HOMOGENIZATION OF A LOCALLY PERIODIC TIME-DEPENDENT DOMAIN

. We consider the homogenization of a Robin boundary value problem in a locally periodic perforated domain which is also time-dependent. We aim at justifying the homogenization limit, that we derive through asymptotic expansion technique. More exactly, we obtain the so-called corrector homog- enization estimate that speciﬁes the convergence rate. The major challenge is that the media is not cylindrical and changes over time. We also show the existence and uniqueness of solutions of the microscopic problem.


1.
Introduction. Several mathematical models arising from systems biology, material sciences, and other applied sciences, give rise to partial differential equations with a complex structure because of impurities and inhomogeneity in these models, see [8,13]. Some kind of these problems involve spatial inhomogeneities of the underlying material at microscopic scales and hence a detailed analytical or even numerical approach becomes infeasible due to these heterogeneities [36]. A natural and well-established methodology is to average out these impurities in the medium through some mechanism, appropriate for the model equation. Such mechanism is commonly referred to as finding the effective, homogenized, model of the equation. There are two general strategies to achieve this goal. One strategy is the phenomenological approach, in which one directly establishes the equations governing the macroscopic behavior without inquiring the detailed structure of the microscale. In this case, the parameters of the macroscopic model are fitted by the experiment. Another strategy is to start from the microscale model of heterogeneities and deduce the effective equations based on averaging methods such as asymptotic expansion technique, and the effective coefficients are obtained mathematically [8,13]. From the analytical perspective, we ask for a rigorous justification of the effective model and, if available, error estimates describing the difference to the original microscopic model [22].
In recent years there has been a renewed interest in mathematical models where the bulk domain changes over time, giving rise to free and moving boundary value problems [4,6,9,12,24,32,37]. A concrete example of such a model is the diffusion in the brain extracellular space. The effective diffusion of the PDE model of the brain cell microenvironment describes the average behavior of the physiology of neuronal populations [19,20]. The extracellular space of the brain is a heterogeneous complex medium in which several factors impose constraints on the diffusion process [20,37]. The primary constraints are mainly consequences of the geometrical structure of the medium, that changes in time due to specialized features and physiological conditions in the brain, such as transport of water in brain ischemia [37]. In such cases, the microscopic problem is a diffusion equation in bounded micro-domain that evolves in time [37]. From the mathematical point of view, we deal with a Robin boundary problem that allows boundary to evolve with time [37].
As a first step before handling the homogenization of a moving boundary problem, we consider in this paper a parabolic problem in a non-cylindrical domain. We are also concerned with a problem that arises in the locally periodic microstructure. Indeed, the geometry of the heterogeneities is such that the perforation is periodic in space, but not the shape and size of them, and they may as well vary in time. For more details, the reader is referred to Section 2.1, where we explain our concept of time-dependent locally periodic domain. Although various homogenization results have been achieved for periodic and locally periodic fixed microstructures over time [2,7,18,26,27], we study the asymptotic behavior of the model in a non-cylindrical domain.
To capture the macrostructure in the homogenized form, an extension of the formal two-scale asymptotic expansion to the level set framework was introduced in [34]. This method was recently applied to mathematical models of evolving brain extracellular space, colloid dynamics, drug delivery systems, and biofilm growth [28,29,31,37]. In these works the corresponding upscaled form is derived, leaving out the convergence aspects of the model. Our main goal in this paper is to discuss this convergence problem, by a new and simple constructive technique. Indeed, we obtain corrector estimates which show the speed of convergence as −→ 0 based on a suitable norm (See Theorem 4.1). Such estimates justify the homogenization limit which in the literature are usually called error and corrector estimates and provide a method to evaluate the accuracy of the upscaled model.
