1. 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 [1-4]. 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 [7-10] some Lp-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 L2-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 Lp-estimations of scalar product in the weighted functional spaces.
2. Main Functional Spaces
Let
be an open subset of
space (particularly
).
Let
be a Banach space of functions
, summable with power
, where a norm is
![](https://www.scirp.org/html/8-7400656\b27fd1b2-f213-406d-b729-f87a2211c0e8.jpg)
Let
be a Banach space of vector-functions
,
![](https://www.scirp.org/html/8-7400656\8e0bba12-5760-45ee-8f4a-9df102da895e.jpg)
where
(
), with a norm
.
Let
and
be Banach spaces
![](https://www.scirp.org/html/8-7400656\c68bcb00-c843-42a6-903b-8416f542ad55.jpg)
![](https://www.scirp.org/html/8-7400656\a01f261f-81e5-4a39-818f-95a877190e66.jpg)
with norms
![](https://www.scirp.org/html/8-7400656\410ff890-216a-44c9-b713-c8e39d46790f.jpg)
respectively.
We denote by
and
the closures of the set of test vector-functions in
and
, respectively.
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].
Lemma 2.1. Let
,
. There exists a constant
, that for any
and ![](https://www.scirp.org/html/8-7400656\9f4b5ee2-4bbf-403a-a4c5-6942d830a007.jpg)
![](https://www.scirp.org/html/8-7400656\1407728c-899f-49da-908e-c069485641d0.jpg)
Lemma 2.2. Let
,
. There exists a constant
, that for any
, ![](https://www.scirp.org/html/8-7400656\23372ca4-f7c2-4366-abe0-cf43d3960e1d.jpg)
![](https://www.scirp.org/html/8-7400656\89a9d178-35f2-4e40-af06-9dc2b7850863.jpg)
Lemma 2.3. Let
,
. There exists a constant
, that for any
and ![](https://www.scirp.org/html/8-7400656\b8b4fa34-e35f-45d7-b977-0b4ad6968f72.jpg)
![](https://www.scirp.org/html/8-7400656\283080c3-a03f-417c-a6e7-fa58773c834a.jpg)
The main result of this paper is a proof of similar estimates for
.
Let
. For each
and
we define Banach spaces of vector-functions:
![](https://www.scirp.org/html/8-7400656\3c93a394-91e0-40df-a862-32933ef2d556.jpg)
![](https://www.scirp.org/html/8-7400656\922f7b41-07c3-48f2-8cd9-021593c22593.jpg)
with the corresponding norms
,
.
For
[12] these spaces are defined as:
![](https://www.scirp.org/html/8-7400656\21f43802-daf5-4ad9-9329-971473fa7676.jpg)
3. Estimations of Scalar Products
The main result of the current article is Theorem 3.1. Let
,
,
,
. Then there exists a positive constant
, which does not depend on vector-functions
and
, and the inequality
(1)
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 ![](https://www.scirp.org/html/8-7400656\f663df31-4345-40f3-9219-0d6600f85b34.jpg)
(2)
(3)
Let
,
,
. Then the identities (2) and (3) are equivalent to
(4)
(5)
Proof (Theorem 3.1). Let
. Let
and
be smooth vector-functions on ![](https://www.scirp.org/html/8-7400656\66e47ba6-bf14-48c9-8f1a-25be12e672d3.jpg)
![](https://www.scirp.org/html/8-7400656\7fefc2cd-91cd-4c08-95b0-a48ae392f62a.jpg)
Let
denote a closed solid sphere with radius
centered at the origin and with the boundary
. Consider the integral
(6)
where
is a function of scalar argument
![](https://www.scirp.org/html/8-7400656\b40e86c6-fbf3-4fda-a825-d374b058c25e.jpg)
We use the representation (3) for the vector-function
in the integral (6)
![](https://www.scirp.org/html/8-7400656\2c1de07b-b5db-4c50-af7a-1b3b5f17ee3b.jpg)
For the first of resulting integrals (
) we use a vector field relation
![](https://www.scirp.org/html/8-7400656\f16abe1b-4f79-4f46-9577-6e85b59d113f.jpg)
and then we invoke the Gauss-Ostrogradsky theorem and use the fact that
when
. So
