.

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

and, respectively

If we get

At last, when, then

Thus, we obtain

and therefore

Bringing together the constructed estimates, we derive the following inequality for integral (6)

where

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

(7)

(8)

(9)

(10)

Here. The conductivity of the atmosphere is denoted as. Let denotes a bounded open star-shaped subset of defined by conditions

(11)

(12)

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

(13)

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

(14)

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 [13]). 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

for all.

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

Hense

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.

5. Conclusion

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.

6. Acknowledgements

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).

REFERENCES

  1. E. Byhovskii and N. Smirnov, “Orthogonal Decomposition of the space of vector functions square-summable on a given domain, and the operators of vector analysis,” (Russian) Trudy Matematicheskogo Instituta Steklov, Vol. 59, 1960, pp. 5-36.
  2. G. Duvaut and J. Lions, “Inequalities on Mechanics and Physics,” Springer-Verlag, Berlin, 1976. doi:10.1007/978-3-642-66165-5
  3. R. Temam, “Navier-Stokes Equations: Theory and Numerical Analysis,” North-Holland Publishing Company, Amsterdam, 1977.
  4. H. Weil, “The method of orthogonal Projection in potential theory,” Duke Mathematics Journal, Vol. 7, No. 1, 1940, pp. 411-444. doi:10.1215/S0012-7094-40-00725-6
  5. V. Girault and P.-A. Raviart, “Finite Element Approximation of the Navier-Stokes Equations,” Springer-Verlag, New-York, 1979.
  6. V. Maslennikova, “Lp estimates, and the asymptotics at t→∞ of the solution of the Cauchy problem for a Sobolev system,” (Russian) Trudy Matematicheskogo Instituta Steklov, Vol. 103, 1968, pp. 117-141.
  7. A. Kalinin, “Some estimations in the theory of vector fields,” (Russian) Vestnik UNN Series Mathematical Modeling and Optimal Control, Nizhny Novgorod, No. 20, 1997, pp. 32-38.
  8. A. Kalinin and A. Kalinkina, “Estimates of vector fields and stationary set of Maxwell equations,” (Russian) Vestnik UNN Series Mathematical Modeling and Optimal Control, No. 1, 2002, pp. 95-107.
  9. A. Kalinin and A. Kalinkina, “Lp-estimates for Vector fields,” Russian Mathematics (Izvestiya Uchebnykh Zavedenii Matematika), Vol. 48, No. 3, 2004, pp. 23-31.
  10. A. Kalinin, S. Morozov, “Stationary problems for the set of Maxwell equations in heterogeneous areas,” (Russian) Vestnik UNN Series Mathematical Modeling and Optimal Control, No. 20, 1997, pp. 24-31.
  11. A. Kalinin, “Estimations of scalar products for vector fields and their application in some problems of mathematical physics,” (Russian) Izvestiya of Institution of Mathematics and Infomatics UdSU, Vol. 3, No. 37, 2006, pp. 55-56.
  12. A. Zhidkov, “Estimates of the scalar products of vector fields in unbounded regions,” (Russian) Vestnik UNN, Nizhny Novgorod, No. 1, 2007, pp. 162-166.
  13. P. Lax and A. Milgram, “Parabolic Equations,” Annals of Mathematics Studies, Vol. 33, 1954, pp. 167-190.

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