Elliptic boundary value problem - AbsoluteAstronomy.com

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, an elliptic boundary value problem is a special kind of boundary value problem

Boundary value problem

In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions...

which can be thought of as the stable state of an evolution problem. For example, the Dirichlet problem

Dirichlet problem

In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation in the interior of a given region that takes prescribed values on the boundary of the region....

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

Heat equation

The heat equation is an important partial differential equation which describes the distribution of heat in a given region over time...

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

Catenary

In physics and geometry, the catenary is the curve that an idealised hanging chain or cable assumes when supported at its ends and acted on only by its own weight. The curve is the graph of the hyperbolic cosine function, and has a U-like shape, superficially similar in appearance to a parabola...

. 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.

It is not possible to discuss elliptic boundary value problems in more detail without referring to calculus

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

in multiple variables.

Unless otherwise noted, all facts presented in this article can be found in .

The main example

In two dimensions, let

be the coordinates. We will use the notation

for the first and second partial derivative

Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...

s of

with respect to

, and a similar notation for

. We will use the symbols

and

for the partial differential operators in

and

. The second partial derivatives will be denoted

and

. We also define the gradient

, the Laplace operator

Laplace operator

In mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or Δ...

and the divergence

. Note from the definitions that

.

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

where

is a region in the plane and

is the boundary of that region. The function

is known data and the solution

is what must be computed. This example has the same essential properties as all other elliptic boundary value problems.

The solution

can be interpreted as the stationary or limit distribution of heat in a metal plate shaped like

, if this metal plate has its boundary adjacent to ice (which is kept at zero degrees, thus the Dirichlet boundary condition

Dirichlet boundary condition

In mathematics, the Dirichlet boundary condition is a type of boundary condition, named after Johann Peter Gustav Lejeune Dirichlet who studied under Cauchy and succeeded Gauss at University of Göttingen. When imposed on an ordinary or a partial differential equation, it specifies the values a...

.) The function

represents the intensity of heat generation at each point in the plate (perhaps there is an electric heater resting on the metal plate, pumping heat into the plate at rate

, which does not vary over time, but may be nonuniform in space on the metal plate.) After waiting for a long time, the temperature distribution in the metal plate will approach

Nomenclature

Let

where

and

are constants.

is called a second order differential operator

Differential operator

In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...

. If we formally replace the derivatives

and

, we obtain the expression

.

If we set this expression equal to some constant

, then we obtain either an ellipse

Ellipse

In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

(if

are all the same sign) or a hyperbola

Hyperbola

In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...

(if

and

are of opposite signs.) For that reason,

is said to be elliptic when

and hyperbolic if

. Similarly, the operator

leads to a parabola

Parabola

In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...

, and so this

is said to be parabolic.

We now generalize the notion of ellipticity. While it may not be obvious that our generalization is the right one, it turns out that it does preserve most of the necessary properties for the purpose of analysis.

General linear elliptic boundary value problems of the second degree

Let

be the space variables. Let

be real valued functions of

. Let

be a second degree linear operator. That is,

(divergence form).

(nondivergence form)

We have used the subscript

to denote the partial derivative

Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...

with respect to the space variable

. The two formulae are equivalent, provided that

.

In matrix notation, we can let

be an

matrix valued function of

and

be a

-dimensional column vector-valued function of

, and then we may write

(divergence form).

One may assume, without loss of generality, that the matrix

is symmetric (that is, for all

. We make that assumption in the rest of this article.

We say that the operator

is elliptic if, for some constant

, any of the following equivalent conditions hold:

(see eigenvalue).
.
.

An elliptic boundary value problem is then a system of equations like

(the PDE) and

(the boundary value).

This particular example is the Dirichlet problem

Dirichlet problem

. The Neumann problem is

and

where

is the derivative of

in the direction of the outwards pointing normal of

. In general, if

is any trace operator

Trace operator

In mathematics, the concept of trace operator plays an important role in studying the existence and uniqueness of solutions to boundary value problems, that is, to partial differential equations with prescribed boundary conditions...

, one can construct the boundary value problem

and

.

In the rest of this article, we assume that

is elliptic and that the boundary condition is the Dirichlet condition

Sobolev spaces

The analysis of elliptic boundary value problems requires some fairly sophisticated tools of functional analysis

Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

. We require the space

, the Sobolev space

Sobolev space

In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space...

of "once-differentiable" functions on

, such that both the function

and its partial derivatives

are all square integrable. 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.) The space

is a Hilbert space

Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

, which accounts for much of the ease with which these problems are analyzed.

