Laplace equation

The scalar form of Laplace's equation is the partial differential equation

(1)

where is the Laplacian.

Note that the operator is commonly written as by mathematicians (Krantz 1999, p. 16). Laplace's equation is a special case of the Helmholtz differential equation

(2)

with , or Poisson's equation

(3)

with .

The vector Laplace's equation is given by

(4)

A function which satisfies Laplace's equation is said to be harmonic. A solution to Laplace's equation has the property that the average value over a spherical surface is equal to the value at the center of the sphere (Gauss's harmonic function theorem). Solutions have no local maxima or minima. Because Laplace's equation is linear, the superposition of any two solutions is also a solution.

A solution to Laplace's equation is uniquely determined if (1) the value of the function is specified on all boundaries (Dirichlet boundary conditions) or (2) the normal derivative of the function is specified on all boundaries (Neumann boundary conditions).

Coordinate System	Variables	Solution Functions
Cartesian		exponential functions, circular functions, hyperbolic functions
circular cylindrical		Bessel functions, exponential functions, circular functions
conical		ellipsoidal harmonics, power
confocal ellipsoidal		ellipsoidal harmonics of the first kind
elliptic cylindrical		Mathieu function, circular functions
oblate spheroidal		Legendre polynomial, circular functions
parabolic		Bessel functions, circular functions
parabolic cylindrical		parabolic cylinder functions, Bessel functions, circular functions
paraboloidal		circular functions
prolate spheroidal		Legendre polynomial, circular functions
spherical		Legendre polynomial, power, circular functions

Laplace's equation can be solved by

separation of variables in all 11 coordinate systems that the Helmholtz differential equation can. The form these solutions take is summarized in the table above. In addition to these 11 coordinate systems, separation can be achieved in two additional coordinate systems by introducing a multiplicative factor. In these coordinate systems, the separated form is

(5)

and setting

(6)

where are scale factors, gives the Laplace's equation

(7)

If the right side is equal to , where is a constant and is any function, and if

(8)

where is the Stäckel determinant, then the equation can be solved using the methods of the Helmholtz differential equation. The two systems where this is the case are bispherical and toroidal, bringing the total number of separable systems for Laplace's equation to 13 (Morse and Feshbach 1953, pp. 665-666).

In two-dimensional bipolar coordinates, Laplace's equation is separable, although the Helmholtz differential equation is not.

Zwillinger (1997, p. 128) calls

(9)

the Laplace equations.