Poisson Equation

Poisson Equation


a partial differential equation of the form Δu = f, where Δ is the Laplace operator:

When n = 3, the equation is satisfied by the potential u(x, y, z) due to a mass distribution with volume density f(x, y, z)/4π (in regions where f = 0, u satisfies the Laplace equation) and by the potential due to a charge distribution. The density of distribution f here must satisfy certain smoothness conditions—for example, continuity of the partial derivatives.

Suppose f is nonzero only in a finite region G and suppose it is bounded and has continuous first-order partial derivatives. When n = 2, the Poisson equation then has the particular solution

and when n = 3,

Here, r(A, P) is the distance between a variable point of integration A and some point P. In a more detailed form,

The solution of boundary value problems for the Poisson equation reduces, by means of the substitution u = ν + w, to the solution of boundary value problems for the Laplace equation Δw = 0. The Poisson equation was first studied by S. D. Poisson in 1812.

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Elliptic PDEs, Measures and Capacities: From the Poisson Equation to Nonlinear Thomas-Fermi Problems
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His major contributions were in probability theory and electrostatics, where he developed the well-known Poisson equation governing the electrostatic potential arising from an arbitrary charge distribution.
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Consequently, the scalar potential satisfies the Poisson equation