In this paper we obtain a weak Harnack inequality for nonnegative bounded supersolutions of
where D is a bounded Lipschitz domain in , , , and is a Caratheodory function (that is measurable in for all , and continuous in ξ for a.e. ) satisfying the coercivity and growth conditions
for a.e. and all , where and are positive constants. The variable exponent p belongs to and satisfies
For and by we denote the open cube centered at z with edge length and faces parallel to the coordinate hyperplanes. We shall also assume the log-condition
Let denote the set of functions such that for a.e. and . Smooth functions are dense in in the sense that for any there exists a sequence such that
as , see [1–3]. We say that is a supersolution of (1.1) in Q if
for any and nonnegative with , which vanish near the lateral boundary of Q. For more on definitions and basic properties of solutions to parabolic equations with - and -Laplacian and corresponding functional spaces we refer the reader to [1–3, 12].
The main result of this paper is a weak Harnack inequality for nonnegative bounded supersolutions of (1.1). For and we denote
Further , , . For a function and a measurable set we denote
Below u is a bounded nonnegative supersolution of (1.1) in , satisfying , , and we denote and .
Letsatisfy (
1.2
), and. There exist positive numbers,,,,, which depend only on n,,,, L, ν, such that if,or, and, where, thena.e. in.
With respect to the parameters , , , L, ν the constants , , are decreasing, and , are increasing.
The by now classical results on regularity of solutions to the parabolic p-Laplace type equations, in particular the Hölder continuity of solutions, can be found in [5]. The Harnack inequality for the parabolic p-Laplace type equation with constant exponent was obtained in [8], and the weak Harnack inequality for the same range of p is due to [13] and [10, Chapter 5, Theorem 7.1]. Corresponding results for were proved in [6, 7]. In the papers [19, 20] the results of [8] were extended to weighted degenerate parabolic equations. The Hölder continuity for weighted degenerate parabolic equations with constant exponent was obtained earlier in [4, 16]. Note that while the weighted equations with constant exponent bear certain resemblance to the variable exponent case in that the intrinsic time varies from point to point, the source of this phenomenon and methods to treat it are different. The Hölder continuity of solutions to parabolic equations with the parabolic -Laplacian was established in [2]. The Harnack inequality was extended to the parabolic -Laplacian in [21]. For various properties of functional spaces with variable exponent we refer the reader to [11]. Recently a number of powerful results for parabolic equations of general structure, which covers situations with variable exponent, double phase, weights, were obtained in [17, 18] by a different method.
Theorem 1.1 generalizes the weak Harnack inequality of [13] and [10, Chapter 5, Theorem 7.1] to the variable exponent case. This result states [13] that if then (in the situation of Theorem 1.1) for any there holds
where and
where and depend only on n, , , and . This result essentially states that if then Theorem 1.1 is true with (clearly, in this case ). Indeed, if then inequality (1.3) is trivial and does not give any information about u in (the LHS is dominated by the first term on the RHS). This corresponds to the case when there is not enough time for the initial average positivity to fully cover the extended region. Otherwise we get the required inequality, but with the additional term . One can get rid of this term by continuing u to infinite times, say, by solving an additional initial-boundary value problem for (1.1), so that , or examining the proofs provided in [10, 13].
The difference between Theorem 1.1 and the results of [13], [10, Chapter 5, Theorem 7.1] is that in the constant exponent case for the homogeneous equation there is no requirement for the initial average λ to be quantitatively separated from zero, which in our case we have to require. This feature is characteristic of results for equations with variable exponent and stems from the fact that the equation loses homogeneity. The resulting additional bound for λ is similar to what arises for equations with lower order terms. Note that there are two different cases treated in the statement of Theorem 1.1. The first one, , corresponds to the “almost linear” case, where the equation is treated as a perturbation of the linear parabolic equation. In the second case, when , the technique becomes substantially different which represents the transition to the “degenerate parabolic” scenario. In the latter case there additionally arises the requirement for the smallness of oscillation . Whether this additional condition is artificial or in the essense of things is unknown to the author at present.
