Abstract
In the paper, the necessary and sufficient condition of compact removability is obtained.
1. Introduction
The questions of compact removability for Laplace equation is studied by Carleson [1]. The uniform elliptic equation of the seconds order of divergent structure is studied by Moiseev [2]. The compact removability for elliptic and parabolic equations of nondivergent structure is considered by Landis [3]. Gadjiev, Mamedova [4]. The removability condition of compact in the space of continuous functions are constructed in the papers Harvey and Polking [5], KilpelΓ€inen and Zhong [6]. The different questions of qualitative properties of solutions of uniformly degenerated elliptic equations is studied by Chanillo and Wreeden [7]. In paper [8] the second order uniform divergent elliptic operator is considered.
Let be dimensional Euclidean space of the points . Denote the ball by for and the cylinder by . Further, let for and be an ellipsoid . Let be an bounded domain with the domain . is a such king of ellipsoid that is a set of all functions, satisfying in the uniform Lipschitz condition and having zero near the .
Denote by and the vector .
Denote by the Banach space of the functions given on with the finite norm where Further, let be a degenerated set of all functions from by the norm of the space . Denote by the set of all bounded in functions.
Let be some compact. Denote by the totality of all functions , each of which there exists some neighbourhood of the compact in which .
The compact is called the removable relative to the first boundary value problem for the operator in the space if all generalized solution of the equation in formed in zero on and belonging to the space identically equal to zero. We will say that the function is nonnegative on the set , in the sense if there exists the sequence of the functions , such that , for and .
The function is nonnegative and in sense if there exists the sequence of the functions , such that , for and . It is easy to determine the inequalities , , , and also equality on the set in the sense if at the same time and on , in the sense .
Let be measurable function in , finite and positive for a.e. . Denote by the Banach space of the functions given on , with the norm
Let be a Banach space of the functions given on , with the finite norm
Analogously to , it is introduced the subspace for . The space conjugated to we will denote by .
We will consider the elliptic operator in the bounded domain In assumption that is a real symmetric matrix with measurable in elements, moreover, for all and a.e. , the condition Here, is a constant.
The function is called the generalized solution of the equation in , if for any function the integral identity be fulfilled.
Here, is a given function from .
Let be some compact. The function is called generalized solution of the equation in vanishing on if integral identity (1.7) is fulfilled for any function .
We will assume that the coefficients of the operator continued in with saving condition (1.2), (1.6). For this, it is sufficient, for example, to assume that for , , where is a Croneker symbol.
Let , , , , be a given functions. Let us consider the first boundary value problem The function we will call generalized solution of problem (1.8) if for any function , the integral identity is fulfilled.
Our aim to get the necessary and sufficient condition of compact removability in the class of bounded functions.
2. Preliminaries Statements
At first, we introduce some auxiliary statements.
Lemma 2.1. If relative to the coefficients of the operator condition (1.2), (1.6) are fulfilled, then the first boundary value problem (1.8) has a unique generalized solution at any , , , . At this, there exists such that if , , , , , , then solution is continuous in .
Lemma 2.2. Let relative to the coefficients of the operator conditions (1.2), (1.6) be fulfilled. Then, any generalized solution of the equation in is continuous by Holder at each strictly internal domain .
Lemma 2.3. Let relative to the coefficients of the operator , conditions (1.2), (1.6) be fulfilled and . Then, for any positive generalized solution , the equation in the Harnack inequality is true If at this and , then the inequality of form (2.1) is true in ellipsoid .
Lemma 2.4. Let relative to the coefficients of the operator conditions (1.2), (1.6) be fulfilled and generalized solution of the first boundary-value problem (1.8) at , . Then, if is bounded on in the sense , then for solution the following maximum principle is true:
where is an exact lower (upper) bound those numbers , for which ββon in the sense .
These lemmas are proved analogously to paper [7]. Therefore, the proof of these lemmas is not given.
Let be some compact and a set of all functions such that on , in the sense . Let one considers the functional
is a compact capacity relative to ellipsoid and is called the value and denoted by . In case , the corresponding value is called capacity of the compact and denoted by .