The main idea is that we approximate the microscopic problem while considering time slices of the time-dependent domain. Next, we solve a family of approximating problems corresponding to cylindrical domains. The corrector estimates for each cylindrical problem can be obtained with classical methods. Finally, we show that these estimates give us the desired corrector estimates for the original problem. Such estimates obtained in this framework are especially interesting from a computational point of view. Indeed, in the context of Multiscale Finite Element Method, these estimates are needed to ensure the convergence of the method [3,14]. This paper is organized as follows: Section 2 provides a detailed description of the model at the microscale level in the time and space dependent microstructure. In Section 3 we show the existence and uniqueness results for microscale equations in non-cylindrical domains. In Section 4, we introduce a macroscopic model in twoscale limit form and prove the convergence rate theorem for the homogenization problem. (i) G(t, x, 0) < cons. < 0, and G(t, x, y)| y∈∂Y > cons. > 0 for all x ∈ Ω, t ∈ (0, T ), The perforated domain will be defined by are holes in the domain. The Assumption 1 insures that the holes Q ,j do not intersect. Furthermore, we study the case that the holes are getting bigger (condition (iii)) during the diffusion process. In Figure 1, a schematic picture of the domain Ω (t) is depicted for a fixed time t.
In this study, the following system is considered x ∈ Ω (0), where, n (t, x) is the outward normal vector on Γ (t) and N is the normal vector on ∂Ω. Furthermore, we assume that q (x) := q(x, x ) in which q(x, y) is a Y -periodic function in the second variable.
Assumption 2. We assume the following conditions on initial data and parameters of the problem (2.1): Furthermore, we assume that there is a function f ∈ L ∞ (Ω) ∩ H 2 (Ω) such that Here, we present an expansion of the normal vector n in a power series in . It will be necessary to prove our main result in Section 4. This expansion can be done in terms of the level set function G , which we assume to be sufficiently regular so that all the following computations make sense. We have In the same fashion, we get and x ∈ Ω(0).
In this section, we suppose that the time-dependent domain, Ω(t), defined by in which the smooth function G satisfies: Moreover, we shall use standing conditions on the functions q and u I as follow: H3. q ∈ C 1 (O) and u I ∈ H 1 (Ω(0)). It is noteworthy that H1 will be concluded by the condition (ii) in Assumption 1 for small enough parameter . Then the result of this section can be applied to the locally periodic perforated domain defined in the previous section. It is also clear that Ω(s) ⊂ Ω(t) for s > t in our case according to the assumption H2. Before going to the existence result, we state some spaces of functions and define the weak solution of the problem.
3.1. Spaces of functions. Here, we construct Lebesgue and Sobolev spaces of functions defined on non-cylindrical domains. These function spaces are needed in order to state the variational formulation of problem (3.1). We refer the reader to [16] for similar definitions, further references and more details on the construction of spaces.
3.1.3. The trace space of L 2 (0, T ; L 2 (Γ(t))). From now on, we assume that the Lebesgue measure |Ω(t)| and (d − 1)-dimensional Hausdorff measure |Γ(t)| are bounded away from zero (uniformly in t). This condition can be easily deduced from the assumption H1. Now, let us introduce and define the trace space , is a norm for this space. The next proposition defines the trace operator on L 2 (0, T ; H 1 (Ω(t))).
We are not going to prove here the uniform boundedness of the trace operator. However, the next lemma and the forthcoming discussion will show why Proposition 1 must be correct. (see Remark 3.) Indeed, we need the result of the following lemma later to prove the existence and uniqueness of the solution for problem (3.1).
Remark 2. For β > 0 the constant C in Lemma 3.1 has the form of C = 1/β λ 1 (1/β) in which for a parameter α > 0, value λ 1 (α) is the first eigenvalue of the problem: Definition 3.2. We say that a bounded domain D ⊂ R d satisfies a uniform interior sphere condition, if there exists an r 0 > 0, such that for each point x ∈ ∂D one can find a ball B of radius r 0 , satisfying B ⊂ D and x ∈ ∂B.
Lemma 3.3 (Theorem 2.4, [11]). If D ⊂ R d , satisfies a uniform interior sphere condition of radius 2r 0 , then Remark 3. There exists a fixed value of r 0 such that for all time 0 ≤ t ≤ T , the time-dependent domain Ω(t) with the assumption H1 satisfies the uniform interior sphere condition of radius r 0 . This fact proves Proposition 1. Furthermore, we can conclude that the constant C in Lemma 3.1 is uniformly bounded when D = Ω(t) and 0 ≤ t ≤ T .

Weak form of solution.
In the following, we want to construct the weak form of the solution of (3.1). To this end, we provide some remarks and lemmas which will be needed to obtain the weak formulation.
)) ) will be the dual space of V (see [23]). We define the Banach space in which u t represents the distributional derivative of u with respect to time.
