So, we get an estimation for:
We use Hölder’s inequality for the second integral again.
Let’s estimate the following integral
Denote and consider
When (i.e.), then
If we get
At last, when, then
Thus, we obtain
Bringing together the constructed estimates, we derive the following inequality for integral (6)
Going to the limit for in the last inequality, we will obtain estimation (1).
Note, that for the theorem may be proved similarly using the Equivalence (2).
4. Discussion of the Stationary Problem of Electromagnetic Theory
As an example of using the estimations proved in Section 3, we will consider a problem of determining the magnetic field stretch in the whole space with a bounded conducting subdomain.
Stationary electromagnetic field is described by the set of stationary Maxwell’s equations
Here. The conductivity of the atmosphere is denoted as. Let denotes a bounded open star-shaped subset of defined by conditions
Functions are permeability and permittivity. They satisfy the following conditions
The is a vector-function of the external electromotive force, which is asumed given and satisfying the condition
Function equals zero for almost all .
We introduce the necessary functional spaces
Denote. It is readily proved that this functional space will be Hilbert space relatively to scalar product
We name the solution of the Problem (7)-(10) the functions, and satisfying condition for almost all.
The validity of (10) implies the distibution for all defined by the formula
Equation (7) in conducting subdomain will be
and in nonconducting subdomain () it becomes an identity.
Multiplying the last equation by, , integrating along, and using or
It becomes obvious that the problem of determining the stationary magnetic field can be formulated as follows:
Determine vector-function satisfying the integral identity
for all functions.
We need the following statement to prove the theorem of solvability (Theorem 4.2) for the Problem (14).
Lemma 4.1 (Lax-Milgram ). Let be a Hilbert space over the field of real numbers. Let be a symmetric bilinear bounded coercive form,—linear bounded functional. Then there exists a unique element satisfying the equality
Theorem 4.2 (Solvability of the Problem (14)). Let
satisfy (11), (12), and
for almost all. Then the solution of the generalized Problem (14) exists and is unique.
Proof. Let’s verify the conditions of the Lax-Milgram lemma.
Let us denote
Obviously, is a bilinear and symmetric form. The finiteness is easily proved by condition (11):
Using the Cauchy-Bunyakovsky-Schwarz inequality, we obtain
Let’s show coercivity of the form.
Whereas thus for each, i.e. the vector-function satisfies estimation
The following notation is used. Let’s use Estimation (1)
Since, the last summand is zero. Then using the Hölder’s inequality, we obtain
where. The estimates show the coercivity of the bilinear form, because
Now we verify the conditions for functional. The linearity is obvious. Let’s show the finiteness using the Cauchy-Bunyakovsky-Schwarz inequality
Thus, all the constraints of the Lax-Milgram lemma are satisfied, and the solution of the Problem (14) exists and is unique.
Remark. The solvability of the studied problem is also true when is a positive-definite tensor. The scheme of the proof is similar to Theorem 4.2.
Let satifies relation (14) for all . Let’s show that other indefinite functions will be defined in from equaitons (7)-(10) as values depending on.
We determine function in the conductivity area by equality
Let. Let’s extend by zero in. According to the Lax-Milgram lemma there is the unique function, which for each satisfies the equality
Then and as, then. Therefore we obtain that
This shows that.
The function is defined by relation (13) as shown above.
The paper was devoted to the proof of Lp-estimation of vector fields in weighted functional spaces. Also we discussed a solvability of the problem of determinig the magnetic field stretch in the whole space. The proof of solvability is based on the proved estimation.
This work was supported by Analytical Departmental Program “Highschool scientific potential growth” (2009- 2011) Russian Ministry of Education and Science (reg. no. 2.1.1/3927), Federal Target Program “Scientific and Scientific-Pedagogical Personnel of Innovative Russia” (2009-2013) (project NK-13P-13) and RFBR Grant (project 09-01-97019-r_povolzhie_a).