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10 Variational Problems of Functionals with Vector, Tensor …
10.29 Find the Euler equation of the functional
J [u]
=
S
g
2
|∇u|
2
+ (1 − g)|∇u| +
λ
2
(I − u)
2
dS.
10.30 Find the Euler equation and boundary condition of the functional J [T ] =
V
k
2
∇T · ∇T − q v T + ρc p
∂ T
∂t
T
dV +
S α
1
2
T
2
− T f T
dS.
10.31 The functional for the optimal membrane shape of the cardiac valve prostheses is J =
D (
√
1 + ∇u · ∇u + λu)dx dy +
C udl, determine its Euler
equation and boundary condition.
10.32 Find the Euler equation and boundary condition of the functional J =
V [ p(∇u)
2
+ qu
2
− 2 f u]dV +
S (σ u
2
− 2ku)dS.
10.33 Find the Euler equations and boundary conditions of the functional
J [u, v] =
V
αu
2 (v str − c)
2
1
|∇v| 2 + ε
+ β|∇u × ∇v|
2 + γ [δ + (1 − δ)|∇v|](u − u ind )
2
dV
10.34 Find the Euler equations and boundary condition of the functional J =
V F(u, a, ∇ · e
u a)dV .
10.35 There is the following variational problem of the functional for diffusional
processes of heat and matter
J =
V
σ (∇T )
2
2
+ T
∇ · σ ∇T − ρC v
∂ T
∂t
dV
(1)
where, σ is the given function, find its Euler equation.
10.36 Find the Euler equation of the functional J =
V [((u)
2
− 2 f u]dV .
10.37 Find the Euler equation and boundary condition of the functional J =
V
1
2ρC V
(∇ · H)
2
+
H
k
·
∂ H
∂t
dV +
S θ H · n dS.
10.38 Find the Euler equations of the energy functional J [E, E
∗
] =
V (∇ × E
∗
· μ
−1
∇ × E − ω
2 E
∗
· ε E)dV .
10.39 In the electromagnetic field problem of the linear, static, inhomogeneous,
anisotropic medium, the Lagrange density function is
L =
H −
B
˜
μ
·
∇ × E +
∂ B
∂t
+ K
−
E −
D
˜
ε
·
∇ × H −
∂ D
∂t
− J
where, B is the magnetic induction; D the electric displacement vector; E is
the electric field intensity; H is the magnetic field intensity; J is the current
density; K is the magnetic current density; The magnetic permeability ˜
μ and
the permittivity (or dielectric constant) ˜
ε are only the functions of coordinates,
and B = ˜
μH, D = ˜
ε E. Find the corresponding Euler equations.
10.40 Find the Euler equations and boundary conditions expressed by A, the
functional is
Π =
V
1
2
(∇ × A ∗ · v∇ × A + ∇ × A · v∇ × A ∗ ) + A ∗ · σ
∂ A
∂t
− A · σ
∂ A ∗
∂t
dV
10 Variational Problems of Functionals with Vector, Tensor …
10.29 Find the Euler equation of the functional
J [u]
=
S
g
2
|∇u|
2
+ (1 − g)|∇u| +
λ
2
(I − u)
2
dS.
10.30 Find the Euler equation and boundary condition of the functional J [T ] =
V
k
2
∇T · ∇T − q v T + ρc p
∂ T
∂t
T
dV +
S α
1
2
T
2
− T f T
dS.
10.31 The functional for the optimal membrane shape of the cardiac valve prostheses is J =
D (
√
1 + ∇u · ∇u + λu)dx dy +
C udl, determine its Euler
equation and boundary condition.
10.32 Find the Euler equation and boundary condition of the functional J =
V [ p(∇u)
2
+ qu
2
− 2 f u]dV +
S (σ u
2
− 2ku)dS.
10.33 Find the Euler equations and boundary conditions of the functional
J [u, v] =
V
αu
2 (v str − c)
2
1
|∇v| 2 + ε
+ β|∇u × ∇v|
2 + γ [δ + (1 − δ)|∇v|](u − u ind )
2
dV
10.34 Find the Euler equations and boundary condition of the functional J =
V F(u, a, ∇ · e
u a)dV .
10.35 There is the following variational problem of the functional for diffusional
processes of heat and matter
J =
V
σ (∇T )
2
2
+ T
∇ · σ ∇T − ρC v
∂ T
∂t
dV
(1)
where, σ is the given function, find its Euler equation.
10.36 Find the Euler equation of the functional J =
V [((u)
2
− 2 f u]dV .
10.37 Find the Euler equation and boundary condition of the functional J =
V
1
2ρC V
(∇ · H)
2
+
H
k
·
∂ H
∂t
dV +
S θ H · n dS.
10.38 Find the Euler equations of the energy functional J [E, E
∗
] =
V (∇ × E
∗
· μ
−1
∇ × E − ω
2 E
∗
· ε E)dV .
10.39 In the electromagnetic field problem of the linear, static, inhomogeneous,
anisotropic medium, the Lagrange density function is
L =
H −
B
˜
μ
·
∇ × E +
∂ B
∂t
+ K
−
E −
D
˜
ε
·
∇ × H −
∂ D
∂t
− J
where, B is the magnetic induction; D the electric displacement vector; E is
the electric field intensity; H is the magnetic field intensity; J is the current
density; K is the magnetic current density; The magnetic permeability ˜
μ and
the permittivity (or dielectric constant) ˜
ε are only the functions of coordinates,
and B = ˜
μH, D = ˜
ε E. Find the corresponding Euler equations.
10.40 Find the Euler equations and boundary conditions expressed by A, the
functional is
Π =
V
1
2
(∇ × A ∗ · v∇ × A + ∇ × A · v∇ × A ∗ ) + A ∗ · σ
∂ A
∂t
− A · σ
∂ A ∗
∂t
dV