Dirichlet problem
Encyclopedia
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 problem can be solved for many PDEs, although originally it was posed for Laplace's equation
. In that case the problem can be stated as follows:
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
.
, who proposed a solution by a variational method which became known as Dirichlet's principle. 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
, determine an electrical potential as solution.
However, Weierstrass found a flaw in Dirichlet's argument, and a rigorous proof of existence was found only in 1900 by Hilbert
. It turns out that the existence of a solution depends delicately on the smoothness of the boundary and the prescribed data.
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.
is continuous. More precisely, it has a solution when
for
, where 
denotes the Hölder condition
.
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
and chosen such that
for 
.
, and the Laplace equation in particular. Other examples include the biharmonic equation
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 problems and Cauchy problem
s.
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...
, a Dirichlet problem is the problem of finding a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
which solves a specified partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
(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
Laplace's equation
In mathematics, 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:where ∆ = ∇² is the Laplace operator and \varphi is a scalar function...
. 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 Rn, is there a unique continuous functionContinuous functionIn 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...
u twice continuously differentiable in the interior and continuous on the boundary, such that u is harmonicHarmonic functionIn mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R which satisfies Laplace's equation, i.e....
in the interior and u = f on the boundary?
This requirement is called 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 main issue is to prove the existence of a solution; uniqueness can be proved using the maximum principle
Maximum principle
In mathematics, the maximum principle is a property of solutions to certain partial differential equations, of the elliptic and parabolic types. Roughly speaking, it says that the maximum of a function in a domain is to be found on the boundary of that domain...
.
History
The Dirichlet problem is named after Lejeune DirichletJohann Peter Gustav Lejeune Dirichlet
Johann Peter Gustav Lejeune Dirichlet was a German mathematician with deep contributions to number theory , as well as to the theory of Fourier series and other topics in mathematical analysis; he is credited with being one of the first mathematicians to give the modern formal definition of a...
, who proposed a solution by a variational method which became known as Dirichlet's principle. 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
Electrostatics is the branch of physics that deals with the phenomena and properties of stationary or slow-moving electric charges....
, determine an electrical potential as solution.
However, Weierstrass found a flaw in Dirichlet's argument, and a rigorous proof of existence was found only in 1900 by Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
. 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 bywhere
is the Green's functionGreen's function
In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions...
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 measureMeasure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
. The function is given by the unique solution to the Fredholm integral equationFredholm 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.-Equation of the first kind :...
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 whenfor
, where denotes the Hölder conditionHölder condition
In mathematics, a real or complex-valued function ƒ on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants C, \alpha , such that...
.
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
is a continuous function on the boundary of the open unit disk , then the solution to the Dirichlet problem is given byThe solution
is continuous on the closed unit disk and harmonic on The integrand is known as the Poisson kernel
Poisson kernel
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 disc. The kernel can be understood as the derivative of the Green's function for the Laplace equation...
; this solution follows from the Green's function in two dimensions:
where
is harmonicand chosen such that
for .Generalizations
Dirichlet problems are typical of elliptic partial differential equations, and potential theoryPotential theory
In mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" was coined in 19th-century physics, when it was realized that the fundamental forces of nature could be modeled using potentials which...
, and the Laplace equation in particular. Other examples include the biharmonic equation
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...
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 problems and Cauchy problem
Cauchy problem
A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions which are given on a hypersurface in the domain. Cauchy problems are an extension of initial value problems and are to be contrasted with boundary value problems...
s.