Lemma 2.5. There exists the unique function such that on in the sense and .
Proof. It is easy to see that is convex closed set in . From the fact that is a Hilbert space, it follows the existence of unique function , which achieved an exact lower bound of the functional . Next,
It is clear that . Moreover, . Denote by . We have
On the other side, according to (1.2),
From (2.5) and (2.6), we conclude
that is, . From uniqueness extreme function, it follows that , and lemma is proved.
The function , which achieved an exact lower bound of the functional on the set is called capacity of the compact potential relative to the ellipsoid .
Lemma 2.6. Let be a capacity potential of the compact relative to which is a generalized solution of the equation in , vanishing on 0 and in 1 on in the sense .
Proof. It is sufficient to show the truth of the first part of assertion of lemma. Let and on in the sense . Then, for any . Therefore,
Thus,
that is,
Tending to zero, we conclude
It is easy to see as in (2.11), we can take any function from with compact support in . Then,
Substituting on , we arrive to
Lemma is proved.
Let be a charge of bounded variation, given on . We will say that the function is a weak solution of the equation , equaling to zero on if for any function , the integral identity is fulfilled.
According to Lemma 2.1 (at ), there exists the continuous linear operator from in such that for any functional , the function is unique in generalized solution of the equation .
The operator is called Green operator.
By Lemma 2.1, this operator at we transform to . It is easy to see that the function is weak solution of the equation , equaling to zero on if and only if for any function the integral identity is fulfilled.
By analogy with [8], we can show that for each measure on , there exists the unique weak solution of the equation equaling to zero on .
Let us say that the charge if there exists the vector , , , for any function , the integral identity is true.
At this, it is evident that
Lemma 2.7. The weak solution of the equation , equaling to zero on , belongs to , if and only if .
Proof. At first, we will show that if the function satisfies the integral identity
for any function zero on , then it is weak solution of the equation , equaling to zero on . Really, assuming that , , we obtain
and now, it is sufficient to use the identity (2.15). We will show that . For this, it is sufficient to prove that if , then , ,. Assume in condition (2.18) that , .
We will obtain
Let . Assuming that at and , , , we will obtain
Using (2.20), we conclude
From (2.20) and (2.22), it follows that
Thus, from (2.23), take out for
So, . Inversely, if is a weak solution of the equation , vanishing on , then there exists such that
for any function , .
Then, from Lemma 2.1, we obtain that . The lemma is proved.
Let now be Dirac measure, concentrated at the point 0, and an arbitrary fixed point .
The weak solution of the equation , vanishing on , is called Green function of the operator in .
In case the corresponding function is called the fundamental solution of the operator and denoted by .
According to the above proved, if is an arbitrary function from , then the generalized solution of the equation can be introduced in the following form: We can show that is nonnegative in moreover; .
Lemma 2.8. For any charge of bounded variation on , the integral exists, finite a.e. in and is weak solution of the equation , equaling to zero on .
Proof. Without losing generality, we will assume that the charge is the measure in . Let , in . Denote by the generalized solution of the equation . Then, according to Lemma 2.1 and according to Lemma 2.4. At this,
Then, by the Fubini theorem, we conclude that the integral there exists for almost all ; moreover,
Let us note that (2.29) is fulfilled for weak nonnegative and continuous in function . Now, it is sufficient to remember the identity (2.15) and lemma is proved.
Let us consider now -capacity of the potential of the compact relative to the ellipsoid . Before, it was proved that satisfies (2.11) at any nonnegative on the function . By the Schwartz theorem [9, 10], there exists the measure on such that
Further, since on in the sense , then the carrier of the measure is situated on . The measure is called -capacity of the compact .
