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 ...
, a Dirichlet problem is the problem of finding a
function which solves a specified
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to h ...
(PDE) in the interior of a given region that takes prescribed values on the boundary of the region.
The Dirichlet problem can be solved for many PDEs, although originally it was posed for
Laplace's equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \na ...
. In that case the problem can be stated as follows:
:Given a function ''f'' that has values everywhere on the boundary of a region in R
''n'', is there a unique
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
''u'' twice continuously differentiable in the interior and continuous on the boundary, such that ''u'' is
harmonic
A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', t ...
in the interior and ''u'' = ''f'' on the boundary?
This requirement is called the
Dirichlet boundary condition. The main issue is to prove the existence of a solution; uniqueness can be proved using the
maximum principle
In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations. ...
.
History
The Dirichlet problem goes back to
George Green, who studied the problem on general domains with general boundary conditions in his ''Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism'', published in 1828. He reduced the problem into a problem of constructing what we now call
Green's functions, and argued that Green's function exists for any domain. His methods were not rigorous by today's standards, but the ideas were highly influential in the subsequent developments. The next steps in the study of the Dirichlet's problem were taken by
Karl Friedrich Gauss, William Thomson (
Lord Kelvin) and
Peter Gustav Lejeune Dirichlet, after whom the problem was named, and the solution to the problem (at least for the ball) using the
Poisson kernel was known to Dirichlet (judging by his 1850 paper submitted to the Prussian academy). Lord Kelvin and Dirichlet suggested a solution to the problem by a
variational method based on the minimization of "Dirichlet's energy". According to Hans Freudenthal (in the ''Dictionary of Scientific Biography'', vol. 11),
Bernhard Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...
was the first mathematician who solved this variational problem based on a method which he called
Dirichlet's principle
In mathematics, and particularly in potential theory, Dirichlet's principle is the assumption that the minimizer of a certain energy functional is a solution to Poisson's equation.
Formal statement
Dirichlet's principle states that, if the funct ...
. The existence of a unique solution is very plausible by the "physical argument": any charge distribution on the boundary should, by the laws of
electrostatics
Electrostatics is a branch of physics that studies electric charges at rest ( static electricity).
Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for a ...
, determine an
electrical potential as solution. However,
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
found a flaw in Riemann's argument, and a rigorous proof of existence was found only in 1900 by
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
, using his
direct method in the calculus of variations. It turns out that the existence of a solution depends delicately on the smoothness of the boundary and the prescribed data.
General solution
For a domain
having a sufficiently smooth boundary
, the general solution to the Dirichlet problem is given by
:
where
is the
Green's function for the partial differential equation, and
:
is the derivative of the Green's function along the inward-pointing unit normal vector
. The integration is performed on the boundary, with
measure . The function
is given by the unique solution to the
Fredholm integral equation of the second kind,
:
The Green's function to be used in the above integral is one which vanishes on the boundary:
:
for
and
. Such a Green's function is usually a sum of the free-field Green's function and a harmonic solution to the differential equation.
Existence
The Dirichlet problem for harmonic functions always has a solution, and that solution is unique, when the boundary is sufficiently smooth and
is continuous. More precisely, it has a solution when
:
for some
, where
denotes the
Hölder condition
In mathematics, a real or complex-valued function ''f'' on ''d''-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants ''C'', α > 0, such that
: , f(x) - f(y) , \leq C ...
.
Example: the unit disk in two dimensions
In some simple cases the Dirichlet problem can be solved explicitly. For example, the solution to the Dirichlet problem for the unit disk in R
2 is given by the
Poisson integral formula.
If
is a continuous function on the boundary
of the open unit disk
, then the solution to the Dirichlet problem is
given by
:
The solution
is continuous on the closed unit disk
and harmonic on
The integrand is known as the
Poisson kernel; this solution follows from the Green's function in two dimensions:
:
where
is
harmonic
A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', t ...
(
) and chosen such that
for
.
Methods of solution
For bounded domains, the Dirichlet problem can be solved using the
Perron method, which relies on the
maximum principle
In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations. ...
for
subharmonic functions. This approach is described in many text books. It is not well-suited to describing smoothness of solutions when the boundary is smooth. Another classical
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 ...
approach through
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 ...
s does yield such information. The solution of the Dirichlet problem using
Sobolev spaces for planar domains can be used to prove the smooth version of the
Riemann mapping theorem
In complex analysis, the Riemann mapping theorem states that if ''U'' is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping ''f'' (i.e. a bijective holomorphi ...
. has outlined a different approach for establishing the smooth Riemann mapping theorem, based on the
reproducing kernels of Szegő and Bergman, and in turn used it to solve the Dirichlet problem. The classical methods of
potential theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gra ...
allow the Dirichlet problem to be solved directly in terms of
integral operators, for which the standard theory of
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
and
Fredholm operators is applicable. The same methods work equally for 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 ...
.
[See:
*
* ]
Generalizations
Dirichlet problems are typical of
elliptic partial differential equation
Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form
:Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\,
whe ...
s, and
potential theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gra ...
, and the
Laplace equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \n ...
in particular. Other examples include the
biharmonic equation
In mathematics, the biharmonic equation is a fourth-order partial differential equation which arises in areas of continuum mechanics, including linear elasticity theory and the solution of Stokes flows. Specifically, it is used in the modeling of ...
and related equations in
elasticity theory.
They are one of several types of classes of PDE problems defined by the information given at the boundary, including
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 ...
s and
Cauchy problems.
Example: equation of a finite string attached to one moving wall
Consider the Dirichlet problem for the
wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and ...
describing a string attached between walls with one end attached permanently and the other moving with the constant velocity i.e. the
d'Alembert equation on the triangular region of the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\t ...
of the space and the time:
:
:
:
As one can easily check by substitution, the solution fulfilling the first condition is
:
Additionally we want
:
Substituting
:
we get the condition of
self-similarity
:
where
:
It is fulfilled, for example, by the
composite function
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
: