

be the coordinates. We will use the subscript notation for the first and second partial derivatives, the Laplace operator.

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

is what must be computed.

.

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.

be an open, bounded. Representing the PDE as a partial differential operator.

is said to be in divergence form in case of the former and in nondivergence form are continuously differentiable-dimensional column vector-valued function One may assume, without loss of generality. We make that assumption in the rest of this article.

is elliptic, any of the following equivalent conditions hold:

1. (see eigenvalue).

2. .

3. .

is elliptic, then the associated BVP is called an elliptic boundary-value problem.

Boundary conditions

The above BVP is a particular example of a Dirichlet problem. The Neumann problem is

1. and

is any trace operator, one can construct the boundary value problem

1. and

2. .

.

Sobolev spaces

The analysis of elliptic boundary value problems requires some fairly sophisticated tools of functional analysis, the Sobolev space are all square integrable is a Hilbert space, which accounts for much of the ease with which these problems are analyzed.

th derivative is continuous.

Weak or variational formulation

and integrate by parts using Green's theorem to obtain

1. .

term, yielding

1. (*)

where

1. and

2. .

is a general elliptic operator, the same reasoning leads to the bilinear form

1. .

We do not discuss the Neumann problem but note that it is analyzed in a similar way.

Continuous and coercive bilinear forms

. There are many possible choices, but for the purpose of this article, we will assume that

1. is continuously differentiable

2. and

3. is bounded.

is furthermore bilinear and continuous is linear is square integrable.

is coercive,

is elliptic.)

Existence and uniqueness of the weak solution

One may show, via the Lax–Milgram lemma to the weak problem (*).

), one can show the same result using the Riesz representation theorem instead.

, which itself depends on Poincaré's inequality.

Strong solutions

solves the strong system

apparently meaningless. There are many ways to remedy the situation, the main one being regularity.

Regularity

A regularity theorem for a linear elliptic boundary value problem of the second order takes the form

Theorem , the space of "twice differentiable" functions whose second derivatives are square integrable.

There is no known simple condition necessary and sufficient for the conclusion of the theorem to hold, but the following conditions are known to be sufficient:

1. , or

2. is convex.

, but that is unfortunately false.

Almost everywhere solutions

are defined almost everywhere almost everywhere.

Strong solutions

is a smooth manifold with the strong definition of the derivative.

, together with a Sobolev imbedding theorem.

Numerical solutions

While in exceptional circumstances, it is possible to solve elliptic problems explicitly, in general it is an impossible task. The natural solution is to approximate the elliptic problem with a simpler one and to solve this simpler problem on a computer.

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.)

Eigenvalues and eigensolutions

is a compact linear map. Equipped with the spectral theorem for compact linear operators, one obtains the following result.

Theorem is a compact linear map. It has a basis of eigenvectors and matching eigenvalues such that

1. as,

2. ,

3. whenever and

4. for all

Series solutions and the importance of eigensolutions

,

via the formula

where

(See Fourier series.)

. Implemented on a computer using numerical approximations, this is known as the spectral method.

An example

Consider the problem

1. (Dirichlet conditions).

The reader may verify that the eigenvectors are exactly

with eigenvalues

. Therefore,

yielding the solution

Maximum principle

There are many variants of the maximum principle. We give a simple one.

Theorem. . In other words, the maximum is attained on the boundary.

is constant.

Notes

References