Laplace's Equation and the Maximum Principle

A harmonic function solves Laplace's equation u_xx plus u_yy equals 0. Every such function has the mean value property: its average over any circle whose disc lies inside the region equals its value at the center, whatever the radius. For example, u(x,y) equals 3x plus 4 is harmonic, and its average over any circle centered at (1,1) is just u(1,1), which is 7. Not every formula qualifies: x squared plus y squared has Laplacian 4, while x squared minus y squared has Laplacian 0.

From averaging comes the maximum principle: on a bounded region, a nonconstant harmonic function that is continuous up to the boundary attains its maximum and minimum only on the boundary, never inside. Uniqueness for the Dirichlet problem follows. If two such solutions share the same boundary data, their difference is harmonic with zero boundary values, so its max and min are both zero and the solutions agree everywhere. Boundary values between 0 and 5 likewise trap the whole interior between 0 and 5.

Harmonic functions are smooth at every interior point, even when the boundary data is rough or jumps. Averaging is what smooths: each interior value blends the whole boundary, so no interior point inherits a jump or a corner. At the centre of a disc the value is the plain average of the boundary values. Think of electrostatics: with no interior charge, the boundary voltages fix one gentle landscape inside.