Laplace's equation

Laplace's equation states that a function's Laplacian is zero. Its solutions, called harmonic functions, have a mean-value property and describe equilibrium fields without sources or sinks.

Laplacian: A second-order differential operator that sums a function's unmixed second derivatives in Euclidean coordinates. Setting this operator to zero is the equation's defining condition.
Electrostatics: The study of stationary electric charges and the electric fields and potentials they produce. In a charge-free region, electric potential satisfies Laplace's equation.
Partial differential equation: An equation involving an unknown function of multiple variables and its partial derivatives. Laplace's equation is a central example of this broader equation class.
Poisson's equation: An elliptic partial differential equation in which the Laplacian equals a prescribed source function. It generalizes Laplace's equation by allowing sources within the domain.
Harmonic function: A twice-differentiable function whose Laplacian vanishes throughout its domain. These functions are precisely the classical solutions of Laplace's equation.
Gravitational potential: A scalar field whose gradient determines gravitational acceleration in Newtonian gravity. Where mass density is zero, the Newtonian potential is harmonic.
Elliptic partial differential equation: A partial differential equation whose principal part is elliptic, typically enforcing smooth interior behavior. Laplace's equation is the canonical elliptic equation.
Wave equation: A hyperbolic partial differential equation describing the propagation of disturbances through space and time. Unlike Laplace's equation, it governs evolving waves rather than static equilibrium.
Mean-value theorem for harmonic functions: A property stating that a harmonic function's value at a point equals its average over suitable surrounding spheres or circles. It reveals how the equation ties each interior value to nearby values.
Steady-state heat conduction: Heat transfer in a body whose temperature no longer changes with time. With no internal heat generation, equilibrium temperature satisfies Laplace's equation.

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