744
10 Variational Problems of Functionals with Vector, Tensor …
Problems 10.10
10.1 Find the Euler equations and natural boundary conditions of the functional
J =
V F(u, v, ∇ × (u∇v))dV .
10.2 Derive the Euler equations and natural boundary conditions of the functional
J =
V F(u, a, a × ∇u)dV .
10.3 Write out the Euler equations and natural boundary conditions of the
functional J =
V F(u, ∇u)dV .
10.4 Derive the Euler equations and natural boundary conditions of the functional
J =
V F(u, a, ∇(∇u · a))dV .
10.5 Derive the Euler equations and natural boundary conditions of the functional
J =
V F(u, a, (a · ∇)u)dV . Where, a and ∇ are the inner product relation,
it is not a to take the right divergence.
10.6 Write out the Euler equations and natural boundary conditions of the
functional J =
V F(u, f (|∇u|))dV .
10.7 Derive the Euler equations and natural boundary conditions of the functional
J =
V F(a, b, ∇|a × b|)dV .
10.8 Write out the Euler equations and natural boundary conditions of the
functional J =
V F(u, ∇ · u∇u)dV .
10.9 Find the Euler equations and natural boundary conditions of the functional
J =
V F(u, v, ∇ · v∇u)dV .
10.10 Find the Euler equations and natural boundary conditions of the functional
J =
V F(u, v, ∇ · u∇v)dV .
10.11 Find the Euler equations and natural boundary conditions of the functional
J =
V F(u, v, ,(vu))dV .
10.12 Derive the Euler equations and natural boundary conditions of the functional
J =
V F
u, v, a, ∇ ·
u
v
a
dV .
10.13 Derive the Euler equations and natural boundary conditions of the functional
J =
V F(a, b, ∇ · (a × ∇ × b))dV .
10.14 Write out the Euler equations and natural boundary conditions of the
functional J =
V F(u, ∇ × u∇u)dV .
10.15 Derive the Euler equations and natural boundary conditions of the functional
J =
V F(u, a, ∇ × (∇u × a))dV .
10.16 Derive the Euler equations and natural boundary conditions of the functional
J =
V F(a, b, a × ∇ × b, (∇ × a) × (∇ × b))dV .
10.17 Find the Euler equation of the functional J =
V
1 + (∇u) 2 dV .
10.18 For metal–oxide–semiconductor devices, the functional about the electrostatic potential ϕ of the modified variational principle of the quantum
mechanical effect is
J [ϕ] =
V
ε
2
(∇ϕ)
2
− (1 − e
−x
2 /λ
2 )ρϕ
dV
where, ε is electric permitivity; ρ is the electric charge density; λ is the de
Broglie (Louis, 1892.8.15–1987.3.15) wavelength of a thermally energized
10 Variational Problems of Functionals with Vector, Tensor …
Problems 10.10
10.1 Find the Euler equations and natural boundary conditions of the functional
J =
V F(u, v, ∇ × (u∇v))dV .
10.2 Derive the Euler equations and natural boundary conditions of the functional
J =
V F(u, a, a × ∇u)dV .
10.3 Write out the Euler equations and natural boundary conditions of the
functional J =
V F(u, ∇u)dV .
10.4 Derive the Euler equations and natural boundary conditions of the functional
J =
V F(u, a, ∇(∇u · a))dV .
10.5 Derive the Euler equations and natural boundary conditions of the functional
J =
V F(u, a, (a · ∇)u)dV . Where, a and ∇ are the inner product relation,
it is not a to take the right divergence.
10.6 Write out the Euler equations and natural boundary conditions of the
functional J =
V F(u, f (|∇u|))dV .
10.7 Derive the Euler equations and natural boundary conditions of the functional
J =
V F(a, b, ∇|a × b|)dV .
10.8 Write out the Euler equations and natural boundary conditions of the
functional J =
V F(u, ∇ · u∇u)dV .
10.9 Find the Euler equations and natural boundary conditions of the functional
J =
V F(u, v, ∇ · v∇u)dV .
10.10 Find the Euler equations and natural boundary conditions of the functional
J =
V F(u, v, ∇ · u∇v)dV .
10.11 Find the Euler equations and natural boundary conditions of the functional
J =
V F(u, v, ,(vu))dV .
10.12 Derive the Euler equations and natural boundary conditions of the functional
J =
V F
u, v, a, ∇ ·
u
v
a
dV .
10.13 Derive the Euler equations and natural boundary conditions of the functional
J =
V F(a, b, ∇ · (a × ∇ × b))dV .
10.14 Write out the Euler equations and natural boundary conditions of the
functional J =
V F(u, ∇ × u∇u)dV .
10.15 Derive the Euler equations and natural boundary conditions of the functional
J =
V F(u, a, ∇ × (∇u × a))dV .
10.16 Derive the Euler equations and natural boundary conditions of the functional
J =
V F(a, b, a × ∇ × b, (∇ × a) × (∇ × b))dV .
10.17 Find the Euler equation of the functional J =
V
1 + (∇u) 2 dV .
10.18 For metal–oxide–semiconductor devices, the functional about the electrostatic potential ϕ of the modified variational principle of the quantum
mechanical effect is
J [ϕ] =
V
ε
2
(∇ϕ)
2
− (1 − e
−x
2 /λ
2 )ρϕ
dV
where, ε is electric permitivity; ρ is the electric charge density; λ is the de
Broglie (Louis, 1892.8.15–1987.3.15) wavelength of a thermally energized