Dirichlet Problem
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In 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 ...
(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 ''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'', ...
in the interior and ''u'' = ''f'' on the boundary? This requirement is called 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 differenti ...
. 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 function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differenti ...
s, 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 Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
, William Thomson (
Lord Kelvin William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy at the University of Glasgow for 53 years, he did important ...
) and
Peter Gustav Lejeune Dirichlet Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
, after whom the problem was named, and the solution to the problem (at least for the ball) using the
Poisson kernel In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the deriv ...
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 The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions ...
based on the minimization of "Dirichlet's energy". According to Hans Freudenthal (in the ''Dictionary of Scientific Biography'', vol. 11), Bernhard Riemann 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 func ...
. 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 am ...
, determine an
electrical potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
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 mathematic ...
found a flaw in Riemann's argument, and a rigorous proof of existence was found only in 1900 by David Hilbert, 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 D having a sufficiently smooth boundary \partial D, the general solution to the Dirichlet problem is given by : u(x) = \int_ \nu(s) \frac \,ds, where G(x, y) is the
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differenti ...
for the partial differential equation, and : \frac = \widehat \cdot \nabla_s G (x, s) = \sum_i n_i \frac is the derivative of the Green's function along the inward-pointing unit normal vector \widehat. The integration is performed on the boundary, with measure ds. The function \nu(s) is given by the unique solution to the
Fredholm integral equation In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied by Ivar Fredholm. A useful method to ...
of the second kind, : f(x) = -\frac + \int_ \nu(s) \frac \,ds. The Green's function to be used in the above integral is one which vanishes on the boundary: : G(x, s) = 0 for s \in \partial D and x \in D. 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 f(s) is continuous. More precisely, it has a solution when : \partial D \in C^ for some \alpha \in (0, 1), where C^ 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 R2 is given by the Poisson integral formula. If f is a continuous function on the boundary \partial D of the open unit disk D, then the solution to the Dirichlet problem is u(z) given by : u(z) = \begin \displaystyle \frac \int_0^ f(e^) \frac \,d\psi & \text z \in D, \\ f(z) & \text z \in \partial D. \end The solution u is continuous on the closed unit disk \bar and harmonic on D. The integrand is known as the
Poisson kernel In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the deriv ...
; this solution follows from the Green's function in two dimensions: : G(z, x) = -\frac \log, z - x, + \gamma(z, x), where \gamma(z, x) 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'', ...
(\Delta_x \gamma(z, x) = 0) and chosen such that G(z, x) = 0 for x \in \partial D.


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 natu ...
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 holomorphic ma ...
. 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 operator An integral operator is an operator that involves integration. Special instances are: * The operator of integration itself, denoted by the integral symbol * Integral linear operators, which are linear operators induced by bilinear forms involvi ...
s, 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 British ...
and
Fredholm operator In mathematics, Fredholm operators are certain Operator (mathematics), operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operat ...
s is applicable. The same methods work equally for the Neumann problem.See: * *


Generalizations

Dirichlet problems are typical of elliptic partial differential equations, 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 = \na ...
in particular. Other examples include the biharmonic equation and related equations in
elasticity theory In physics and materials science, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate loads are ...
. They are one of several types of classes of PDE problems defined by the information given at the boundary, including Neumann problems 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 s ...
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\ ...
of the space and the time: : \frac u(x, t) - \frac u(x, t) = 0, : u(0, t) = 0, : u(\lambda t, t) = 0. As one can easily check by substitution, the solution fulfilling the first condition is : u(x, t) = f(t - x) - f(x + t). Additionally we want : f(t - \lambda t) - f(\lambda t + t) = 0. Substituting : \tau = (\lambda + 1) t, we get the condition of
self-similarity __NOTOC__ In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically s ...
: f(\gamma \tau) = f(\tau), where : \gamma = \frac. 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 ...
: \sin
log(e^ x) Log most often refers to: * Trunk (botany), the stem and main wooden axis of a tree, called logs when cut ** Logging, cutting down trees for logs ** Firewood, logs used for fuel ** Lumber or timber, converted from wood logs * Logarithm, in mathe ...
= \sin log(x)/math> with : \lambda = e^ = 1^, thus in general : f(\tau) = g log(\gamma \tau) where g is a
periodic function A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to d ...
with a period \log(\gamma): : g tau + \log(\gamma)= g(\tau), and we get the general solution : u(x, t) = g log(t - x)- g log(x + t)


Notes


References

* * S. G. Krantz, ''The Dirichlet Problem''. §7.3.3 in ''Handbook of Complex Variables''. Boston, MA: Birkhäuser, p. 93, 1999. . * S. Axler, P. Gorkin, K. Voss,
The Dirichlet problem on quadratic surfaces
', Mathematics of Computation 73 (2004), 637–651. * . * Gérard, Patrick; Leichtnam, Éric: Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Math. J. 71 (1993), no. 2, 559–607. * . * . * * . * . * . * . * . * . * . * . * . * . *


External links

* * {{Peter Gustav Lejeune Dirichlet Potential theory Partial differential equations Fourier analysis Mathematical problems Boundary value problems