We consider Dirichlet boundary value problems which are posed either inside
or outside of a sphere with radius
R >
0: our domain Ω is either
Ω
−
:=
{
x
∈
R
3
 
x

< R
}
(
squiggleright
“interior problem”), or
(361)
Ω
+
:=
{
x
∈
R
3
 
x

> R
}
(
squiggleright
“exterior problem”).
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Notice that Ω
−
is a bounded domain, whereas Ω
+
is not. In both cases, the
boundary value problem has the form
Δ
u
= 0
in Ω
,
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u
(
R, ϕ, ϑ
) =
f
(
ϕ, ϑ
)
,
ϕ
∈
[0
,
2
π
)
, ϑ
∈
[0
, π
]
,
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lim
r
→∞
u
(
r, ϕ, ϑ
) = 0
,
ϕ
∈
[0
,
2
π
)
, ϑ
∈
[0
, π
]
.
(365)
Separation of Variables
We write the unknown function in the form
u
(
r, ϕ, ϑ
) =
F
(
r
)
G
(
ϕ, ϑ
). We separate the radial variable from the angular
variables by
r
2
F
′′
F
+ 2
r
F
′
F
=

1
G
Δ
S
2
G
=
k
∈
R
,
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and the separate differential equations are given by
r
2
F
′′
+ 2
rF
′

kF
=
0
,
r < R
or
r > R,
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Δ
S
2
G
+
kG
=
0
,
ϕ
∈
[0
,
2
π
)
, ϑ
∈
[0
, π
]
.
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We solve for
G
first. For that purpose, we separate again,
G
(
ϕ, ϑ
) =
H
(
ϕ
)
Q
(
ϑ
),
and we obtain

H
′′
H
=
sin
ϑ
Q
(sin
ϑQ
′
)
′
+
k
sin
2
ϑ
=
μ
2
∈
R
.
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The separate ODEs are now given by
H
′′
+
μ
2
H
=
0
,
ϕ
∈
[0
,
2
π
)
,
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sin
2
ϑQ
′′
+ sin
ϑ
cos
ϑQ
′
+
(
k
sin
2
ϑ

μ
2
)
Q
=
0
,
ϑ
∈
[0
, π
]
.
(371)
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