10.6 Variational Problems of Functionals with Parallel-Type Inner …
701
P · F
T
= F · P
T
(in V )
(8)
u = u 1 (on A u )
(9)
P · N = P 1N (on A P )
(10)
Example 10.6.8 In 1981, Horn and Schunck proposed the global energy functional
J =
Ω
[( f x u + f y v + f t )
2
+ α(|∇u|
2
+ |∇v|
2
)]dx dy
(1)
where f x , f y and f t are all the given functions, α is a constant. Find the Euler
equations of the functional.
Solution Let the integrand F = ( f x u + f y v + f t )
2
+ α(|∇u|
2
+ |∇v|
2
) =( f x u +
f y v + f t )
2
+ α(∇u · ∇u + ∇v · ∇v), the various partial derivatives are
∂ F
∂u
= 2 f x ( f x u + f y v + f t ),
∂ F
∂v
= 2 f y ( f x u + f y v + f t ),
∂ F
∂∇u
= 2α∇u,
∂ F
∂∇v
= 2α∇v (2)
The Euler equations of the functional are
ααu − f x ( f x u + f y v + f t ) = 0
ααv − f y ( f x u + f y v + f t ) = 0
(3)
Example 10.6.9 The Ginzburg–Landau energy functional is
G(u, A) =
1
2
V
|∇u − i Au|
2
+ |∇ × A − H 0 |
2
+
k
2
2
(1 − |u|
2
)
2
dV
(1)
Find the Euler equations of the functional.
Solution Let the integrand is F =
1
2
|∇u − i Au|
2
+|∇ × A − H 0 |
2
+
k
2
2
(1−|u|
2
)
2 ,
taking note that |∇u − i Au|
2
= (|∇u − i Au|)·(|∇u + i Au|), find the various partial
derivatives
∂ F
∂u
= Au · A − k
2 u(1 − |u|
2
),
∂ F
∇u
= ∇u,
∂ F
∂ A
= Au
2
,
∂ F
∂∇ × A
= ∇ × A − H 0
(2)
The Euler equations of the functional are
u − Au · A + k
2 u(1 − |u|
2
) = 0
∇ × ∇ × A + Au
2
= 0
(3)