Lemma 3.5 (see [25]). Suppose assumption H1 and consider a function ρ := in which v n = v · n is the normal velocity of the boundary.
We can conclude obviously the following proposition from Lemma 3.5.
To get the weak formulation, multiply (3.1) by a test function φ ∈ C ∞ (R × R d ), we will have: By using Proposition 2 for the functions u and φ, we get: 3), we can get the following weak formulation: Therefore, our concept of weak solution will be as follow.
Definition 3.6. A weak solution of (3.1) is a function u that satisfies the following conditions: We iteratively solve a parabolic problem in the cylindrical domain Ω(t k ) × I k : Since u k ∈ C(t k , t k+1 ; L 2 (Ω(t k ))), we can define the traces where the limit is taken in L 2 (Ω(t k )). This is the reason why the initial condition makes sense in (3.6). Now we are ready to define the approximate solution which is defined in the domain We denote the boundary of Ω P with Indeed, Ω P makes an approximation of the domain Ω such that Ω ⊂ Ω P . See Figure  2 for a schematic picture.
Lemma 3.9 (see [5]). Assume H1. The domain Ω P converges to Ω, and Γ P converges to Γ in the Hausdorff metric. As a consequence, In the two next lemmas, we can see some estimates on the approximate solution u P . Lemma 3.10. There holds for some constant C > 0 depending only on Ω, u I and T .
Proof. Fix k and notice that the pairing u P with u P t on (t k , t k+1 ) × Ω(t k ) makes sense. After integration by parts we get: Integrating the former equality on [t k , t k+1 ], we obtain: where we have used the relation u k (t k , ·) = u k−1 (t k , ·)χ Ω(t k ) . According to Lemma 3.1 for suitable parameter β, (see also Remark 3) there is a constant C > 0 such that where Ω P (t) = {x : (x, t) ∈ Ω P }, and Ω P defines in (3.8). Let which is a piecewise C 1 function. Then by (3.9), we have ψ (t) ≤ Cψ(t) + u I L 2 which implies that ψ(t) is bounded for 0 ≤ t ≤ T . Again use (3.9) to prove statement (ii). The proof of statement (i) will also accomplish by (3.9) and the trace inequality in Lemma 3.1.
Lemma 3.11. There exists constant C > 0 depending only on Ω, u I and T , such that Proof. Due to the classical result in the regularity theory for parabolic PDE, we know that u k t ∈ L 2 (t k , t k+1 ; L 2 (Ω(t k ))) and we have Let us define On the other hand by the trace inequality in Lemma 3.1, we conclude that Now consider the solution of the first-order linear PDE, ∇ψ · ∇G = q|∇G| with the Cauchy data ψ = 0 on ∂Γ(0). The condition H1 insures that ψ can obtain by integral where X(t) is a characteristic solving X = ∇G(t, X). Reminding that n = ∇G |∇G| , the solution ψ satisfies the condition ∂ n ψ = q, on Γ(t). Now denote Σ = Ω P \ Ω and notice that Σ(t k ) = Ω(t k−1 ) \ Ω(t k ), although Σ(t) converges to ∂Ω(t k ), when t → t + k . Then where constant C is an upper bound for ∆ψ which depends only on G and q. We also have used this fact that u k = u k−1 in Ω(t k ) and so their traces on ∂Ω(t k ) are equal. Then for simplicity, we denote them by u P . Remind that Σ(t + k−1 ) = ∂Ω(t k−1 ) and ) . Combine the result with (3.10) and (3.11), to get |u P t | 2 dxdt ≤ C( u I H 1 (Ω(0)) + u P (T ) L 2 (Ω(T )) + u P L 2 (0,T ;H 1 (Ω(t))) ). Now apply Lemma 3.10 to complete the proof.
Recalling the definition of u P , we obtain the following result from Lemma 3.10 and Lemma 3.11. Corollary 1. There exists a constant C > 0 depending only on Ω, u I and T and independent of any partition in P such that u P L 2 (0,T ;H 1 (Ω P (t))) ≤ C, u P t L 2 (0,T ;L 2 (Ω P (t))) ≤ C. 3.4. Existence of solutions. In this section we prove the existence of weak solutions for (3.1).
Lemma 3.12. There is a functions u such that the following statement holds (up to extracting a subsequence) when P → 0, u P | Ω u weakly in L 2 (0, T ; H 1 (Ω(t))). Proof.