The log-Hölder condition, written here in the form (1.2) and (clearly, if then the latter is implied by the former, with some different L) is also typical for equations with nonstandard growth and coercivity conditions. It was introduced in [22] as a sufficient condition for density of smooth functions in variable exponent Sobolev-Orlicz spaces. Later it was discovered that under the log-Hölder condition other properties of Sobolev spaces and of solutions to elliptic and parabolic equations are transferred to the variable exponent case, see for instance [11, 23]. This is best explained on the following example: consider (1.1) with (the “pure” parabolic -Laplacian) in . Then rescaling , , gives the same equation with in . The coefficient is quantitavely separated from zero and infinity if in for some . This leads to , . For technical reasons, further in this paper we do not use the spatial change of variables.
The proof of Theorem 1.1 follows the general outline of the proof of the weak Harnack inequality in [10, Chapter 5, §7–13]. Section 2 gathers some well-known facts. In Section 3 we set up a De Giorgi type toolbox (for the elliptic version see [15], for the linear parabolic setting see [14], and for the constant exponent degenerate parabolic equations it is provided in [5]). In Section 4 we obtain the expansion of positivity result by the method of [8]. In Section 5 the Local Clustering Lemma [9] leads to an improvement of the expansion of positivity result. Sections 6 and 7 are devoted to reverse Hölder inequalities for supersolutions and gradient estimates implied by them (for such estimates can be found in [5]). In Section 8 we prove Theorem 1.1 by considering the “hot” and “cold” alternative [13]. The final Section 9 provides two variations of Theorem 1.1.
Further we denote , , and . Clearly . For a set A by we denote the indicator function of this set.
Technical lemmas
We start with the classical De Giorgi inequality (see [5]), [10, 14, 15]. Denote the -volume of the unit sphere in by .
Letandbe real numbers. Thenwhere the constant.
We need the following measure-theoretical lemma from [9] (see also [10]). We reproduce the proof of this simple albeit powerful statement for the benefit of the reader.
Letsatisfyfor someand. Let. Then for anysuch thatandthere exists a cubesuch that
First, we partition the cube into congruent subcubes , , with edge , . Let be the set of cubes such that and be the number of such cubes. Clearly, . Therefore, .
Let be the set of cubes from such that and be the number of cubes from this set. For each cube by the De Giorgi inequality of Lemma 2.1 we have
Summing over cubes from we get
Therefore,
Clearly, provided that
Thus for any we have , the set is non-empty, and each cube from the set can be taken as from the statement of the lemma. The proof of Lemma 2.2 is complete. □
We shall use the Sobolev inequality in the following form [14]:
where Ω is a domain in , , and . As a corollary, for there holds
The following two standard estimates can be found for instance in [5].
Let,be a sequence of nonnegative numbers satisfying, where,. Thenasprovided that.
Let,be a sequence of equibounded nonnegative numbers satisfying, where,, and. Then.
De Giorgi–Ladyzhenskaya–Uraltseva toolbox
In this section we assume that u is a nonnegative bounded supersolution of (1.1) in the cylinder , which satisfies , , , and .
Recall that , , for and we denote .
For two concentric cubes we shall use Lipschitz cut-off functions of the form . Clearly on , outside , on , and . Further we shall state the existence of such function without specifying it. The same is used for functions used to cut off time intervals — this will be a piecewise linear function, for instance when , when , and for (or take for cutting off the other direction).
The proofs of statements in this section will be based on the following energy estimate, which is similar to the so-called classes of Ladyzhenskaya and Uraltseva [14].
Let η be a Lipschitz function equal to zero on the lateral boundary ofsatisfying. Denote. For anyandthere holdsIfis independent of t then for anyandthere holds
First we use the test function (for the rigorous proof see [1–3]) to obtain
for all . Then apply the estimates
and
Since , we estimate further . After the obvious cancellation we arrive at the estimate.
For the first integral on the RHS we use the inequality
where we used that
This gives the second estimate of the Lemma when . Otherwise we estimate further the second integral on the LHS using the inequalities
and
This gives
Now the required estimate is obvious. □
For anyand, such thatthere holdsif
Let , and be the Lipshitz function such that outside , on , and . From the second inequality of Lemma 3.1, using , and so , for we get
Denote
For and it follows that
Now,
Take . Then . Set and denote
where is the constant from the statement of the Lemma. Since , for there holds , and thus
which is the required estimate. The proof of Lemma 3.2 is complete. □
We shall also use a simple corollary of this lemma
Assume that u satisfies (
3.1
) and. For anythere holdsif, whereis the constant from Lemma
3.2
.
