Poisson's equation
/ pwah-SON /
Laplace's equation describes balance in empty space; Poisson's equation describes balance when there is a source feeding the field. Put electric charge in a region and the electrostatic potential is no longer harmonic — it bends in response to the charge. Poisson's equation says exactly how: the imbalance of the potential at each point is set by how much source sits there. It is Laplace's equation with the right-hand side switched on.
Written out, the equation is Laplacian u = f, where f is a given source term (a known function of position) — for example u_xx + u_yy = f(x, y) in the plane. Where f = 0 the equation reduces to Laplace's, so f is precisely the local departure from harmonic balance. In electrostatics it reads Laplacian u = -rho/epsilon_0 with rho the charge density; in gravity Laplacian u = 4 pi G rho with rho the mass density. The general solution is any one particular solution (a convolution of f with the fundamental solution) plus any harmonic function — and that free harmonic piece is fixed by the boundary conditions.
Poisson's equation is the workhorse of potential theory and of equilibrium modelling generally: steady heat with internal heating, the deflection of a loaded elastic membrane, the pressure in incompressible flow, the gravitational field of a planet. Solving it on a domain with prescribed boundary values is the Dirichlet problem for Poisson's equation; on simple domains it is solved by Green's functions, and numerically it is the canonical test problem for finite-difference and finite-element methods.
A uniformly charged ball produces a potential satisfying Laplacian u = constant inside (where charge sits) and Laplacian u = 0 outside (empty space). Solving each piece and matching them at the surface gives the familiar field that is constant-curvature inside and falls off like 1/r outside.
Source present means Poisson; source absent means Laplace — the same field obeys both in different regions.
A particular solution is not unique: you may add any harmonic function and still satisfy Laplacian u = f. Uniqueness comes only after you impose boundary conditions. The sign convention (Laplacian u = f versus Laplacian u = -f) varies by textbook and physics context — always check it.