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In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an elliptic boundary value problem is a special kind of
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 constraints, called the boundary conditions. A solution to a boundary value problem is a solution to t ...
which can be thought of as the stable state of an
evolution problem. For example, the
Dirichlet problem
In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region.
The Dirichlet prob ...
for the
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
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
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for t ...
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
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
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
In physics and geometry, a catenary (, ) is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends in a uniform gravitational field.
The catenary curve has a U-like shape, superficia ...
.
[Swetz, Faauvel, Bekken, "Learn from the Masters", 1997, MAA , pp.128-9] 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
be the coordinates. We will use the notation
for the first and second
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 (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
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
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
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
In the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differential ...
.) 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
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 that accepts a function and return ...
. If we formally replace the derivatives
by
and
by
, we obtain the expression
:
.
If we set this expression equal to some constant
, then we obtain either an
ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
(if
are all the same sign) or a
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...
(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
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descript ...
, 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
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 (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
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
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
).
#
.
#
.
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
In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region.
The Dirichlet prob ...
. The
Neumann problem
In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann.
When imposed on an ordinary or a partial differential equation, the condition specifies the values of the derivative appli ...
is
:
and
:
where
is the derivative of
in the direction of the outwards pointing normal of
. In general, if
is any
trace operator
In mathematics, the trace operator extends the notion of the restriction of a function to the boundary of its domain to "generalized" functions in a Sobolev space. This is particularly important for the study of partial differential equations with ...
, 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 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 (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
. We require the space
, the
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 together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense t ...
of "once-differentiable" functions on
, such that both the function
and its partial derivatives
,
are all
square integrable
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value i ...
. 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
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
, 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
In vector calculus, Green's theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by . It is the two-dimensional special case of Stokes' theorem.
Theorem
Let be a positively orient ...
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
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
on
for
#
is continuous on
for
#
is continuous on
and
#
is bounded.
The reader may verify that the map
is furthermore
bilinear and
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
, and that the map
is
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
in
, and continuous if (for instance)
is square integrable.
We say that the map
is
coercive
Coercion () is compelling a party to act in an involuntary manner by the use of threats, including threats to use force against a party. It involves a set of forceful actions which violate the free will of an individual in order to induce a desi ...
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 Weak formulations are important tools for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential equations. In a weak formulation, equations or c ...
, 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
:''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem.''
The Riesz representation theorem, sometimes called the R ...
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
in
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
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
, and in that case
almost everywhere.
Strong solutions
One may further prove that if the boundary of
is a
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
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
is
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
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
,
finite difference method
In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are di ...
and
spectral method
Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations. The idea is to write the solution of the differential equation as a sum of certain " basis functio ...
for examples.)
Eigenvalues and eigensolutions
Another Sobolev imbedding theorem states that the inclusion
is a compact linear map. Equipped with the
spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix (mathematics), matrix can be Diagonalizable matrix, diagonalized (that is, represented as a diagonal matrix i ...
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 may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
of
eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s
and matching
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s
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
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
.)
The series converges in
. Implemented on a computer using numerical approximations, this is known as the
spectral method
Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations. The idea is to write the solution of the differential equation as a sum of certain " basis functio ...
.
An example
Consider the problem
:
on
:
(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.
References
Further reading
*
{{DEFAULTSORT:Elliptic Boundary Value Problem
Mathematical analysis
Partial differential equations
Boundary value problems