
816 SECOND-ORDER ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES
◮ Domain: 0 ≤ x < ∞, 0 ≤ y < ∞ . Second boundary value problem.
A quadrant of the plane is considered. Boundary conditions are prescribed:
∂
x
w = f
1
(y) at x = 0, ∂
y
w = f
2
(x) at y = 0.
Solution:
w(x, y) = −
Z
∞
0
f
1
(η)G(x, y, 0, η) dη −
Z
∞
0
f
2
(ξ)G(x, y, ξ, 0) dξ
+
Z
∞
0
Z
∞
0
Φ(ξ, η)G(x, y, ξ, η) dξ dη.
1
◦
. The Green’s function for λ = −s
2
< 0:
G(x, y, ξ, η) =
1
2π
K
0
(s̺
1
) + K
0
(s̺
2
) + K
0
(s̺
3
) + K
0
(s̺
4
)
,
̺
1
=
p
(x −ξ)
2
+ (y − η)
2
, ̺
2
=
p
(x − ξ)
2
+ (y + η)
2
,
̺
3
=
p
(x + ξ)
2
+ (y − η)
2
, ̺
4
=
p
(x + ξ)
2
+ (y + η)
2
.
2
◦
. The Green’s function for λ = k
2
> 0:
G(x, y, ξ, η) = −
i
4
H
(2)
0
(k̺
1
) + H
(2)
0
(k̺
2
) + H
(2)
0
(k̺
3
) + H
(2)
0
(k̺
4
)
.
◮