Lp-Estimations of Vector Fields in Unbounded Domains

Some new estimations of scalar products of vector fields in unbounded domains are investigated. Lp-estimations for the vector fields were proved in special weighted functional spaces. The paper generalizes our earlier results for bounded domains. Estimations for scalar products make it possible to investigate wide classes of mathematical physics problems in physically inhomogeneous domains. Such estimations allow studying issues of correctness for problems with non-smooth coefficients. The paper analyses solvability of stationary set of Maxwell equations in inhomogeneous unbounded domains based on the proved Lp-estimations.


Introduction
The estimations of scalar products of vector fields and their norms play a significant role in proving the solvability of mathematical physics problems.Many researches are devoted to the study of estimates of the norms of vector functions in different functional spaces [1][2][3][4].But in the most cases such estimations require the homogeneous areas when their parameters don't depend on space coordinates [5,6].
For inhomogeneous areas we suggest using estimations of scalar products of vector fields for the mathematical physics problems.In the publications [7][8][9][10] some L p -estimations of scalar product of vector fields in the limited areas were obtained and was investigated the possibility of their application to study the solvability of different problems of electromagnetic theory.
It is natural to study problem formulations in non-homogeneous unbounded domains for most problems of mathematical physics.In the publications [11,12] we proved L 2 -estimations of scalar products of vector fields in unlimited areas.
The paper is dedicated to solvability of a stationary set of Maxwell equations in the whole space, based on the proved L p -estimations of scalar product in the weighted functional spaces.

Main Functional Spaces
Let be an open subset of space (particu- ), with a norm = 1, 2,3 i  and be Banach spaces rot; respectively.We denote by

div;
p H  The following estimates for scalar products of vector fields in the bounded star-shaped domain with the regular boundary were obtaned in [8,9,11].

 
= q p p1 .There exists a constant , that for any and + div div rot . 3 The main result of this paper is a proof of similar estimates for . 3 we define Banach spaces of vector-functions: with the corresponding norms For [12] these spaces are defined as:

Estimations of Scalar Products
The main result of the current article is is correct.
In proving Theorem 3.1 the following statement is used.
Lemma 3.2 [7].Let  be an open set in (particularly, ) star-shaped on .Then the following identities are true for all and each function ( . Let and be smooth vector-functions on Let R B denote a closed solid sphere with radius centered at the origin and with the boundary where We use the representation (3) for the vector-function in the integral ( 6) For the first of resulting integrals ( 1 I ) we use a vector field relation and then we invoke the Gauss-Ostrogradsky theorem and use the fact that We estimate the first integral.Applying Hölder's inequality to Applying Hölder's inequality to the second inner integral, we have: We can write the estimation as , Applying Hölder's inequality several times, we get The following estimation is obvious Next we construct an estimation for integral 2 I .We apply Hölder's inequality to So, we get an estimation for 2 I : We use Hölder's inequality for the second integral again.

 
Thus, we obtain , and therefore 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).

R  
Note, that for 1 < < 3 2 p the theorem may be proved similarly using the Equivalence (2).

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

 
x H in the whole space with a bounded conducting subdomain.
Here .The conductivity of the atmosphere is denoted as  

 
, L      3 are permeability and per- mittivity.They satisfy the following conditions 3 is a vector-function of the external electromotive force, which is asumed given and satisfying the condition Ker div; = : div = 0 , Ker rot; = : rot = 0 , Ker div ; = : Ker div; , . 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 for almost all . The validity of (10) implies the distibution  for all Equation (7) in conducting subdomain will be and in nonconducting subdomain ( 3 \    ) it becomes an identity.
Multiplying the last equation by , rotψ , and using or It becomes obvious that the problem of determining the stationary magnetic field can be formulated as follows: Determine vector-function 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 Proof.Let's verify the conditions of the Lax-Milgram lemma.
Let us denote 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 .

 
, the last summand is zero.Then using the Hölder's inequality, we obtain The estimates show the coercivity of the bilinear form, because Now we verify the conditions for functional   f  .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.

Conclusion
The paper was devoted to the proof of L p -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.

larly 3    3  3 L
 be a Banach space of functions : u    , summable with power , where a norm is p

3 
Stationary electromagnetic field is described by the set of stationary Maxwell's equations Problem (14) exists and is unique.
The scheme of the proof is similar to Theorem 4.2.