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The Uniformly Parabolic Equations of Higher Order with Discontinuous Data in Generalized Morrey Spaces and Elliptic Equations in Unbounded Domains

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Tair Gadjiev and Konul Suleymanova

Submitted: October 20th, 2020 Reviewed: February 23rd, 2021 Published: March 22nd, 2021

DOI: 10.5772/intechopen.96781

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We study the regularity of the solutions of the Cauchy-Dirichlet problem for linear uniformly parabolic equations of higher order with vanishing mean oscillation (VMO) coefficients. We prove continuity in generalized parabolic Morrey spaces Mp,φ of sublinear operators generated by the parabolic Calderon-Zygmund operator and by the commutator of this operator with bounded mean oscillation (BMO) functions. We obtain strong solution belongs to the generalized Sobolev-Morrey space Wp,φm,1∘Q. Also we consider elliptic equation in unbounded domains.


  • higher order parabolic equations
  • generalized Morrey spaces
  • sublinear operators
  • Calderon-Zygmund integrals
  • VMO
  • Cauchy-Dirichlet problem
  • elliptic equations
  • unbounded domain

1. Introduction

We consider the higher order linear Cauchy-Dirichlet problem in Q=Ω×0T, being a cylinder in Rn+1, ΩRn be a bounded domain 0<T<


where pQ=∂Ω×0TΩ×t=0 stands for the parabolic boundary of Q and Dαβ=αx1α1,,xnαnβy1β1,,ynβn, α=k=1nαk, β=k=1nβk.

The unique strong solvability of this type problem was proved in [1]. In [2] the regularity of the solution in the Morrey spaces Lp,λRn+1 with p1, λ0n+2 and also its Hölder regularity was studied. In [3] Nakai extend these studies on generalized Morrey spaces Mp,φRn+1 with a weight φ satisfying the integral condition


The generalized Morrey space is then defined to be the set of all fLp,locRn+1 such that


where the supremum is taken over all parabolic balls E with respect to the parabolic distance.

The main results connected with these spaces is the following celebrated lemma: let DfLp,nλ even locally, with nλ<p, then u is Holder continuous of exponent α=1nλp. This result has found many applications in theory elliptic and parabolic equations. In [2] showed boundedness of the maximal operator in Lp,λRn+1 that allows them to prove continuity in these spaces of some classical integral operators. So was put the beginning of the study of the generalized Morrey spaces Mp,φ,p>1 with φ belonging to various classes of weight functions. In [3] proved boundedness of maximal and Calderon-Zygmund operators in Mp,φ imposing suitable integral and doubling conditions on φ. These results allow to study the regularity of the solutions of various linear elliptic and parabolic value problems in Mp,φ (see [4, 5, 6]). Here we consider a supremum condition on the weight which is optimal and ensure the boundedness of the maximal operator in Mp,φ. We use maximal inequality to obtain the Calderon-Zygmund type estimate for the gradient of the solution of the problem (1) and (2) in the Mp,φ.

The results presented here are a natural extension of the previous paper [7] to parabolic equations. Here we study the boundedness of the sublinear operators, generated by Calderon-Zygmund operators in generalized Morrey spaces and the regularity of the solutions of higher order uniformly elliptic boundary value problem in local generalized Morrey spaces where domain is bounded. Also hear we study higher order uniformly elliptic boundary value problem where domain is unbounded.

In paper [8] Byun, Palagachev and Wang is study the regularity problem for parabolic equation in classical Lebesgue classes and of Byun, Palagchev and Softova [9, 10] where the problem studied in weighted Lebesgue and Orlicz spaces with a Muckenhoupt weight and the classical Morrey spaces Lp,λQ with λ0n+2.

In papers [11, 12] the authors studied second order linear elliptic and parabolic equations with VMO coefficients.

Denote by a the coefficient axt=aαβxt:QMn×n and by fxt nonhomogeneous term. Suppose that the operator is uniformly parabolic.

The paper is organized as follows. In section 2 we introduce some notations and give the definition of the generalized Morrey spaces Mp,φQ. In section 3 we study sublinear operators generated by parabolic singular integrals in generalized Morrey spaces. In section 4 we is consider sublinear operators generated by non-singular integrals, in section 5 singular and non-singular integrals in generalized Morrey spaces. In section 6 we consider uniformly parabolic equations of higher order with VMO coefficients and proved regularity of solutions. In section 7 we study uniformly elliptic equations in unbounded domains.


2. Some notation and definition

The following notations are used in this paper:


is the Euclidean metric, x=i=1nxi2+t212; Brx=yRn:xy<r, Br=crn; Irxt=yRn+1:xy<rtτ<r2,Irxt=crn+2; Qr=IrxτQ for each xτQ, 2Irxτ=I2rxτ.

Sn is the unit sphere in Rn+1;


the letter C is used for various positive constants.

In the following, besides the standard parabolic metric ρxt=maxxt12. We use the equivalent one


considered by Fabes and Riviere in [13]. The topology induced by ρxt consists of the ellipsoids


It is easy to see that the this metrics ore equivalent. In fact, for each Er there exist parabolic cylinders I¯ and I¯ with measure comparable to rn+2 such that I¯ErI¯.

Let Q=Ω×0T,T>0, be a cylinder in R+n+1. We give the definitions of the functional spaces that we are going to use. Let aL1,loc(Rn+1 and let aEr=Er1Eraydy be the mean value of the integral of a. Denote


where Er ranges over all ellipsoids in Rn+1. We say aBMO (bounded mean oscillation [14]) if


is finite. is a norm in a BMO constant functions.

