Laplace's Equation

see also:

• Fundamental Partial Differential Equations
• Poisson’s Equation
• Harmonic Analysis
• Morse Theory
• Invariants
• Conservation Laws

Laplace’s equation is a second-order elliptic PDE used in potential theory, electrostatics, and fluid dynamics, among other fields. It represents stationary (time-independent) solutions where there is no net change in a quantity:

${\nabla }^{2}u=0$

The solutions to Laplace’s equation, known as harmonic functions, have no local maxima or minima.