
904 SECOND-ORDER ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES
◮ Domain: 0 ≤ x ≤ a, 0 ≤ y ≤ b, −∞ < z < ∞ . First boundary value problem.
An infinite cylindrical domain of a rectangular cross-section is considered. Boundary con-
ditions are prescribed:
w = f
1
(y, z) at x = 0, w = f
2
(y, z) at x = a,
w = f
3
(x, z) at y = 0, w = f
4
(x, z) at y = b.
Solution:
w(x, y, z) =
Z
b
0
Z
∞
−∞
f
1
(η, ζ)
∂
∂ξ
G(x, y, z, ξ, η, ζ)
ξ=0
dζ dη
−
Z
b
0
Z
∞
−∞
f
2
(η, ζ)
∂
∂ξ
G(x, y, z, ξ, η, ζ)
ξ=a
dζ dη
+
Z
a
0
Z
∞
−∞
f
3
(ξ, ζ)
∂
∂η
G(x, y, z, ξ, η, ζ)
η=0
dζ dξ
−
Z
a
0
Z
∞
−∞
f
4
(ξ, ζ)
∂
∂η
G(x, y, z, ξ, η, ζ)
η=b
dζ dξ
+
Z
a
0
Z
b
0
Z
∞
−∞
Φ(ξ, η, ζ)G(x, y, z, ξ, η, ζ) dζ dη dξ.
Green’s function:
G(x, y, z, ξ, η, ζ) =
2