7.8 Variational Problems of Equivalence with Boundary Value Problem …
429
u| S = g
(7.8.2)
∂u
∂n
S
= h
(7.8.3)
p
∂u
∂n
+ σ u
S
= k
(7.8.4)
where, S is the space closed surface, V is the space open domain surrounded by S, In
V p = p(x, y, z) > 0, q = q(x, y, z) 0, σ = σ (x, y, z) 0, f = f (x, y, z), and
p ∈ C
1 , q, f , σ are all belong to C, g = g(x, y, z), h = h(x, y, z), k = k(x, y, z) are
all on S. Equations (7.8.1) and (7.8.2) are called the first boundary value problem
of elliptic equation; Eqs. (7.8.1) and (7.8.3) are called the second boundary value
problem of elliptic equation; Eqs. (7.8.1) and (7.8.4) are called the third boundary
value problem of elliptic equation. When p(x, y, z) ≡ 1, q(x, y, z) ≡ 0, Eq. (7.8.1)
is transformed to the Poisson equation
−∇ · ∇u = −u = f (x, y, z ∈ V )
(7.8.5)
The boundary conditions are still the three kinds of cases of Eqs. (7.8.2), (7.8.3)
and (7.8.4), note that at this point p = 1 in Eq. (7.8.4). If the free term f = 0 in
Eq. (7.8.5), then it is called the Laplace(’s) equation.
The Poisson equation combines with the boundary condition (7.8.2)~boundary
condition (7.8.4) respectively, they are called the first, second and third boundary
value problems for the Poisson(’s) equation of the three-dimensional problem
respectively. If f = 0, then they are called the first, second and third boundary
value problems for the Laplace(’s) equation of the three-dimensional problem
respectively. The first problem for the Poisson equation is also called the Dirichlet
problem, the second problem for the Poisson equation is also called the Neumann
problem, the third problem for the Poisson equation is also called the Robin
problem.
If z = 0 in Eq. (7.8.1), then the space problem is transformed into a plane problem
−∇ · ( p∇u) + qu = f (x, y ∈ D)
(7.8.6)
It has also the following three kinds of boundary conditions
u| Γ = g
(7.8.7)
∂u
∂n
Γ
= h
(7.8.8)
p
∂u
∂n
+ σ u
Γ
= k
(7.8.9)
429
u| S = g
(7.8.2)
∂u
∂n
S
= h
(7.8.3)
p
∂u
∂n
+ σ u
S
= k
(7.8.4)
where, S is the space closed surface, V is the space open domain surrounded by S, In
V p = p(x, y, z) > 0, q = q(x, y, z) 0, σ = σ (x, y, z) 0, f = f (x, y, z), and
p ∈ C
1 , q, f , σ are all belong to C, g = g(x, y, z), h = h(x, y, z), k = k(x, y, z) are
all on S. Equations (7.8.1) and (7.8.2) are called the first boundary value problem
of elliptic equation; Eqs. (7.8.1) and (7.8.3) are called the second boundary value
problem of elliptic equation; Eqs. (7.8.1) and (7.8.4) are called the third boundary
value problem of elliptic equation. When p(x, y, z) ≡ 1, q(x, y, z) ≡ 0, Eq. (7.8.1)
is transformed to the Poisson equation
−∇ · ∇u = −u = f (x, y, z ∈ V )
(7.8.5)
The boundary conditions are still the three kinds of cases of Eqs. (7.8.2), (7.8.3)
and (7.8.4), note that at this point p = 1 in Eq. (7.8.4). If the free term f = 0 in
Eq. (7.8.5), then it is called the Laplace(’s) equation.
The Poisson equation combines with the boundary condition (7.8.2)~boundary
condition (7.8.4) respectively, they are called the first, second and third boundary
value problems for the Poisson(’s) equation of the three-dimensional problem
respectively. If f = 0, then they are called the first, second and third boundary
value problems for the Laplace(’s) equation of the three-dimensional problem
respectively. The first problem for the Poisson equation is also called the Dirichlet
problem, the second problem for the Poisson equation is also called the Neumann
problem, the third problem for the Poisson equation is also called the Robin
problem.
If z = 0 in Eq. (7.8.1), then the space problem is transformed into a plane problem
−∇ · ( p∇u) + qu = f (x, y ∈ D)
(7.8.6)
It has also the following three kinds of boundary conditions
u| Γ = g
(7.8.7)
∂u
∂n
Γ
= h
(7.8.8)
p
∂u
∂n
+ σ u
Γ
= k
(7.8.9)