Since Ω ⊂ Ω P , the statement follows from Corollary 1.
In the next lemma, we state convergence on the approximated boundary.
Lemma 3.13. Let u be the function defined in Lemma 3.12. Then the following convergence holds for every φ ∈ C ∞ (R × R d ), when P → 0, Proof. Let Σ P = Ω P \ Ω, then similar to the proof of Lemma 3.11, we can find a function ψ ∈ H 2 (Σ P (t)) such that ∂ n ψ = φ on Γ(t) and Γ P (t). This is equivalent to solve the first-order linear equation ∇ψ · ∇G = φ|∇G| in the domain Σ P (t). Furthermore, the estimate ψ H 2 (Σ P (t)) ≤ C φ H 2 (Σ P (t)) is valid where constant C depends only on G. Now we can write u P φ dσ dt ≤ C u P L 2 (0,T ;H 1 (Ω P (t))) φ L 2 (0,T ;H 2 (Σ P (t))) tends to zero because of Corollary 1 and Lemma 3.9. On the other hand, according to the compactness of the trace operator, we know that u P → u strongly in L 2 (0, T ; Γ(t)). This completes the proof of the lemma. Now, we are ready to state and prove the existence theorem.
Theorem 3.14 (Existence). The function u defined in Lemma 3.12 is a weak solution of problem (3.1) in the sense of Definition 3.6.
In order to obtain the statement (iii) in Definition 3.6, we must show that the following term vanishes when P → 0, (remind the weak convergence of u P ) To prove this, note that for a fixed value s, we have (Ω P (s)) + u P t L 2 (Ω P (s)) ) φ H 1 (Σ P (s)) . From Corollary 1 and Lemma 3.9, we conclude that ∂ n u P φ dσ ds ≤ C φ L 2 (0,T ;H 1 (Σ P (t))) , converges to zero. Now apply the boundary condition ∂ n u P + qu P = 0 on Γ P and Lemma 3.13 to show the convergence of (3.12) to zero. According to Definition 3.6, we need to verify the initial condition. Consider a cylinder K × [0, t 1 ] ⊂⊂ Ω, so the regularity theory of parabolic equations in cylindrical domains prove that u ∈ C(0, t 1 ; L 2 (K)). Obviously, the statement (ii) in Definition 3.6 is valid.

MORTEZA FOTOUHI AND MOHSEN YOUSEFNEZHAD
Substitute φ m in (3.5), and apply Proposition 2, it follows that quφ m dxdt, and then passing to the limit as m → ∞, where Again apply Proposition 2, it yields that where we have used the relation (3.14) in the last equality. Note that by the trace inequality in Lemma 3.1 (see also Remark 3), there is a constant C > 0 such that and then From Gronwall's inequality, we obtain u(a) 2 L 2 (Ω(a)) ≤ −J δ e Ca Now, let δ → 0 to get u(a) = 0.
4. Convergence rate of homogenization. In this section, we study heterogeneous problem (2.1). First, we introduce some mathematical notation for effective and two-scale model, and then make a formulate for the homogenized model. Finally, we find a suitable corrector estimate in the time-dependent locally periodic micro-domain.

Macroscopic model (upscale form).
A formal locally periodic asymptotic method is applied to derive the upscale model. In order to formulate the upscaled equations and to obtain a closed formula for the effective transport coefficients, we use the notation: We will prove that the solution of (2.1) converges to the solution of the following upscaled model as → 0, x ∈ Ω, where the porosity θ(t, x) of the medium is given by q(x, y) dσ(y).
Moreover, the effective diffusivity tensor D(t, x) is defined by x, y)) dy.
Here M := (M 1 , M 2 , M 3 , · · · , M d ) is a vector function which is obtained from the x and t-dependent cell problem We should remark that n 0 is the outward normal vector on the boundary ∂B(t, x) and satisfies the relation (2.2).
We define the corrector term Obviously, we can see that it satisfies the following problem x ∈ Ω, y ∈ Y (t, x), ∇ y u 1 (t, x, y) · n 0 = −∇ x u 0 · n 0 , x ∈ Ω, y ∈ ∂B(t, x), u 1 (t, x, y) is Y − periodic. Remark 5. The averaged diffusion tensor D is symmetric (D ij = D ji ) and (uniformly in x and t) positive definite, i.e. there is some θ > 0 such that: (See Appendix in [28] for the proof) Remark 6. By asymptotic expansion technique, one can find the macroscopic model (4.1) from the heterogeneous problem (2.1) similar to what has been shown in [37,34]. Here, we will show the convergence rate and present a corrector estimate (see Theorem 4.1 for details). Here, we state some regularity results for the solutions of problem (4.1) which need to the proof of main results.
