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10 Variational Problems of Functionals with Vector, Tensor …
Example 10.4.3 The functional with the Laplace operator acting on the continued
multiplication of n scalars. Find the Euler equations and natural boundary conditions
of the functional
J =
V
F(u 1 , u 2 , · · · , u n , ,
n
i=1
u i )dV
(1)
Solution Taking the variation to the functional, we get
δ J =
V
⎛
⎜
⎜
⎝
n
i=1
∂ F
∂u i
δu i +
∂ F
∂∂
n
i=1
u i
δδ
n
i=1
u i
⎞
⎟
⎟
⎠ dV
(2)
Putting ϕ =
∂ F
∂∂
n
i=1
u i
, the integral of the latter term in Eq. (2) can be written as
ϕδδ
n
i=1
u i = ϕδ∇ · ∇
n
i=1
u i = ϕ∇ · ∇δ
n
i=1
u i = ∇ · ϕ∇δ
n
i=1
u i − ∇ϕ · ∇δ
n
i=1
u i
= ∇ · ϕ∇δ
n
i=1
u i − ∇ ·
(∇ϕ)δ
n
i=1
u i
+ ϕ
n
i=1
n
i=1
u i
δu i
u i
(3)
Substituting Eq. (3) into Eq. (2), we get
δ J =
V
⎧
⎨
⎩
n
i=1
⎛
⎝ ∂ F
∂u i
+
ϕ
u i
n
i=1
u i
⎞
⎠ δu i + ∇ ·
⎛
⎝ ϕ∇δ
n
i=1
u i
⎞
⎠ − ∇ ·
⎡
⎣ (∇ϕ)δ
n
i=1
u i
⎤
⎦
⎫
⎬
⎭
dV
(4)
The Euler equations of the functional are
∂ F
∂u i
+
1
u i
n
i=1
u i
∂ F
∂∂
n
i=1
u i
= 0
( 5 )
The natural boundary conditions are
∂ F
∂∂
n
i=1
u i
S
= 0, ∇
∂ F
∂∂
n
i=1
u i
· n
S
= 0
( 6 )