Harmonic function

A twice-differentiable function whose Laplacian vanishes throughout its domain. Its value at each point equals the average of its values on every sufficiently small sphere centered there.

Laplace equation: The partial differential equation stating that a function’s Laplacian is zero. Harmonic functions are precisely the classical solutions of this equation.
Laplacian: The differential operator obtained by summing a function’s second derivatives in the coordinate directions. Its vanishing is the defining condition for a harmonic function.
Electrostatics: The study of stationary electric charges, electric fields, and electric potentials. In charge-free regions, electrostatic potential is harmonic.
Subharmonic function: A function whose value is no greater than its averages over sufficiently small surrounding spheres. Its mean-value inequality relaxes the equality required of harmonic functions.
Potential theory: The mathematical study of potentials and the differential equations they satisfy. Harmonic functions form its central class of solutions.
Mean value property: A property in which a function’s value at a point equals its average over spheres centered there. For harmonic functions, this averaging identity characterizes local behavior.
Partial differential equation: An equation involving an unknown function and its partial derivatives. The Laplace equation places harmonic functions within the theory of differential equations.
Gravitational potential: A scalar field describing gravitational potential energy per unit mass in a gravitational field. Away from mass, the Newtonian gravitational potential satisfies Laplace’s equation.
Superharmonic function: A function whose value is no less than its averages over sufficiently small surrounding spheres. It reverses the mean-value inequality associated with harmonic functions.
Pierre-Simon Laplace: A French mathematician and astronomer who developed the equation now bearing his name. His work on gravitational potential established the equation defining harmonic functions.

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