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E
2
)
(1
− E
2
sin
2
B
0
)
9
/
2
cos
B
0
sin
B
0
(4
A
1
E
2
(1
=
1
8
−
15
E
2
+22
E
2
sin
2
B
0
−
20
E
4
sin
2
B
0
+9
E
4
sin
4
B
0
)
.
−
Let us now give the solution of the Korn-Lichtenstein equations with respect to the ellipsoid-of-
revolution and subject to the integrability condition of the type of the vectorial Laplace-Beltrami
equation in the function space of bivariate polynomials of type (
15.29
)-(
15.32
) and restricted to
the coecient constraints given by (
15.69
)-(
15.73
). The quoted result is collected in the following
Box
15.3
.
Box 15.3 (Vanishing and non-vanishing polynomial coecients
x
ij
and
y
ij
:
n
=1
...n
=4).
n
=1:
x
01
=0
,
y
01
given
,
(15.80)
[(
15.69
)]
x
10
=
1
s
0
y
01
.
y
10
=0
.
n
=2:
x
02
=0
,
y
02
given
.
1
2
s
0
(2
r
0
y
02
+
r
1
y
01
)
,
[(
15.70
)]
x
20
=0
,
[(
15.53
)]
y
20
=
−
[(
15.52
)]
x
11
=
s
0
(2
y
02
−
s
1
x
10
)
.
[(
15.70
)]
y
11
=0
.
(15.81)
n
=3:
x
03
=0
,
y
03
given
,
1
6
s
0
[(
15.54
)]
x
30
=
−
(2
r
0
x
12
+
r
1
x
11
)
,
[(
15.71
)]
y
30
=0
,
(15.82)
[(
15.55
)]
x
21
=0
,
[(
15.71
)]
y
21
=2
s
0
x
30
,
[(
15.71
)]
x
12
=
1
s
0
(3
y
03
−
s
1
x
11
−
s
2
x
10
)
.
[(
15.71
)]
y
12
=0
.
n
=4:
x
04
=0
,
y
04
given
,
1
4
[(
15.58
)]
x
40
=0
,
[(
15.72
)]
y
40
=
−
r
0
x
31
,
1
3
s
0
(2
y
22
−
[(
15.72
)]
x
31
=
3
s
1
x
30
)
,
[(
15.72
)]
y
31
=0
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