
914 SECOND-ORDER ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES
◮ Domain: 0 ≤ x ≤ a, 0 ≤ y ≤ x, 0 ≤ z ≤ c. First boundary value problem.
A right prism whose base is an isosceles right-angled triangle is considered. Boundary
conditions are prescribed:
w = f
1
(y, z) at x = 0, w = f
2
(x, z) at y = 0, w = f
3
(x, z) at y = x,
w = f
4
(x, y) at z = 0, w = f
5
(x, y) at z = c.
Eigenvalues of the homogeneous problem:
λ
nmk
=
π
2
a
2
(n + m)
2
+ m
2
+
π
2
k
2
c
2
; n, m, k = 1, 2, 3, . . .
Eigenfunctions:
w
nmk
=
sin
π
a
(n+m)x
sin
πmy
a
−(−1)
n
sin
πmx
a
sin
π
a
(n+m)y
sin
πkz
c
.
◮ Domain: 0 ≤ x ≤ a, 0 ≤ y ≤ x, 0 ≤ z ≤ c. Second boundary value problem.
A right prism whose base is an