Laplace's 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.

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 17, 1972.

Byerly, W. E. An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, 1959.

Eisenhart, L. P. "Separable Systems in Euclidean 3-Space." Physical Review 45, 427-428, 1934.

Eisenhart, L. P. "Separable Systems of Stäckel." Ann. Math. 35, 284-305, 1934.

Eisenhart, L. P. "Potentials for Which Schroedinger Equations Are Separable." Phys. Rev. 74, 87-89, 1948.

Krantz, S. G. "The Laplace Equation." §7.1.1 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 16 and 89, 1999.

Moon, P. and Spencer, D. E. "Recent Investigations of the Separation of Laplace's Equation." Proc. Amer. Math. Soc. 4, 302, 1953.

Moon, P. and Spencer, D. E. "Eleven Coordinate Systems." §1 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 1-48, 1988.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 125-126 and 271, 1953.

Valiron, G. The Geometric Theory of Ordinary Differential Equations and Algebraic Functions. Brookline, MA: Math. Sci. Press, pp. 306-315, 1950.

Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 417, 1995.

Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 128, 1997.