The discussion in details of Sobolev spaces is beyond the scope of this article, but we will quote required results as they arise.

Unless otherwise noted, all derivatives in this article are to be interpreted in the weak, Sobolev sense. We use the term "strong derivative" to refer to the classical derivative of calculus. We also specify that the spaces

consist of functions that are

times strongly differentiable, and that the

th derivative is continuous.

Weak or variational formulation

The first step to cast the boundary value problem as in the language of Sobolev spaces is to rephrase it in its weak form. Consider the Laplace problem

. Multiply each side of the equation by a "test function"

and integrate by parts using Green's theorem

Green's theorem

In mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C...

to obtain

.

We will be solving the Dirichlet problem, so that

. For technical reasons, it is useful to assume that

is taken from the same space of functions as

is so we also assume that

. This gets rid of the

term, yielding

(*)

where

and

.

If

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.

Continuous and coercive bilinear forms

The map

is defined on the Sobolev space

of functions which are once differentiable and zero on the boundary

, provided we impose some conditions on

and

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

is continuously differentiable on for
is continuous on for
is continuous on and
is bounded.

The reader may verify that the map

is furthermore bilinear

Bilinear

Bilinear may refer to:* Bilinear sampling, a method in computer graphics for choosing the color of a texture* Bilinear form* Bilinear interpolation* Bilinear map, a type of mathematical function between vector spaces...

and continuous

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

, and that the map

is linear

Linear

In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...

, and continuous if (for instance)

is square integrable.

We say that the map

is coercive if there is an

for all

This is trivially true for the Laplacian (with

) and is also true for an elliptic operator if we assume

and

. (Recall that

when

is elliptic.)

Existence and uniqueness of the weak solution

One may show, via the Lax–Milgram lemma, that whenever

is coercive and

is continuous, then there exists a unique solution

to the weak problem (*).

If further

is symmetric (i.e.,

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

Riesz representation theorem

There are several well-known theorems in functional analysis known as the Riesz representation theorem. They are named in honour of Frigyes Riesz.- The Hilbert space representation theorem :...

instead.

This relies on the fact that

forms an inner product on

, which itself depends on Poincaré's inequality.

Strong solutions

We have shown that there is a

which solves the weak system, but we do not know if this

solves the strong system

Even more vexing is that we are not even sure that

is twice differentiable, rendering the expressions

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 If (some condition), then the solution is in , 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:

The boundary of is , or
is convex.

It may be tempting to infer that if

is piecewise

then

is indeed in

, but that is unfortunately false.

Almost everywhere solutions

In the case that

then the second derivatives of

are defined almost everywhere

Almost everywhere

In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...

, and in that case

almost everywhere.

Strong solutions

One may further prove that if the boundary of

is a smooth manifold and

is infinitely differentiable in the strong sense, then

is also infinitely differentiable in the strong sense. In this case,

with the strong definition of the derivative.

The proof of this relies upon an improved regularity theorem that says that if

and

, then

, together with a Sobolev imbedding theorem saying that functions in

are also in

whenever

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 element method

The finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...

, finite difference method

Finite difference method

In mathematics, finite-difference methods are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives.- Derivation from Taylor's polynomial :...

and spectral method

Spectral method

Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain Dynamical Systems, often involving the use of the Fast Fourier Transform. Where applicable, spectral methods have excellent error properties, with the so called "exponential...

for examples.)

Eigenvalues and eigensolutions

Another Sobolev imbedding theorem states that the inclusion

is a compact linear map. Equipped with the spectral theorem

Spectral theorem

In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...

for compact linear operators, one obtains the following result.

Theorem Assume that is coercive, continuous and symmetric. The map from to is a compact linear map. It has a basis
Basis
Basis may refer to* Cost basis, in income tax law, the original cost of property adjusted for factors such as depreciation.* Basis of futures, the value differential between a future and the spot price...

of eigenvectors and matching eigenvalues such that

as ,
,
whenever and
for all

Series solutions and the importance of eigensolutions

If one has computed the eigenvalues and eigenvectors, then one may find the "explicit" solution of

via the formula

where

(See Fourier series

Fourier series

In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...

.)

The series converges in

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

Spectral method

An example

Consider the problem

(Dirichlet conditions).

The reader may verify that the eigenvectors are exactly

with eigenvalues

The Fourier coefficients of

can be looked up in a table, getting

. Therefore,

yielding the solution

Maximum principle

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

Theorem. (Weak maximum principle.) Let , and assume that . Say that in . Then . In other words, the maximum is attained on the boundary.

A strong maximum principle would conclude that

for all

unless

is constant.

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