430
7 Variational Principles
where, Γ is the plane closed curve, D is the open domain surrounded by Γ , In D,
p = p(x, y) > 0, q = q(x, y) 0, σ = σ (x, y) 0, f = f (x, y), and p ∈ C
1 , q,
f , σ are all belong to C, g = g(x, y), h = h(x, y), k = k(x, y) are all on Γ . Other
definitions related to the space problem are applicable for the plane problem, it need
only change the three-dimensional problem into the two-dimensional problem. the
two-dimensional and three-dimensional problems are similar, just changing V into
D, changing S into Γ , A two-dimensional problem and three dimensional problem
can be solved by the same kind of method.
If σ = 0, then the third boundary value problem can be converted into the second
boundary value problem, therefore, the second boundary value problem is the special
case of the third boundary value problem, the three boundary value problems can be
converted into the first value problem and the third boundary value problem to solve
by two kinds of cases. In brief, whether it is a three-dimensional or two-dimensional
problem, which can be converted into the third boundary value problem to solve
According to the variational principle of the symmetric positive definite operator,
only the differential equation of homogeneous boundary condition has the variational
principle. In fact, the differential equation of nonhomogeneous boundary condition
has also the variational principle. The variational principle of an elliptic equation is
given below.
Theorem 7.8.1 Let there be the elliptic Eq. (7.8.6), its boundary conditions are
u| Γ 1 = g
(7.8.10)
p
∂u
∂n
+ σ u
Γ 2
= k
(7.8.11)
where, Γ = Γ 1 +Γ 2 is a piecewise smooth plane closed curve, D is an open domain
surrounded by Γ , on D + Γ , p = p(x, y) > 0, q = q(x, y) 0, f = f (x, y),
σ = σ (x, y) 0, and p(x, y) ∈ C
1 , q(x, y), f (x, y), σ (x, y) are all belong to
C, on the boundary Γ 1 , g = g(x, y), on the boundary Γ 2 , k = k(x, y). Then the
necessary and sufficient conditions satisfied by the solution u = u 0 of the equation
is that the functional
J [u] =
D
[ p(∇u)
2
+ qu
2
− 2u f ]d D +
Γ 2
(σ u
2
− 2ku)dΓ
(7.8.12)
takes the minimum at u = u 0 under the condition of u| Γ 1 = g.
Proof Construct a function u 1 , it is smooth enough in the domain D, and it satisfies
the nonhomogeneous boundary conditions on the boundary
u 1 | Γ 1 = g,
p
∂u 1
∂n
+ σ u 1
Γ 2
= k
7 Variational Principles
where, Γ is the plane closed curve, D is the open domain surrounded by Γ , In D,
p = p(x, y) > 0, q = q(x, y) 0, σ = σ (x, y) 0, f = f (x, y), and p ∈ C
1 , q,
f , σ are all belong to C, g = g(x, y), h = h(x, y), k = k(x, y) are all on Γ . Other
definitions related to the space problem are applicable for the plane problem, it need
only change the three-dimensional problem into the two-dimensional problem. the
two-dimensional and three-dimensional problems are similar, just changing V into
D, changing S into Γ , A two-dimensional problem and three dimensional problem
can be solved by the same kind of method.
If σ = 0, then the third boundary value problem can be converted into the second
boundary value problem, therefore, the second boundary value problem is the special
case of the third boundary value problem, the three boundary value problems can be
converted into the first value problem and the third boundary value problem to solve
by two kinds of cases. In brief, whether it is a three-dimensional or two-dimensional
problem, which can be converted into the third boundary value problem to solve
According to the variational principle of the symmetric positive definite operator,
only the differential equation of homogeneous boundary condition has the variational
principle. In fact, the differential equation of nonhomogeneous boundary condition
has also the variational principle. The variational principle of an elliptic equation is
given below.
Theorem 7.8.1 Let there be the elliptic Eq. (7.8.6), its boundary conditions are
u| Γ 1 = g
(7.8.10)
p
∂u
∂n
+ σ u
Γ 2
= k
(7.8.11)
where, Γ = Γ 1 +Γ 2 is a piecewise smooth plane closed curve, D is an open domain
surrounded by Γ , on D + Γ , p = p(x, y) > 0, q = q(x, y) 0, f = f (x, y),
σ = σ (x, y) 0, and p(x, y) ∈ C
1 , q(x, y), f (x, y), σ (x, y) are all belong to
C, on the boundary Γ 1 , g = g(x, y), on the boundary Γ 2 , k = k(x, y). Then the
necessary and sufficient conditions satisfied by the solution u = u 0 of the equation
is that the functional
J [u] =
D
[ p(∇u)
2
+ qu
2
− 2u f ]d D +
Γ 2
(σ u
2
− 2ku)dΓ
(7.8.12)
takes the minimum at u = u 0 under the condition of u| Γ 1 = g.
Proof Construct a function u 1 , it is smooth enough in the domain D, and it satisfies
the nonhomogeneous boundary conditions on the boundary
u 1 | Γ 1 = g,
p
∂u 1
∂n
+ σ u 1
Γ 2
= k