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unsatisfying results if we were to develop the subject in a thorough manner. Appen-
dix D in [AgoM05] has a little bit to say about that integral and gives references where
one can find more information. The hypotheses of some theorems in this chapter
would be cleaner if we were to use the Lebesgue integral. We shall not do so, but
needed to point this out for the mathematically minded reader because, as we have
done throughout this topic, we always want to state results carefully with all the
correct hypotheses.
21.2
The Ubiquitous Laplace Equation
One of the most important differential equations in mathematics and science is the
Laplace equation
2
2
∂
∂
u
x
∂
∂
u
y
+
=
0
(21.1)
2
2
for a function u(x,y) of two variables. The solution of a great many problems lead to
this equation or some variant of it. Mathematically, one is lead to this equation right
from the start when studying analytic functions of a complex variable. If the function
()
=
()
+
()
fz
uz
i
vz
is an analytic function, then u(z) and v(z) satisfy the
Cauchy-Riemann equations
∂
∂
u
x
∂
∂
v
y
∂
∂
u
y
=-
∂
∂
v
x
.
=
and
(Recall that
C
=
R
2
so that we can switch back and forth between thinking of a func-
tion as a function of a complex variable z or as a function of two real variables x and
y.) Therefore, by taking partial derivatives of these two equations we get
2
2
2
2
∂
∂
u
x
∂
∂∂
v
xy
∂
∂
u
y
∂
∂∂
v
yx
.
=
and
=-
2
2
Since an analytic function is infinitely differentiable, one can show that the mixed
partials are equal, which leads immediately to the Laplace equation. A similar argu-
ment shows that the function v also satisfies the Laplace equation.
Definition.
Any function u(x,y) of two real variables that has continuous partials up
to order two that satisfies the Laplace equation is called a
harmonic function
.
Harmonic functions satisfy a maximum principle like analytic functions, which
we should mention in passing.
21.2.1 Theorem.
(The Maximum Principle) A harmonic function defined on a
closed and bounded set assumes its maximum and minimum value on the boundary
of this set.
