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Chapter 10

Laplace’s Equation

10.1 Introduction

This chapter is devoted to the Laplace equation. We introduce one of its important
properties the maximum principle. Then we solve the equation in series form in rectangles,
circles, and related shapes. The case of a circle leads to the beautiful Poisson formula.
If a diffusion or wave process is stationary (independent of time), then u t = 0 and
u tt = 0. Therefore, both the diffusion and the wave equations reduce to the Laplace
equation:

u xx = 0 in one dimension
∇ · ∇u = ∆u = u xx + u yy = 0 in two dimensions
∇ · ∇u = ∆u = u xx + u yy + u zz = 0 in three dimensions.

A solution of the Laplace equation is called a harmonic function.
In one dimension, we have simply u xx = 0, so the only harmonic functions in one
dimension are u(x) = A + Bx. But this is so simple that it hardly gives us a clue to what
happens in higher dimensions.
The inhomogeneous version of Laplace’s equation

∆u = f

with f a given function, is called Poisson’s equation.
Besides stationary diffusions and waves, some other instances of Laplace’s and Pois-
son’s equations include the following.
Electrostatics. From Maxwell’s equations, one has curl E = 0 and div E = 4πρ,
where ρ is the charge density. The first equation impliesE = − grad φ for a scalar function
φ (called the electric potential). Therefore,

∆φ = div(grad φ) = − div E = −4πρ,

which is Poisson’s equation (with f = −4πρ).
Steady fluid flow. Assume that the flow is irrotational (no eddies) so that curl v = 0,
where v = v(x, y, z) is the velocity at the position (x, y, z), assumed independent of

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