
924 SECOND-ORDER ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES
◮ Domain: 0 ≤ r ≤ R, 0 ≤ ϕ ≤ ϕ
0
, 0 ≤ z ≤ a. First bound ary value problem.
A cylindrical sector of fi nite thickness is considered. Boundary conditions are prescribed:
w = 0 at ϕ = 0, w = 0 at ϕ = ϕ
0
, w = 0 at r = R,
w = 0 at z = 0, w = 0 at z = a.
Eigenvalues:
λ
nmk
=
π
2
k
2
a
2
+
µ
2
nm
R
2
; n, m, k = 1, 2, 3, . . .
Here, the µ
nm
are positive roots of the transcendental equation J
nπ/ϕ
0
(µ) = 0.
Eigenfunctions:
w
nmk
= J
nπ/ϕ
0
µ
nm
r
R
sin
nπϕ
ϕ
0
sin
kπz
a
.
The square of the norm of an eigenfunction is defined as
kw
nmk
k
2
=
1
8
aR
2
ϕ
0
J
′
nπ/ϕ
0
(µ
nm
)
2
.
◮ Domain: 0 ≤ r ≤ R, 0 ≤ ϕ ≤ ϕ
0
, 0 ≤ z ≤ a. Mixed boundary val ...