Note that (3.1) implies the same estimate with λ replaced by . □
Now we prove a De Giorgi type lemma. Recall that .
Let,. For any, andthere existsdepending only on n,, Γ, ϰ, δ,,, and ξ, such thatandimplya.e. in. One can takewith the positive constant.
For denote
Let be the Lipschitz functions vanishing outside , such that , on , . Let be the Lipschitz functions such that when , when , , .
Let be the Lipschitz functions vanishing outside , such that , on , . Let be the Lipschitz functions such that when , when , , .
Denote
Using the first estimate of Lemma 3.1 for the level and the cut-off function we get
The Young inequality
and
give
Denote
In this notation we have
By the embedding inequality (2.1),
On the other hand,
Thus,
where . By Lemma 2.3, as provided that
Dividing this inequality by we see that as if
Recalling the definition of we complete the proof of Lemma 3.4. □
The next lemma is an analogue of Lemma 3.4 involving initial data.
Letand assume thata.e. in. For anyandthere holdsa.e. inifandwhereis the same constant as in the previous lemma.
We repeat the proof of Lemma 3.4, setting , and taking the cut-off functions and depending only on x. Using the notation of Lemma 3.4 we get (3.3) with . The same argument as above gives as provided that
To complete the proof of Lemma 3.5 it remains to note that the LHS of the last inequality does not exceed 1. □
Leta.e. in. Ifthen for allandthere holdswherewith, provided that the right-hand side of this estimate is greater than.
Apply in the cylinder Lemma 3.5 with λ replaced by , , , and the given . If then a.e. in for all
If and we use . If we use . The proof of Lemma 3.6 is complete with . □
Now we prove the “concentration of positivity” lemma using the “telescopic argument”.
LetandFor any,, andsuch that, in the cylinderthere holdswhere, provided that.
Let be the Lipshitz function vanishing outside such that on , , , and let be the Lipschitz function such that for , for , , and .
Denote , . Let
From Lemma 3.1 with the level and the cut-off function we get
Denote
By Lemma 2.1 and assumption (3.4),
for almost all . Integrating this in t and using , we get
Raising this to the power and using (3.6) we have
Summing these inequalities over (assume that (3.5) holds for ), we get (note that )
which yields
where . By the condition of the lemma, (3.5) holds for all , which justifies the above argument. The proof of Lemma 3.7 is complete. □
Expansion of positivity
Here we obtain expansion of positivity results following [8, 10]. In this section , , is a nonnegative bounded supersolution of (1.1) in , such that , , , , and , . As above, we denote , for we denote , and . We start from a special form of the expansion of positivity result for p close to 2.
(Almost linear case).
Assume that u satisfies (
3.1
) with. For anythere exist positive constants,, and, such that if,, andthenHere, and the constantsanddepend only on,, n,, ϰ, Γ, and α.
We assume from the beginning that . By Lemma 3.2,
where with given in (3.2). Therefore,
where . Denote . Since , we have . Set and . By Lemma 3.7, for any and there holds
provided that and
Since , , and , recalling the definition of we can replace the last condition by
Denote . By Lemma 3.4 in with , , and , there holds a.e. in if and
where
Thus it is sufficient to find and ρ satisfying (4.1) and
It is clear that
and
since (see (3.2)) and . Using (4.4)and we get
Take , then , and denote
Take , then
Finally, restrict s to the set . Then (4.1) is satisfied by the choice of ρ and (4.3) follows from (4.5) and (4.6), so we obtain in . Thus we have proved Lemma 4.1 with , , and . □
We shall also use the following simple corollary of Lemmas 3.5 and 3.7.
Assume thata.e. in,. For anyandthere exist positive constants,, and, which depend only on,,,, n,, Γ, ϰ, such thatandprovided that,, and.
The proof repeats the proof of the previous lemma with the only difference that we use Lemma 3.5 instead of Lemma 3.2. Set
By Lemma 3.5 with and we get
Set , then
Denote . Since , we have . By Lemma 3.7 with and , for any and there holds
provided that and
Since and the last condition holds if
Denote . By Lemma 3.4 with , , and , there holds a.e. in if and
Thus it is sufficient to find j and ρ satisfying (4.7) and
One easily finds that
From (4.4), for j and s such that
we get
Set , then and (4.9) implies (4.7). Denote
Take . Then
Restrict s to the set . Then condition (4.9) is satisfied, so is (4.7), from (4.11) we get (4.8), and finally arrive at in . Thus we have proved the statement of Lemma 4.2 with , , and defined in the beginning of the proof. The proof of Lemma 4.2 is complete. □
For p not close to 2 the proof of the expansion of positivity lemma is more involved and requires a special trick which was the cornerstone of the breakthrough paper [8].
