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G 22 /G 11
+ , G 11 /G 22
+ .
R
R
(15.26)
Here, we are interested in a local solution of the ellipsoidal Korn-Lichtenstein equations around a
point
such that the relations L = L 0 + l and B = B 0 + b hold. A polynomial setup of the
local solution of the ellipsoidal Korn-Lichtenstein equations subject to the ellipsoidal vectorial
Laplace-Beltrami equations,
y l = G 11 /G 22 x b , b = G 22 /G 11 x l ,
{
L 0 ,B 0
}
(15.27)
G 22 /G 11 x l l + G 11 /G 22 x b b =0 ,
(15.28)
G 22 /G 11 y l l + G 11 /G 22 y b b =0 ,
is
x ( l,b )= x 0 + x 10 l + x 01 b + x 20 l 2 + x 11 lb + x 02 b 2 + x 30 l 3 + x 21 l 2 b + x 12 lb 2 + x 03 b 3 +
+O(4) ,
(15.29)
y ( l, b )= y 0 + y 10 l + y 01 b + y 20 l 2 + y 11 lb + y 02 b 2 + y 30 l 3 + y 21 l 2 b + y 12 lb 2 + y 03 b 3 +
+O(4) ,
or
x ( l,b )=
P n ( l,b ) ,
n =0
(15.30)
y ( l,b )=
Q n ( l,b ) ,
n =0
with
P 0 ( l,b ):= x 0 ,
P 1 ( l,b ):= x 10 l + x 01 b =
α + β =1
x αβ l α b β ,
P 2 ( l,b ):= x 20 l 2 + x 11 lb + x 02 b 2 =
α + β =2
x αβ l α b β ,
(15.31)
.
P n ( l,b ):=
α + β = n
x αβ l α b β ,
and
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