7.5.
Applications of Conformal Mapping.
Let us now apply what we have learned about analytic/conformal maps. We begin
with boundary value problems for the Laplace equation, and then present some applications
in fluid mechanics.
We conclude by discussing how to use conformal maps to construct
Green’s functions for the twodimensional Poisson equation.
Applications to Harmonic Functions and Laplace’s Equation
We are interested in solving a boundary value problem for the Laplace equation on a
domain Ω
⊂
R
2
. Our strategy is to map it to a corresponding boundary value problem on
the unit disk
D
that we know how to solve. To this end, suppose we know a conformal map
ζ
=
g
(
z
) that takes
z
∈
Ω to
ζ
∈
D
. As we know, the real and imaginary parts of an analytic
function
F
(
ζ
) defined on
D
are harmonic. Moreover, according to Proposition 7.31, the
composition
f
(
z
) =
F
(
g
(
z
)) defines an analytic function whose real and imaginary parts
are harmonic functions on Ω. Thus, the conformal mapping can be regarded as a change of
variables that preserves the property of harmonicity. In fact, this property does not even
require the harmonic function to be the real part of an analytic function, i.e., we need not
assume the existence of a harmonic conjugate.
Proposition 7.37.
If
U
(
ξ,η
)
is a harmonic function of
ξ,η
, and
ζ
=
ξ
+ i
η
=
ξ
(
x,y
) + i
η
(
x,y
) =
g
(
z
)
(7
.
81)
is any analytic function, then the composition
u
(
x,y
) =
U
(
ξ
(
x,y
)
,η
(
x,y
))
(7
.
82)
is a harmonic function of
x,y
.
Proof
: This is a straightforward application of the chain rule:
∂u
∂x
=
∂U
∂ξ
∂ξ
∂x
+
∂U
∂η
∂η
∂x
,
∂u
∂y
=
∂U
∂ξ
∂ξ
∂y
+
∂U
∂η
∂η
∂y
,
∂
2
u
∂x
2
=
∂
2
U
∂ξ
2
parenleftbigg
∂ξ
∂x
parenrightbigg
2
+ 2
∂
2
U
∂ξ∂η
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 Fall '10
 Olver
 Differential Equations, Equations, Partial Differential Equations, Boundary value problem, Green's function, Laplace's equation, Potential theory, Conformal mapping

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