V7.
Laplace's Equation
and Harmonic Functions
In this section, we will show how Green's theorem is closely connected with solutions to
Laplace's partial differential equation in two dimensions:
where w(x, y) is some unknown function of two variables, assumed to be twice differentiable.
Equation (1) models a variety of physical situations, as we discussed in Section P of these
notes, and shall briefly review.
1.
The Laplace operator and harmonic functions.
The two-dimensional Laplace operator, or laplacian as it is often called, is denoted by
V2 or lap, and defined by
The notation V2 comes from thinking of the operator as a sort of symbolic scalar product:
In terms of this operator, Laplace's equation (1) reads simply
Notice that the laplacian is a linear operator, that is it satisfies the two rules
(3)
v2(u
+
v)
=
v2u
+
v2v
,
v2
(cu)
=
c(v2u),
for any two twice differentiable functions u(x, y) and v(x, y) and any constant c.
Definition.
A function w(x, y) which has continuous second partial derivatives and
solves Laplace's equation (1) is called a harmonic function.
In the sequel, we will use the Greek letters q5 and
$
to denote harmonic functions;
functions which aren't assumed to be harmonic will be denoted by Roman letters
f,
g, u, v,
etc.
. According to the definition,
(4)
4(x, y) is harmonic
H
v2q5
=
0
.
By combining (4) with the rules (3) for using Laplace operator, we see
(5)
q5 and
$
harmonic
+
q5
+
$
and cq5 are harmonic (c constant).
Examples of harmonic functions.
Here are some examples of harmonic functions.
The verifications are left to the Exercises.