(Expansion of positivity).
Let u satisfy (
3.1
) with, and,. For anythere exist positive constantsandsuch thatprovided that,, and. The constantdepends only on,, n,, ϰ, Γ, α, and ϵ. The constantsdepends only on,, n,, ϰ, Γ, and α.
Denote , where the constant is given in (3.2). By Lemma 3.3, for there holds
provided that . Introduce the functions
Then
Introduce the new time variable by and , that is
Denote
Then the function satisfies
and for we have
It is easy to see that
Further on in the proof of this lemma we assume that
Then
for all , , . Thus, all the statements from Section 3 remain valid for w with replaced with and Γ replaced by .
Let , . For denote
By Lemma 3.7 with , , and , for any there holds
provided that and
By Lemma 3.4 with , , and ,
if and
where , , and
Let in and , then , and there holds
By (4.4),
Denote
and take . Since , we have , thus
and condition (4.15) is satisfied.
Finally we need (4.13) to hold. Thus we must have
The above conditions are satisfied, we get (4.15) and so (4.14). Returning to the original time variable we obtain
Denote
By Lemma 3.6 with , for any there holds
provided that the right-hand side of the last inequality is greater than . Put
Then a.e. in , provided that . From (4.17) we get the condition on from the statement of the Lemma. The proof of Lemma 4.3 is complete. □
The dependence of on α in Lemma 4.3 has the form , . Following [10] this can be improved to make this dependence power-like, which is the subject of the next section.
We need a statement which unifies the results of Lemma 4.1 and Lemma 4.3.
Assume that u satisfies (
3.1
) with, and. For anythere exist positive constants,,,, and, which depend only on,, n,, ϰ, Γ, and α, such that iforandthenprovided thatand.
Let . Let , , and , be the constants claimed by Lemma 4.1 for the given values of α and by . Let , , and , be the constants claimed by Lemma 4.2 for the given value of , , and . Put .
Let and be the constants claimed by Lemma 4.3 for the given valued of α and , and . Note that from the proofs of Lemmas 4.1 and 4.3 it is clear that .
If and then by Lemma 4.1 there holds
Set
If , , and , then by Lemma 4.3 we have
Set . If apply N times Lemma 4.2 with the parameters given in the beginning of the proof. We obtain
provided that and
Let be the minimal integer greater than or equal to . If then (4.18) is satisfied if . Choose the maximal satisfying and . Then
for . Thus for all inequality (4.18) is satisfied with . The proof of Lemma 4.4 is complete with and . □
Improved expansion of positivity
In Lemmas 4.1 and 4.3 the dependence of on α is of the order of for some . Following [10] we improve this dependence to be power-like, which provides the foundation for the main result of this paper.
Let , , , , and , be the constants from Lemma 4.4 with the parameters , , , where e is the base of the natural logarithm. We assume that , otherwise we replace it by .
Further in this section and u is a nonnegative supersolution of (1.1) in satisfying , where , and , .
Let , , , , and , be the constants from Lemma 4.4 for , , and .
Let be the constant claimed by Lemma 4.2 for , , , .
In the following lemma we assume also the log-condition (1.2), and this is the only place in this paper where this condition is critical.
Letand u satisfy (
3.1
) with,. Assume also that the log-condition (
1.2
) holds. There exist positive constants,,, d,,,, which depend only on,, n,, L, Γ, ν, such thatif,, andor.
From the very beginning we assume that , and set , .
By Lemma 3.2 there holds
where with given in (3.2).
Let be the Lipschitz function such that on , φ vanishes outside , , and . Let be the Lipshitz function such that for , , and . Apply the energy estimate of Lemma 3.1 for the cut-off function and . This gives
Thus,
and there exists a time instant such that
Apply Lemma 2.2 with , . There exists and such that
and
Let be the minimal number such that . Denote
Apply Lemma 4.4 starting from the cube and time instant . Then after m iterations we get
provided that for , there holds
and
We only have to check (5.2) since the other two conditions are satisfied by the assumptions of the Lemma. Note that . If then (5.2) is satisfied if . If , then we need to check that
Since by our assumption , and , we see that (5.3) holds if
If and , let be such that . For condition (5.2) requires that
for . This leads to condition (5.4). For condition (5.2) is satisfied if . Since for we have , condition (5.2) is satisfied for if (5.5) holds for . Since , this again leads to condition (5.4).