It deduce the first relation in (4.4). The second can obtain from the relation Lu 0 = −∂ t u 0 = −v ∈ L 2 (0, T ; H 1 (Ω)) and the regularity results for the elliptic operator L.

4.2.
Main result. Now we are ready to state the main result of our paper. Recall that u is the solution of microscopic problem (2.1), u 0 is the solution for the macroscopic problem (4.1) and u 1 is the solution of the problem (4.3) as well. We are going to prove the following theorem. where u 1, = u 1 | y= x , and the constant C is independent of .

4.3.
Construction of the approximate solution. The main idea for the proof of Theorem 4.1 is that we approximate at the same time all three problems (2.1), (4.1) and (4.3), by considering time slices with a fixed partition 0 = t 0 < t 1 < · · · < t N −1 < t N = T . Then, we show that these approximate solutions satisfy the similar corrector estimates (4.7) uniformly, (i.e. there exists a constant C > 0 for all partitions). So, after passing to the limit, the desired estimate will be obtained for the original problem.
To follow this idea, let us define the locally periodic medium on time slices to approximate the microscale model (2.1). As we mentioned in the Section 2.1, denote We set also and introduce the perforated domain as follows: where Ω k ,l = Ω ,l (t k ) := j∈J Q ,j (t k ), indicates the holes inside the domain. Now define u k to be the solution of the approximat microscale model For t 0 = 0, we let u 0 (x, 0) = f (x). Furthermore, we use the notation Ω ,b = Ω \ j∈J ( (Y + j)), and the smooth cut-off function χ (x) satisfying 0 ≤ χ (x) ≤ 1, for all k. Moreover, |∇χ | ≤ C and 2 |∆χ | ≤ C, with the constant C independent of such that We refer to [7,18] for existence and more insight on the role of such cut-off functions.
In order to analyze the convergence rate of the homogenization, we also discretize the macroscopic problem (4.1) and the corrector problem (4.3) on the time slice. We start with Also, for each j ∈ {1, 2, 3, · · · , d}, consider the following x and k-dependent cell problem The solution to this cell problem allows us to write the results of the formal homogenization procedure in the form of the following two-scale model where the porosity θ k (x) of the medium is given by and the effective diffusivity tensor D k (x) is define by where M k := (M k 1 , M k 2 , M k 3 , · · · , M k d ), and q(x, y) dσ(y). , we can define u P , u P 0 and u P 1 . Moreover, same as the Corollary (1), one can obtain u P L 2 (0,T ;H 1 (Ω P (t))) ≤ C, u P 0 L 2 (0,T ;H 3 (Ω)) ≤ C, u P 1 L 2 ((0,T )×Ω;W 1,s (Y P (t,x)) ≤ C. 4.4. Proof of Theorem 4.1. In this section, we prove the main result of the paper. The auxiliary results which are stated in the following lemmas are essential in the proof. They mainly concern integral estimates for rapidly oscillating functions with prescribed average. We postpone the proof to the Appendix. Lemma 4.2. Let P k (x, y) ∈ H 1 (Ω; L s (Y k (x))) and p k (x, y) ∈ H 1 (Ω; L s (∂B k (x))) be uniformly bounded with respect to k for some s > d (remind that d is the dimension of the space) such that Y k (x) P k (x, y) dy = ∂B k (x) p k (x, y) dσ. Then the inequality holds for every φ ∈ H 1 (Ω (t k )). The constant C does not depend on and k.
holds for all φ ∈ H 1 (Ω (t k )). The constant C does not depend on and k.
The next lemma provides a relation which is necessary to apply Lemma 4.2. We bring it here without proof. Lemma 4.4 (Lemma 5.1, [18]). The following relation holds where n 1 is the tangential vector obtaining by (2.3).

5.
