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10 Variational Problems of Functionals with Vector, Tensor …
The Euler equations of the functional are
⎧
⎪ ⎨
⎪ ⎩
∂ F
∂u
+ a · ∇ · ∇
∂ F
∂∇ · ∇(ua)
= 0
∂ F
∂ a
+ u∇ · ∇
∂ F
∂∇ · ∇(ua)
= 0
(5)
The natural boundary conditions are
∂ F
∂∇ · ∇(ua)
S
= 0, n · ∇
∂ F
∂∇ · ∇(ua)
S
= 0
(6)
Example 10.4.10 The functional with the divergence of product of a scalar and the
gradient of another scalar. Find the Euler equations and natural boundary conditions
of the functional
J =
V
F(u, v, ∇ · (u∇v))dV
(1)
Solution Taking the variation to the functional, we get
δ J =
V
∂ F
∂u
δu +
∂ F
∂v
δv +
∂ F
∂∇ · (u∇v)
δ∇ · (u∇v)
dV
(2)
Putting ϕ =
∂ F
∂∇·(u∇v)
, the integral of the last term in Eq. (2) can be written as
ϕδ∇ · (u∇v) = ϕ∇ · (δu∇v) = ∇ · (ϕδu∇v) − ∇ϕ · (∇vδu + uδ∇v)
= ∇ · (ϕδu∇v) − ∇ϕ · ∇vδu − u∇ϕ · ∇δv
= ∇ · (ϕδu∇v) − ∇ϕ · ∇vδu − ∇ · (u∇ϕδv) + (∇ · u∇ϕ)δv (3)
Substituting Eq. (3) into Eq. (2), we get
δ J =
V
∂ F
∂u
− ∇v · ∇ϕ
δu +
∂ F
∂v
+ ∇ · u∇ϕ
δv + ∇ · (ϕδu∇v) − ∇ · (u∇ϕδv)
dV
(4)
The Euler equations of the functional are
⎧
⎪ ⎨
⎪ ⎩
∂ F
∂u
− ∇v · ∇
∂ F
∂∇ · (u∇v)
= 0
∂ F
∂v
+ ∇ · u∇
∂ F
∂∇ · (u∇v)
= 0
(5)
The natural boundary conditions are