15-2 A fundamental solution for the Korn–Lichtenstein equations 319
15-2 A fundamental solution for the Korn–Lichtenstein equations
A fundamental solution for the Korn–Lichtenstein equations of conformal mapping. The ellipsoidal
Korn–Lichtenstein equations, the ellipsoidal Laplace–Beltrami equations.
For the biaxial ellipsoid E
2
A 1 ,A 1 ,A 2
, we shall construct a fundamental solution for the Korn–Lichtenstein
equations of conformal mapping (15.5) subject to the vectorial Laplace–Beltrami equations (15.7). The
condition of orientation conservation (15.8) is automatically fulfilled.
x L =
G 11 /G 22 y B ,
y L = −
G 11 /G 22 x B ,
or
x B = −
G 22 /G 11 y L ,
y B =
G 22 /G 11 x L ,
(15.23)
G 22 /G 11 x L
L
+
G 11 /G 22 x B
B
= 0 ,
G 22 /G 11 y L
L
+
G 11 /G 22 y B
B
= 0 ,
(15.24)
x L y B − x B y L =
G 22 /G 11 x
2
L +
G 11 /G 22 x
2
B > 0 ,
(15.25)
G 22 /G 11 ∈ R
+ ,
G 11 /G 22 ∈ R
+ .
(15.26)
Here, we are interested in a local solution of the ellipsoidal Korn–Lichtenstein equations around a
point {L 0 , B 0 } 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 , y b =
G 22 /G 11 x l ,
(15.27)
G 22 /G 11 x l
l
+
G 11 /G 22 x b
b
= 0 ,
G 22 /G 11 y l
l
+
G 11 /G 22 y b
b
= 0 ,
(15.28)
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) ,
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) ,
(15.29)
or
x(l, b) =
∞
n=0
P n (l, b) ,
y(l, b) =
∞
n=0
Q n (l, b) ,
(15.30)
with