Laplace's equation in the plane

Many of the most fundamental setups in physics — the steady temperature in a plate, the voltage in a charge-free region, the height of a soap film stretched on a wire loop — all reduce to one and the same equation. It says: at every interior point, the value is the average of its neighbors, with no source pushing it up or down. That equation is Laplace's equation, and its solutions are exactly the harmonic functions.

In the plane, writing u = u(x, y) for the unknown real function, Laplace's equation is u_xx + u_yy = 0, where u_xx and u_yy are the second partial derivatives in x and y. The left side, u_xx + u_yy, is the Laplacian, often written as the Laplacian of u; Laplace's equation is the statement that this Laplacian vanishes everywhere on the region. In polar coordinates (r, theta) the same equation reads u_rr + (1/r) u_r + (1/r^2) u_(theta theta) = 0, a form that is convenient on disks and annuli. The deep link to complex analysis: if f = u + i v is holomorphic, the Cauchy-Riemann equations u_x = v_y, u_y = -v_x force both u and v to satisfy Laplace's equation, so harmonic functions and holomorphic functions are two views of the same world.

Laplace's equation is the prototype of an elliptic partial differential equation — smooth, rigid, and governed entirely by boundary behavior (no time variable, nothing propagating). Its inhomogeneous cousin u_xx + u_yy = rho (a prescribed source rho) is Poisson's equation, which models a region with charge or heat sources. The honest boundary: the full PDE / functional-analytic theory of Laplace's equation belongs to a separate study; here we focus on the complex-analytic side — using holomorphic functions, conformal maps, and the Poisson formula to understand and solve it in the plane.