According to Lemma 2.8,ββ-capacity potential is weak solution of the equation , equaling to zero on and can be represented in the following form:
On the other side, there exists the sequence of the functions ; such that , for and . Assuming in (2.15) instead of , we conclude that it first fart is equal to at any natural , while the left part tends to as . Thus,
Lemma 2.9. Let relative to coefficients of the operator conditions (1.2)β(1.6), , , be fulfilled. Then, for the Green function the following estimations are true: If , then,
Proof. Without loss of generality, we can assume that the coefficients of the operator are continuously differentiable in . The general case is obtained by means of limit passage. Then, at , the function is continuous by and moreover,
Let be a positive number, which will be chosen later, , where is an arbitrary fixed point on . From (2.35), it follows that is internal point of the compact . Then, is capacity potential , represented in the form (2.31), so it means that it equal to zero in it. Thus,
where is a -capacity distribution of the compact . Allowing for the carrier of the measure is situated on , where = and using (2.32), we obtain
Let us assume now that . According to maximum principle, . Therefore, from (2.37), we conclude
If we will assume , then ; that is,
From (2.38) and (2.39), follows that
On the other side, according to Lemma 2.3,
Now, the required estimations (2.33) follows from (2.40) and (2.41). Absolutely analogously the truth of (2.34) is proved.
Corollary 2.10. Let the conditions of the lemma and be fulfilled, , or , , . Then, for fundamental solution , the estimations are true.
3. Removability Criterion of the Compact in the Space
Theorem 3.1. Let relative to the coefficients of the operator , conditions (1.2)β(1.6) be fulfilled. Then, for removability of the compact relative to the first boundary value problem for the operator in the space , it is necessary and sufficient that
Proof. Let the ellipsoid have the same sense as above. It is easy to see that if condition (3.1) is fulfilled, then
Not losing generality, we can limit the case, when the coefficients of the operator is continuously differentiable in . Let us fix an arbitrary and . By virtue of (3.1), there exists the neighbourhood of the compact such that
At this, we can assume that is such small that
Denote by and the -capacity potential of the compact relative to the ellipsoid and -capacity of the distribution , respectively. According to above proved,
moreover, the function is the generalized solution of the equation in , vanishing on 0 and in on 1 in in the sense . Let now be an arbitrary solution of the equation in , vanishing on , . It is easy to see that the function is nonnegative on , in the sense . Hence, it follows, that the function is generalized solution of the equation in , is non-positive on . According to Lemma 2.4,ββ and in particular
By virtue of continuity of the function at and (3.4), we obtain
Thus, from (3.3) and (3.6), we conclude
Using an arbitrariness , we have
Making analogous considerations with the function , we obtain
From (3.8)-(3.9) and an arbitrariness of the point it follows that in . Thereby, the sufficiency of condition (3.1) is proved. Let us prove its necessity. Let us assume that . Denote by the ellipsoid such that , . Assume ββpotential of the compact relative to the ellipsoid and -capacity distribution , respectively. Following to [11], we can give the equivalent definition of Vallee-Poussin type of -capacity of the compact , relative to the ellipsoid . Let be a Green function of the operator in . Let us call the measure on , -admissible if and
The value , where an exact upper boundary is taken on all -admissible measures and is called -capacity of the compact relative to the ellipsoid .
Analogously, the -capacity is determined. At this, by the standard method, we show that there exists the unique measure on which an exact upper boundary of the functional is reached by the set of all -admissible measures . This measure is -capacity distribution of the compact .
According to the above proved, the function is generalized solution of the equation in , equaling to zero on . Besides, from (3.10) and maximum principle, it follows that . On the other side, , as . Theorem is proved.
Lemma 3.2. Let relative to the coefficients of the operator condition (1.2) be fulfilled. Then, if , then
Proof. Let . Then, according to (1.2),
Let , for ; moreover,
Then,
On the other side, as , then , and thereby
Besides, as , then
Thus,
Hence, it follows that
Therefore,
where .
Allowing for (3.14) and (3.20) in (3.15), we obtain
and by virtue of (3.13), the estimation from upper in (3.11) is proved.
For showing the truth of the estimations from lower in (3.11), we note that
Besides, considering the same as in [8], we conclude
where .
Let . Then,
On the other side, if , then we can find , such that , that is;
Besides, as , then
Therefore,
Thereby,
where .
