Dirichlet problem

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 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:

Given a function f that has values everywhere on the boundary of a region in Rn, is there a unique continuous function u twice continuously differentiable in the interior and continuous on the boundary, such that u is harmonic 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.

Contents

History

The Dirichlet problem is named after Lejeune Dirichlet, 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.

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_{\partial D} \nu(s) \frac{\partial G(x,s)}{\partial n} ds

where G(x,y) is the Green's function for the partial differential equation, and

\frac{\partial G(x,s)}{\partial n} = \widehat{n} \cdot \nabla_s G (x,s) = \sum_i n_i \frac{\partial G(x,s)}{\partial s_i}

is the derivative of the Green's function along the inward-pointing unit normal vector \widehat{n}. The integration is performed on the boundary, with measure ds. The function ν(s) is given by the unique solution to the Fredholm integral equation of the second kind,

f(x) = -\frac{\nu(x)}{2} + \int_{\partial D} \nu(s) \frac{\partial G(x,s)}{\partial n} 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^{(1,\alpha)}

for 0 < α, where C(1,α) denotes the Hölder condition.

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{cases} \frac{1}{2\pi}\int_0^{2\pi} f(e^{i\psi})
\frac {1-\vert z \vert ^2}{\vert 1-ze^{-i\psi}\vert ^2} d \psi & \mbox{if }z \in D \\
 f(z) & \mbox{if }z \in \partial D. \end{cases}

The solution u is continuous on the closed unit disk \bar{D} and harmonic on D.

The integrand is known as the Poisson kernel; this solution follows from the Green's function in two dimensions:

G(z,x) = -\frac{1}{2\pi} \log \vert z-x\vert + \gamma(z,x)

where γ(z,x) is harmonic

Δxγ(z,x) = 0

and chosen such that G(z,x) = 0 for x\in \partial D.

Generalizations

Dirichlet problems are typical of elliptic partial differential equations, and potential theory, 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 problems.

See also

  • Perron method

References

External links


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

  • Dirichlet problem — ▪ mathematics       in mathematics, the problem of formulating and solving certain partial differential equations (partial differential equation) that arise in studies of the flow of heat, electricity, and fluids. Initially, the problem was to… …   Universalium

  • Dirichlet-Problem — Als Dirichlet Randbedingung (nach Peter Gustav Lejeune Dirichlet) bezeichnet man im Zusammenhang mit Differentialgleichungen (genauer: Randwertproblemen), Werte, die auf dem jeweiligen Rand des Definitionsbereichs von der Funktion angenommen… …   Deutsch Wikipedia

  • Dirichlet eigenvalue — In mathematics, the Dirichlet eigenvalues are the fundamental modes of vibration of an idealized drum with a given shape. The problem of whether one can hear the shape of a drum is: given the Dirichlet eigenvalues, what features of the shape of… …   Wikipedia

  • Dirichlet , (Peter Gustav) Lejeune — (1805–1859) German mathematician Born in Düren (now in Germany), Dirichlet studied mathematics at Göttingen where he was a pupil of Karl Gauss and Karl Jacobi. He also studied briefly in Paris where he met Joseph Fourier, who stimulated his… …   Scientists

  • Dirichlet-Prinzip — Das Dirichlet Prinzip in der Potentialtheorie besagt, dass Funktionen u in einem Gebiet (mit vorgegebenen Werten u = g auf dem Rand von G), die das „Energiefunktional“ (Dirichlet Integral) mini …   Deutsch Wikipedia

  • Dirichlet boundary condition — In mathematics, the Dirichlet (or first type) boundary condition is a type of boundary condition, named after Johann Peter Gustav Lejeune Dirichlet (1805–1859) who studied under Cauchy and succeeded Gauss at University of Göttingen.[1] When… …   Wikipedia

  • Dirichlet, Peter Gustav Lejeune — ▪ German mathematician born Feb. 13, 1805, Düren, French Empire [now in Germany] died May 5, 1859, Göttingen, Hanover       German mathematician who made valuable contributions to number theory, analysis, and mechanics. He taught at the… …   Universalium

  • Dirichlet eta function — For the modular form see Dedekind eta function. Dirichlet eta function η(s) in the complex plane. The color of a point s encodes the value of η(s). Strong colors denote values close to zero and hue encodes the value s argumen …   Wikipedia

  • Dirichlet-Randbedingung — Als Dirichlet Randbedingung (nach Peter Gustav Lejeune Dirichlet) bezeichnet man im Zusammenhang mit Differentialgleichungen (genauer: Randwertproblemen) Werte, die auf dem jeweiligen Rand des Definitionsbereichs von der Funktion angenommen… …   Deutsch Wikipedia

  • Dirichlet's theorem on arithmetic progressions — In number theory, Dirichlet s theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n ≥ 0. In other… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”