Laplace's Equation
Let the heat equation run forever and the temperature stops changing:
u_t \to 0. What remains is the steady state,
governed by setting the time derivative to zero in u_t = \alpha(u_{xx} + u_{yy}):
\nabla^2 u = u_{xx} + u_{yy} = 0.
This is Laplace's equation, and \nabla^2 (the
Laplacian) is the most important operator in physics — it governs steady
temperatures, electrostatic potentials, gravitational fields, incompressible flow, and more.
Its solutions are called harmonic functions.
The mean-value property
The Laplacian measures how a value compares with the average of its surroundings. Setting it to
zero forces every point to equal the average of its neighbours. Made precise, this is the
mean-value property: the value of a harmonic function at the centre of any circle
equals the average of its values around that circle.
Two consequences follow at once. A harmonic function can have no interior bumps or dips — no local
maxima or minima inside the region (that is the
maximum principle).
And it is completely determined by its values on the boundary: fix the edge, and the smooth
"membrane" stretched across the inside is forced. A simple harmonic function is
u = x^2 - y^2, since u_{xx} + u_{yy} = 2 - 2 = 0.
- \nabla^2 u = u_{xx} + u_{yy} = 0 — the steady state of diffusion; an elliptic PDE.
- Solutions (harmonic functions) satisfy the mean-value property: each value is the average over any surrounding circle.
- No interior maxima or minima; the solution is fixed entirely by its boundary values.
A harmonic saddle
A heatmap of u = x^2 - y^2: warm where u > 0,
cool where u < 0, pale near zero. It rises along the
x-axis and falls along the y-axis — a saddle.
Notice there is no isolated hottest or coldest spot in the interior; the extremes all sit on the
edge, the hallmark of a harmonic function.