Allowing for (3.28) in (3.24), we obtain
Denote by the ball . Let us make in (3.30) the substitution of the variables ; , and let be an image of the point , where . Then, from (3.30), we deduce where by (3.30),
we will denote by Wiener capacity of the compact , relative to the ball . Now, it is sufficient to note that , and required estimation follows from (3.22), (3.23), and (3.31). Lemma is proved.
Lemma 3.3. Let relative to the coefficients of the operator condition (1.2) be fulfilled. Then,
Upper estimation in (3.32) is proved analogously to the estimation in (3.11). For the proofing of the lower estimation, it is sufficient to note that , that is,
where and repeat the consideration of the proofing of the previous lemma.
Corollary 3.4. If conditions (1.2)β(1.6) be fulfilled, then for any , the estimation
is true.
Then, , for
On the other side, arguing the same, as well as in the proof of Lemma 3.2, we have
Now, it is sufficient to take into account that
and from (3.34)-(3.35) follows the required estimation (3.33).
Corollary 3.5. If conditions (1.2)β (1.6) are fulfilled, then at , for the fundamental solution , the estimation
is true.
If , then estimation (3.37) is true for all . Here, .
For proving, at first, let us show that if , then . Really, as
then there exists , such that
Thus,
Thereby,
Now, it is sufficient to note that , and the required assertion is proved. On the other side from (3.38), it follows that for all ,
that is,
So, one will show that if only .
Let now for , and . Denote by the . It is easy to see that there exists , such that
Hence, it follows that
Thus, , where . Analogously, it is proved that . Now, the required estimation (3.37) at follows from (2.42) and Corollary 3.4 from Lemma 3.2. If , then (3.37) immediately follows from (2.42) and Lemma 2.7.
Let be a positive function, determined in , continuous at , moreover (condition (A)).
Further, let be some compact. Let us call the measure on admissible if and , for .
The value , where an exact upper boundary is taken by all admissible measures, is called -capacity of the compact .
Theorem 3.6. Let relative to the coefficients of the operator conditions (1.2)β(1.6) be fulfilled. Then, for removability of the compact relative to the first boundary-value problem for the operator in the space it is sufficient that where .
Proof. We will use the following assertion, which is proved in [11]. Let the function be satisfied condition (A), the compact has zero -capacity, zero measure concentrated on . Then, there exists the point , such that . At this the potential of the measure cannot be bounded on any portion , that is, for any open set at ; the potential is not bound . In particular, if is an arbitrary neighborhood of the point that .
Let the condition (3.46) be fulfilled, an arbitrary measure, concentrated on , is a point, corresponding to the above-stated assertion at . Let us assume at first that . Then, . Further, let be such small neighborhood of the point that if , then
where the number will be chosen later. Let us consider the ellipsoids at . Let us choose such small than for all . Then, according to Corollary 3.5 from Lemma 2.7, we obtain
Hence, it follows that any zero measure , concentrated on cannot be admissible. Thus, , and the required assertion follows from Theorem 3.1.
Let now . Then, using the equality and Corollary 3.5 from Lemma 2.7, we conclude
Theorem is proved.
Remark 3.7. Let conditions of the real theorem be fulfilled and the compact removable relative to the first boundary-value problem for the operator in the space . Then, .
At first, let us note for proofing that the discussion are the same, as at the conclusion of estimation (3.37), we can show that at , , and at the estimations
are true.
Further, analogously to Theorem 3.1, it is shown that if the compact is removable, then according to , where .
Hence, it follows that if mes , then there exists the point , such that , where
Moreover, if is an arbitrary neighborhood of the point , then the potential is not bounded on . Let us consider the case . Choose small neighborhood of the point that at all , the inequality ; are fulfilled. For , we have
where is a ball of the radius with the center origin of the coordinate. Now, it is sufficient to note that according to condition (1.6) , and the assertion the corollary is proved.
Acknowledgment
This paper was performed under the financial support of Science Foundation under the President of Azerbaijan.