Conclusions. We studied the concept of homogenization for a parabolic problem with a Robin boundary condition in a time-dependent microstructure. More precisely, we dealt with a locally periodic microstructure in a formulation that allowed the domain to dependent on time and use a level set function to define it. First, we presented a simple approach to construct solutions of the microscopic problem which consist in performing a time slicing of the domain and solving a family of approximating problems in the cylindrical domain. Similar to the periodic case, we applied the idea of oscillating function, which is synchronous with oscillations in either microstructure or coefficients of microscopic problem. However, we focused on locally periodic media. Finally, we provided a family of the approximated macroscopic problem and corresponding corrector results. Next, using a limiting procedure we obtained the desired corrector estimate between the original microscopic problem and its macroscopic approximation. 6. Appendix. In the last part of the appendix, we will prove Lemma 4.2 and Lemma 4.3. Although similar results have been proved in other literature, the main challenge here is that we have a sequence of domains Ω (t k ) and we desire to obtain some estimation independent of k. First of all, we introduce the method which can help to overcome this challenge. We recall a basic definition of the coordinate transformations in continuum mechanics, see e.g. [10,15,17]. For each x ∈ Ω and t ∈ (0, T ), the function Ψ(t, x, ·) : Y (0) −→ Y (t, x) := Ψ(t, x, Y (0)) is bijective. 3. There exist positive constants C 1 and C 2 such that C 1 ≤ det∇ y Ψ(t, x, y) ≤ C 2 , for all (t, x, y) ∈ (0, T ) × Ω × Y (0). Remark 9. The existence of regular C 2 -motion mapping is ensured by Assumption 1, the fact that G ∈ C 2 ([0, T ] × Ω × Y ) and ∂ y G(t, x, y) = 0. (See [35]). Lemma 6.2. Let P (x, y) ∈ H 1 (Ω; L s (Y k (x))) and p(x, y) ∈ H 1 (Ω; L s (∂B k (x))) be Y -periodic functions satisfying Y k (x) P (x, y) dy = ∂B k (x) p(x, y) dσ. Then there exists the Y -periodic solution v ∈ H 1 (Ω; W 2,s (Y k (x))) for the problem ∆ y v(x, y) = P (x, y), y ∈ Y k (x), ∇ y v(x, y) · n 0 = p(x, y), y ∈ ∂B k (x), (6.1) which satisfies the following estimate for a universal constant C > 0 independent of k, v H 1 (Ω;W 2,s (Y k (x))) ≤ C( P H 1 (Ω;L s (Y k (x))) + p H 1 (Ω;L s (∂B k (x))) ).
Proof. First, consider the regular C 2 -motion mapping Ψ : (0, T ) × Ω × Y (0) −→ (0, T ) × Ω × Y (t, x) which is introduced in Definition 6.1. Then reformulate the problem (6.1) by this map to obtain a new equation for v = v • Ψ in the referential domain Y (0). The transformed function v satisfies in a second-order linear PDE L y v = P • Ψ, in which the coefficients of L y are C 2 regular with respect to the x parameter and continuous with respect to the y variable. Then v ∈ H 1 (Ω; W 2,s (Y (0))) and satisfies v H 1 (Ω;W 2,s (Y (0))) ≤ C( P • Ψ H 1 (Ω;L s (Y (0))) + p • Ψ H 1 (Ω;L s (∂B(0))) ), where the constant C depends only on L y and the domain Y (0). Now use the mapping Ψ to come back to the original domain and complete the proof.
Proof of Lemma 4.2. The proof is inspired by the proof of Lemma 5.3 in [18]. The problem ∆ y v k (x, y) = P k (x, y) y ∈ Y k (x) ∇ y v k (x, y) · n 0 = p k (x, y) y ∈ ∂B k (x) (6.2) has a solution which is Y-periodic in y and unique up to an additive constant. Let multiply the equation (6.2) by the function φ ∈ H 1 (Ω (t k )) and integrate it over the domain Ω (t k ), (Ω (t k ));C(Y k (x))) φ H 1 (Ω (t k )) In the last inequality, Lemma 6.3 is applied. Notice that according to Lemma 6.2, we know that v k H 1 ((Ω (t k ));C 1,α (Y k (x))) is independent of and k.
Proof of Lemma 4.3. The result is similar to Lemma 4 in [7]. Although, the proof show that the constant C is independent of , by the existence of C 2 -motion mapping (Remark 9) we will get the related constant does not depend on k as well.