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10 Variational Problems of Functionals with Vector, Tensor …
10.4 The Euler Equations of Divergence Type Functionals
A functional that the variable can finally be represented as the divergence is called
the divergence type functional, or if a function on which a functional is dependent
can finally expressed by the divergence of the function, then such a functional is
called the divergence type functional. The variational problems of the divergence
type functional are discussed in this section.
Example 10.4.1 The functional with the Laplace operator. Write the Euler equation
and natural boundary conditions of the functional
J =
V
F(u, ∇ · ∇u)dV =
V
F(u, ,u)dV
(1)
Solution Taking the variation to the functional, we get
δ J =
V
∂ F
∂u
δu +
∂ F
∂∇ · ∇u
δ∇ · ∇u
dV
(2)
Let ϕ =
∂ F
∂∇·∇u
=
∂ F
∂∂u
, for the second term on the right side in Eq. (2), there is
ϕδ∇ · ∇u = ϕ∇ · ∇δu = ∇ · ϕ∇δu − ∇ϕ · ∇δu = ∇ · ϕ∇δu − ∇ · (∇ϕδu) + (∇ · ∇ϕ)δu
(3)
Substituting Eq. (3) into Eq. (2), we get
δ J =
V
∂ F
∂u
+ ϕ
δu + ∇ · (ϕ∇δu) − ∇ · (∇ϕδu)
dV
(4)
The Euler equation of the functional is
∂ F
∂u
+
∂ F
∂∂u
= 0
( 5 )
The natural boundary conditions are
∂ F
∂∂u
S
= 0, ∇
∂ F
∂∂u
· n
S
= 0
( 6 )
Example 10.4.2 The functional with dual Laplace operators, its expression is
J =
V
F(u, ,,u)dV
(1)
Write the Euler equation and natural boundary conditions of the functional.