![](https://www.scirp.org/html/8-7400656\2fd05113-a6b1-4099-81a2-1845fef984cf.jpg)
or passing to spherical coordinates the operator ![](https://www.scirp.org/html/8-7400656\c3c0d3dd-a6e1-4b7a-91c1-57f65076b017.jpg)
![](https://www.scirp.org/html/8-7400656\6d912bfb-80df-4c49-978a-0bc6b3b9675f.jpg)
We estimate the first integral. Applying Hölder’s inequality to
we get
![](https://www.scirp.org/html/8-7400656\adf0de4b-9eb0-44df-971e-709d9871f434.jpg)
then
![](https://www.scirp.org/html/8-7400656\7fde5087-617a-406d-a407-d090b1ee0a80.jpg)
Applying Hölder’s inequality to the second inner integral, we have:
![](https://www.scirp.org/html/8-7400656\2794063e-607c-4a98-8b6f-a5df4ce2e4f8.jpg)
We can write the estimation as
![](https://www.scirp.org/html/8-7400656\4189243c-5cab-4987-9a03-c4797629b27d.jpg)
It is obvious that if
an expression
, then
![](https://www.scirp.org/html/8-7400656\b5f8e987-b086-41ec-926b-77bf36e26a32.jpg)
![](https://www.scirp.org/html/8-7400656\19ab48e0-7409-4adc-a92f-7488f7a4e5dc.jpg)
Let us estimate the integral
. It is evident the the
, where
![](https://www.scirp.org/html/8-7400656\5fd4b327-925e-4f19-b280-872350f46f3a.jpg)
Applying Hölder’s inequality several times, we get
![](https://www.scirp.org/html/8-7400656\b4e31183-14d4-4063-b959-18a0cf866770.jpg)
The following estimation is obvious
![](https://www.scirp.org/html/8-7400656\11d1a670-7849-4fe8-a62e-e16a6cc6c2f5.jpg)
Hense
![](https://www.scirp.org/html/8-7400656\c5b584ad-8d4d-4947-a10d-575af1b971d1.jpg)
then
when
.
Next we construct an estimation for integral
. We apply Hölder’s inequality to
.
![](https://www.scirp.org/html/8-7400656\621482e1-368d-4d2c-b37b-a451cb33adc7.jpg)
So, we get an estimation for
:
![](https://www.scirp.org/html/8-7400656\6df0da94-67e9-49e2-8512-0fb1d58d16a0.jpg)
We use Hölder’s inequality for the second integral again.
![](https://www.scirp.org/html/8-7400656\5610b7ca-fa89-4e5d-b3ff-fba4cc5bbf72.jpg)
Let’s estimate the following integral
![](https://www.scirp.org/html/8-7400656\c6f0a0e5-6616-4e27-9ac8-fe2d82020886.jpg)
![](https://www.scirp.org/html/8-7400656\d65ab049-a88c-4a72-91d2-d04f34248866.jpg)
Denote
and consider
![](https://www.scirp.org/html/8-7400656\24803dd5-c65d-4a51-a91f-553bcffd9f07.jpg)
When
(i.e.
), then
![](https://www.scirp.org/html/8-7400656\f6b49d9f-3baa-4c56-b5a9-b54e2db7ba9f.jpg)
and, respectively
![](https://www.scirp.org/html/8-7400656\bcc3bf3a-e401-4441-8fb2-8ce08213aa9c.jpg)
If
we get
![](https://www.scirp.org/html/8-7400656\3389e2c4-db7d-433c-ae6d-913ab2ba73de.jpg)
![](https://www.scirp.org/html/8-7400656\d53838bb-5560-4506-90ac-a7272c18546e.jpg)
At last, when
, then
![](https://www.scirp.org/html/8-7400656\05dc17dc-e6b1-49b3-9f50-59e6532a57a2.jpg)
![](https://www.scirp.org/html/8-7400656\84325079-d58c-488e-8b53-a849e88c07ff.jpg)
Thus, we obtain
![](https://www.scirp.org/html/8-7400656\bac4176d-ea0c-436f-b94d-f918202a5356.jpg)
and therefore
![](https://www.scirp.org/html/8-7400656\999ba9d3-9013-4a04-a931-f7ca7d4bea9f.jpg)
Bringing together the constructed estimates, we derive the following inequality for integral (6)
![](https://www.scirp.org/html/8-7400656\275e32ce-6a46-4c7d-a53e-fedb60d07067.jpg)
where
![](https://www.scirp.org/html/8-7400656\01a87922-d815-4b1f-8ec2-0529eb8157d0.jpg)
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
![](https://www.scirp.org/html/8-7400656\114e478e-332b-492b-bd11-1ec29f215714.jpg)
The
is a vector-function of the external electromotive force, which is asumed given and satisfying the condition
![](https://www.scirp.org/html/8-7400656\a3709d1e-7c82-45ba-a09c-aadd8f88b8e2.jpg)
Function
equals zero for almost all
.