Therefore, to satisfy condition (5.2) it is sufficient to require that
Since , using (5.1) we have
with the positive positive constants and . Thus
where and . So
Let us obtain an upper bound on . Using we have
Here we used that by (5.6) and the assumptions of the Lemma there holds
Now we impose another restriction, , which is equivalent to
Summation over k yields
Using that , (5.7) and the definition of we finally obtain
Note that (5.8) implies
Applying one more time Lemma 4.4 starting from the time level and the cube , we have
where and
and . Here we require that .
If then from (5.8) by Lemma 3.6 with we get
a.e. in , with the positive constant .
If then we apply times Lemma 4.2. This gives
and , are the constants from Lemma 4.2. Let be the minimal integer greater than or equal to . If then we take and with this choice we obtain . If we choose the maximal so that . Then for all we get
and the inequality is satisfied with .
Thus we obtain
To complete the proof of Lemma 5.1 it remains to set , where is the constant from (5.6), , and , and , , defined above. □
From the proof it is clear that the constant d depends only on n, , ϰ, γ, ν, and is independent of , .
Reverse Hölder inequalities
In this section u is a nonnegative bounded supersolution of (1.1) in , such that , , , , , , , and .
Let, and η be a Lipschitz function in Q, vanishing on the lateral boundary of Q, satisfying,,,. For allthere holds
We start from a standard energy estimate which results from testing the integral inequality for supersolutions by and using the Young inequality: there holds
For the first term on the RHS in the case we use
and for we write
Denote
For , we estimate the second term on the RHS using
and for we use
The proof of Lemma 6.1 is complete. □
Under the assumptions of Lemma
6.1
for allthere holds
We use the estimate of Lemma 6.1 multiplied by ϰ. By the Young inequality,
The proof of Lemma 6.2 is complete. □
For denote
Take Lipshitz functions vanishing outside such that on , , ; Lipshitz functions vanishing outside such that on , and ; Lipshitz functions such that when , when , , and ; Lipshitz functions such that when , when , and .
Then the function vanishes on the conjugate parabolic boundary of , , on , , , and the function vanishes on the conjugate parabolic boundary of , , on , , .
Denote
From Lemma 6.2 with cut-off functions and , for , using the estimates
and , we obtain
Applying (2.1) for and we get (recall that )
For , using (6.2) with , where , and denoting , we get
Using the estimate
together with and , we get
Iterating this inequality and using
we obtain the required estimate. For , we take with and denote . This gives (6.3). The rest of the argument goes as above. The proof of Lemma 6.3 is complete. □
The next lemma is similar in nature.
Let,,,, and. Iffor allthenwhere
Let , and be the Lipschitz functions vanishing outside and satisfying on , , . Then by Lemma 6.2 for we have, similar to (6.1),
Using the condition of the lemma we get
where . By (2.1) applied to and , and the condition of the Lemma, there holds
Dividing by and denoting
we obtain
Denote
and let γ be the number defined in the statement of the lemma.
Let us begin with the case . Then by the Young inequality for any we have
Therefore
Iterating this inequality we obtain
with γ given in the statement of the Lemma. Take . Sending and using yields
Recalling the definitions of and and using we complete the proof for .
Now let . For by the Young inequality we get
Therefore
Iterating this inequality we obtain
where γ is given in the statement of the Lemma. Taking and sending we get
Recalling the definitions of and we complete the proof for . □
Gradient estimates and consequences
In this section u is a nonnegative supersolution of (1.1) in , satisfying , , and , , . Let d be the constant from Lemma 5.1 and denote . Let , , we assume here that . Denote .
First we obtain an auxilliary bound for an appropriate Lebesgue norm.
Assume thatandfor alland. Then for anythere holdswhereis monotonically increasing with respect to its parameters.
For any by (7.1) there holds
Thus (recall that ) we have for all . Lemma 6.4 yields
since r also depends only on n, , ϰ, Γ, and .
