
874 SECOND-ORDER ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES
Green’s function:
G(x, y, z, ξ, η, ζ) =
1
4π
∞
X
n=−∞
1
r
n1
−
1
r
n2
,
where
r
n1
=
p
(x − ξ)
2
+ (y − η)
2
+ (z − ζ − 2na)
2
,
r
n2
=
p
(x − ξ)
2
+ (y − η)
2
+ (z + ζ − 2na)
2
.
⊙ Literature: A. G. Butkovskiy (1979), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
◮ Domain: −∞ < x < ∞ , −∞ < y < ∞ , 0 ≤ z ≤ a. Mixed boundary value
problem.
An infinite layer is considered. Boundary conditions are prescribed:
w = f
1
(x, y) at z = 0, ∂
z
w = f
2
(x, y) at z = a.
Solution:
w(x, y, z) =
Z
∞
−∞
Z
∞
−∞
f
1
(ξ, η)
∂
∂ζ
G(x, y, z, ξ, η, ζ)
ζ=0
dξ dη
+
Z
∞
−∞
Z
∞
−∞
f
2
(ξ, η)G(x, y, z, ξ, η, a) dξ dη
+
Z
a
0
Z
∞
−∞
Z
∞
−∞
Φ(ξ, η, ζ)G(x, y, z, ξ, η,