Laplace’s equation governs potentials with no local sources
Laplace’s equation says that the Laplacian of a function, the divergence of its gradient, equals zero. Pierre-Simon Laplace first studied its properties in 1786, and its solutions describe equilibrium states such as steady heat flow, electrostatic potentials, and gravitational fields that do not change with time.
It is a second-order partial differential equation, and the Laplace operator turns one scalar function into another. When the zero on the right is replaced by a given function, the result is Poisson’s equation. Together they are the simplest elliptic partial differential equations, and Laplace’s version falls out of the Helmholtz equation as a special case. Mathematicians call the study of its solutions potential theory.
Twice continuously differentiable solutions are harmonic functions, and all of them are analytic inside their domain. Because the equation is linear and homogeneous, adding solutions gives another solution, so complicated cases can be assembled from simple ones, a rule known as superposition. The equation can be written in Cartesian, cylindrical, or spherical coordinates, or in arbitrary curvilinear ones using the metric tensor.
Boundary problems come in several classic forms. The Dirichlet problem fixes the function’s values on the edge: set the temperature around a region, let heat flow until nothing changes, and the interior temperature is the answer. On a bounded connected region that answer is unique, thanks to the maximum principle. The Neumann problem fixes the normal derivative instead, like prescribing heat flux, so solutions are unique only up to a constant and the total flux must vanish; an insulated boundary has zero normal derivative. The Robin condition prescribes a linear combination of both.
For a unit disk or a half-space, the Poisson integral writes the solution explicitly from boundary data. More generally, Perron’s method builds a candidate as the supremum of subharmonic functions lying below the data; it is harmonic, but whether it meets the boundary values depends on each boundary point. A barrier function guarantees such regularity, and the Wiener criterion characterizes it exactly in terms of capacity.
Source: Laplace's equation
