
884 SECOND-ORDER ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES
◮ Domain: 0 ≤ r ≤R, 0 ≤ ϕ ≤ 2π, −∞ < z < ∞. First boundary value problem.
An infinite circular cylinder is considered. A boundary condition is prescribed:
w = f (ϕ, z) at r = R.
Solution:
w(r, ϕ, z) = −R
Z
2π
0
Z
∞
−∞
f(η, ζ)
∂
∂ξ
G(r, ϕ, z, ξ, η, ζ)
ξ=R
dζ dη
+
Z
R
0
Z
2π
0
Z
∞
−∞
Φ(ξ, η, ζ)G(r, ϕ, z, ξ, η, ζ)ξ dζ dη dξ.
Green’s function:
G(r, ϕ, z, ξ, η, ζ) =
1
2πR
2
∞
X
n=0
∞
X
m=1
A
n
J
n
(µ
nm
r)J
n
(µ
nm
ξ)
J
′
n
(µ
nm
R)
2
µ
nm
cos[n(ϕ − η)] exp
−µ
nm
|z − ζ|
,
where A
0
= 1 and A
n
= 2 for n 6= 0; the J
n
(ξ) are Bessel functions; and the µ
nm
are
positive roots of the transcendental equation J
n
(µR) = 0.
◮ Domain: 0 ≤ r ≤ R, 0 ≤ ϕ