If let be maximal number from such that for some . By the Hölder inequality,
Then by Lemma 6.3 we get
For let be the maximal number from such that
for some . Clearly,
Take , then by the last inequality we have . By the Hölder inequality,
Then
and by Lemma 6.3 we get
The proof of Lemma 7.1 is complete. □
Now we provide an estimate for the gradient norm under assumption (7.1).
Let. Then under the assumptions of Lemma
7.1
, forthere holdswhere.
Let and . By the Young inequality
By Lemma 6.1,
Also,
Now take ε and γ such that
For definiteness, take and the corresponding ε. Since and we have .
By Lemma 7.1 with (since we have ), which gives , we get
If we write with , and use Lemma 7.1 to estimate
If we have
Again by Lemma 7.1 we obtain
Combining these estimates we get
By the Young inequality, for and there holds
Take , where . From (7.2) we have
Put . Then and we arrive at
This proves the claim of Lemma 7.2 with . □
Let. Under the assumptions of Lemma
7.1
, if, thenfor all, and there existssuch that, whereare constants depending only on n,, ϰ, Γ.
Let η be the Lipchitz function vanishing outside such that , on , and . Then
Using the estimate of Lemma 7.2, for , , and recalling that and we get
provided that , where . This proves the first estimate of Lemma 7.3. To prove the second estimate recall that by Lemma 7.1 there holds
for . Let , then by the assumption of the Lemma, , and for . Thus there exists a time instant , , such that
Then for one has
and we can find such that
Together with (7.3) this gives
Splitting this integral into two parts, one over the set where , and the other one over the set where , one easily gets
therefore
The proof of Lemma 7.3 is complete. □
Proof of the weak Harnack inequality
Let d, , , , , , be the constants claimed by Lemma 5.1 for , . We assume from the beginning that , or , that is , .
In this section we fix , , , and denote , , .
Now we are ready to prove the main result of this paper.
We have the following alternative: either (7.1) holds for all (the “cold” alternative of [13]), or it is violated (the “hot” alternative of [13]), that is there exist and such that
In the first case we use Lemma 7.3 to obtain such that
which for implies
Assuming without loss that , in both cases of the alternative we get (8.1) with some . Then by Lemma 5.1 with and we have
Clearly
Denote and .
Let , , be the constants claimed by Lemma 4.2 for , , .
If then from (8.2) and by Lemma 3.6 with we get
a.e. in .
If then we apply times Lemma 4.2. This gives
Let be the minimal integer such that . If then we take and with this choice we obtain
If we choose the maximal so that . Then
Thus for all inequality (8.3) is satisfied with , and for this range of s we get
Combining the estimates obtained for and for we obtain
where . It remains to set , , , , and . The proof of Theorem 1.1 is complete. □
Final remarks
Below u is a bounded nonnegative supersolution of (1.1) in , satisfying . In this section we state without proof two results related to Theorem 1.1.
The ratio in Theorem 1.1 can be easily changed to with any , clearly the constants in this case depend also on a and b. The corresponding changes in the proof are straightforward.
Let,satisfy (
1.2
), and, whereand. There exist positive numbers,,,,, which depend only on n,,,, L, ν, a, b, such that if,or, and, where, thena.e. in.
One can obtain a similar theorem with cubes replaced by “annuli”, the proof essentially repeats the proof of Theorem 1.1. Let are given constants. For denote .
Let,satisfy (
1.2
), and, whereand. There exist positive numbers,,,,, which depend only on n,,,, L, ν,,,,,,, such that if eitheror,, and, where, thena.e. in.
To prove this result, one notes that can be covered by finitely many cubes with edges , then in one of these cubes there holds with some positive . It remains to apply the weak Harnack inequality for cubes (Theorem 9.1 with a suitable choice of a, b), and the standard chain argument.
In this paper we work only with the essentially “autonomous” case where depends only on the spatial variable. The Hölder continuity of solutions [2] is known to be true also for satisfying the log-Hölder condition
We expect Theorem 1.1 to be true also for time-dependent exponents satisfying (9.1).
The author thanks the anonymous referee for their careful reading of this manuscript and useful remarks.
Footnotes
Acknowledgements
This work was supported by Moscow Center of Fundamental and Applied Mathematics, Agreement with the Ministry of Science and Higher Education of the Russian Federation, No. 075-15-2019-1623.
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