10.3 The Euler Equations of Gradient Type Functionals
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Example 10.3.1 The functional with the gradient of continued multiplication of the
n scalars. Find the Euler equation and natural boundary condition 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
· δ∇
n
i=1
u i
⎞
⎟
⎟
⎠ dV
(2)
Putting a =
∂ F
∂∇
n
i=1
u i
, the integral of the latter term in Eq. (2) can be written as
a · δ∇
n
i=1
u i = a · ∇δ
n
i=1
u i = ∇ ·
⎛
⎝ aδ
n
i=1
u i
⎞
⎠ − (∇ · a)δ
n
i=1
u i = ∇ ·
⎛
⎝ aδ
n
i=1
u i
⎞
⎠
− (∇ · a)
n
i=1
n
i=1
u i
u i
δu i
(3)
Substituting Eq. (3) into Eq. (2), we get
δ J =
V
n
i=1
∂ F
∂u i
−
1
u i
n
i=1
u i ∇ · a
δu i + ∇ ·
aδ
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 (i = 1, 2, · · · , n)
(5)
The natural boundary condition is
∂ F
∂∇
n
i=1
u i
· n
S
= 0
( 6 )
Example 10.3.2 The functional with the gradient of a scalar function. Let the
functional