We introduce the necessary functional spaces
![](https://www.scirp.org/html/8-7400656\d8acea55-fb31-456a-a2e4-a2025fe5c016.jpg)
![](https://www.scirp.org/html/8-7400656\69966b55-7ee6-45df-8923-e1fdd1c7efd3.jpg)
![](https://www.scirp.org/html/8-7400656\d600035b-71a1-49e6-8b13-15ebfad0b5d3.jpg)
Denote
. It is readily proved that this functional space will be Hilbert space relatively to scalar product
![](https://www.scirp.org/html/8-7400656\2026493e-ba3f-4d8d-85c9-8ed1bf50498c.jpg)
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
![](https://www.scirp.org/html/8-7400656\c9670944-f8e6-4bb8-9889-59db771a8bd1.jpg)
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
![](https://www.scirp.org/html/8-7400656\bab6e8db-8a41-404e-bf96-68e1d7a383e7.jpg)
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
![](https://www.scirp.org/html/8-7400656\ce077d7b-8dd1-49c9-b210-2163ab349d68.jpg)
Obviously,
is a bilinear and symmetric form. The finiteness is easily proved by condition (11):
![](https://www.scirp.org/html/8-7400656\5bb436bd-8f2e-47ba-941c-ca69e82b19e3.jpg)
Using the Cauchy-Bunyakovsky-Schwarz inequality, we obtain
![](https://www.scirp.org/html/8-7400656\b45a5f69-4be2-4d2c-ba10-de4466131f75.jpg)
Let’s show coercivity of the form
.
Whereas
thus
for each
, i.e. the vector-function
satisfies estimation
![](https://www.scirp.org/html/8-7400656\9c6ea0eb-4418-4f8e-a869-1bb556e75844.jpg)
The following notation is used
. Let’s use Estimation (1)
![](https://www.scirp.org/html/8-7400656\7488efea-c21d-4dd4-a490-803514e984c4.jpg)
Since
, the last summand is zero. Then using the Hölder’s inequality, we obtain
![](https://www.scirp.org/html/8-7400656\fdb2f0a7-d9c4-47d0-8fc6-a4b99150527d.jpg)
Hense
![](https://www.scirp.org/html/8-7400656\2068f3cb-b96d-4335-8ccc-5daeae4622ac.jpg)
where
. The estimates show the coercivity of the bilinear form, because
![](https://www.scirp.org/html/8-7400656\5f1d9b6f-cf21-45ae-bf3b-a891d8673e91.jpg)
Now we verify the conditions for functional
. The linearity is obvious. Let’s show the finiteness using the Cauchy-Bunyakovsky-Schwarz inequality
![](https://www.scirp.org/html/8-7400656\a471e9c9-a750-465f-87c9-34693a01a159.jpg)
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
![](https://www.scirp.org/html/8-7400656\9320464e-87c3-4758-9273-656795a932a8.jpg)
Let
. Let’s extend
by zero in
. According to the Lax-Milgram lemma there is the unique function
, which for each
satisfies the equality
![](https://www.scirp.org/html/8-7400656\e0e7319b-4280-4b48-bd3d-7d007a1b20f6.jpg)
Then
and as
, then
. Therefore we obtain that
![](https://www.scirp.org/html/8-7400656\ac358df7-5464-41cc-bf55-2ac66b1bae56.jpg)
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).