Poisson's equation

The Poisson equation, named after the French mathematician and physicist Siméon Denis Poisson, is an elliptic partial differential equation of second order which is found as part of boundary value problems in much of the physics application.

Mathematical formulation

The Poisson equation is generally

It denotes the Laplace operator and a function. is the desired solution. Is the equation of Laplace 's equation.

To solve the Poisson equation, additional information is to be given, for example in the form of a Dirichlet boundary condition:

With open and bounded.

In this case, one constructs a solution by using the fundamental solution

Laplace's equation. Herein, the volume of the ball unit in the n-dimensional Euclidean space. By folding a solution obtained from.

In order to satisfy the boundary condition, one can use the Green's function

Is a correction function that

Met. It is generally of dependent and only for simple sites easy to find.

If you know, then a solution of the boundary value problem from above by

Given, the surface measure on call.

The solution can also be found using the Perron method, or a variation approach.

Applications in Physics

Poisson's equation, for example, meet the electrostatic potential and the gravitational potential. It is proportional to the electric charge density or the mass density.

For a spatially confined charge density is the solution of the Poisson equation, which goes to zero for large distances, the integral

Each charge at the place in the small area the size contributes additively to the potential at the site with their Coulomb potential (or Kepler potential)

With.

Electrostatics

Since the electrostatic field is a conservative field, it can be expressed in terms of the gradient of potential, with

By applying the divergence follows

According to the first Maxwell equation, however, also applies

Wherein the charge density and the permittivity.

Thus follows the Poisson equation for the electric field

Gravity

The gravitational acceleration g of a mass M follows from the law of gravitation to

In this case, g is the gravitational constant, and r is the distance from the ( point-like ) mass M.

The flow through the surface A of any volume is then

Where the normal vector is. Applies in spherical coordinates

From which it follows:

From a mass distribution described by a mass density, the total mass results to

This follows

With the set of Gaussian results for the integral but also

And thus

Since the shape of the volume is arbitrary, the integrand must be equal, so

Is. The gravitational represents a conservative force field, so that the relationship

Applies. So that the Poisson's equation to the results of gravity

Where the minus sign lifts off.

Swell

• Richard Courant, David Hilbert: Methods of Mathematical Physics. Volume 1 Springer, Berlin and others 1924 ( The basic teachings of the mathematical sciences 12), ( 4th ed. Ibid. 1993, ISBN 3-540-56796-8 ).
• Lawrence C. Evans: Partial Differential Equations. American Mathematical Society, Providence RI 1998, ISBN 0-8218-0772-2 ( Graduate studies in mathematics 19).