We say aVMO (vanishing mean oscillation) [14] if aBMO and


ηaR is called the VMO-modulus of a. For any bounded cylinder Q we define BMOQ and VMOQ taking aL1Q and Qr=QErx,xQ, instead of Er in the definition above. If a function aBMO or VMO, it is possible to extend the function in the whole of Rn+1 preserving its BMO-norm or VMO-modulus, respectively (see [15]). Any bounded uniformly continuous BUC function f with modulus of continuity ωfR belongs to VMO with ηfR=ωfR. Besides, BMO and VMO also contain discontinuous functions, and the following example shows the inclusion Wn+21Rn+1VMOBMO.

Example 2.1. We have that fx=logρxtBMO\VMO;sinfxBMOLRn+1;fαx=logρ(xt)αVMO for any α01;fαWn+21Rn+1 for α011n+2;fαWn+21Rn+1 for α11n+21.

Let φ:Rn+1×R+R+ be a measurable function and p1. The generalized parabolic Morrey space Mp,φRn+1 consists of all fLp,locRn+1 such that


The space Mp,φQ consists of LpQ functions provided the following norm is finite


The generalized weak parabolic Morrey space WM1,αRn+1 consists of all measurable functions such that


where WL1 denotes the weak L1 space. The generalized Sobolev-Morrey space Wp,φ2m,1Q,p1, consists of all Sobolev functions UWp2m,1Q with distributional derivatives DtlDxsuMp,φQ,02l+s2m, endowed by the norm


We also define the space


where Q means the parabolic boundary Ω∂Ω×0T. In problem (1) and (2) the coefficient matrix axt=aα,βxti,j=1n, α,β=m satisfies


for a.e. xtQ,ξRn,ξ=ξαα=mRN, N–number different multiindeks with length equal to m, aα,βxt=aβ,αxt, which implies aα,βxtLQ.

Theorem 2.1.(Main results) LetaxtVMOQwithηα,β=i,j=1nηαβijsatisfy(3), and, for eachp1,letuxtW0p2m,1Qbe a strong solution(1) and (2). IffMp,φQwithφxrbeing a measurable positive function satisfying




with C=Cnpγ∂ΩTηαa;Q.


3. Sublinear operators generated by parabolic singular integrals in generalized Morrey spaces

Let fL1Rn+1 be a function with a compact support and aBMO. For any xsuppf define the sublinear operators T and Ta such that


This operators are bounded in LpRn+1 satisfy the estimates


where constants independent of a and f. Let we have the Hardy operator Hgr=1r0rgsds,r>0.

Theorem 3.1.(see [12]) The inequality


holds for all non-increasing functions g:R+R+ if and only if


Lemma 3.1.(see [12]) LetfLp,locRn+1,p1,be such that


and letTbe a sublinear operator satisfying(6).

i. Ifp>1andTis bounded onLpRn+1, then


ii. Ifp=1andTis bounded fromL1Rn+1onWL1Rn+1,then


where the constants are independent of r,x0 and f.

Theorem 3.2.(see [12]) Letp1andφxrbe a measurable positive function satisfying


and letTbe a sublinear operator satisfying(6).

  1. Ifp>1andTis bounded onLpRn+1, thenTis bounded onMp,φRn+1, and


  1. Ifp=1andTis bounded fromL1Rn+1toWL1Rn+1,then it is bounded fromM1,φRn+1toWM1,φRn+1,and


with constants independent of f.

Our next step is to show boundedness of Ta in Mp,φRn+1. For this we recall some properties of the BMO functions.

Lemma 3.2.John-Nirenberg lemma [[12], Lemma 2.8]. LetaBMOandp1.Then, for anyEr,


As an immediate consequence of (7) we get the following property.

Corollary 3.1.LetaBMO.Then, for all0<2r<s,


Now we estimate the norm of Ta.

Lemma 3.3.(see [12]) LetaBMO.andTabe a bounded operator inLpRn+1,p1,satisfying(7)and(8). Suppose that, for anyfLp,locRn+1,




where C is independent of a,f,x0 and r.

Theorem 3.3.Letp1andφxrbe measurable positive functions such that


forxrRn+1×R+,whereCis independent ofxandr.Suppose thataBMOand letTabe a sublinear operator satisfying(7). IfTais bounded inLpRn+1,then bounded inMp,φRn+1,and


constant C independent of a and f.

Then basic results of the theorem follows by Lemma 3.3 and Theorem 3.1 in the same manner as for Theorem 3.2. For example the functions φxr=rβn+2p,φxr=rβn+2plogml+r with 0<β<n+2p and m1, are weight functions satisfying the condition (20).


4. Non-singular integrals in generalized Morrey spaces

Let xD+n+1, define x¯=xxntDn+1 and x0=x,0,0Rn1. Consider the semi-ellipsoids Er+x0=Er+x0Dn+1. Let fL1D+n+1,aBMOD+n+1, and T¯,T¯a be sublinear operators such that


Let both the operators be bounded in LpD+n+1, satisfy the estimates


constants C independent of a and f.

The following results hold, which can be proved in the some manner as in Section 3 (see [12]).

Lemma 4.1.LetfLp,locD+n+1,p1and for allx0rRn1×R+


IfT¯is bounded onLpD+n+1,then


where the constant c is independent of r,x0 and f.

Theorem 4.1.Supposeφbe a weight function satisfying(14), and letT¯be a sublinear operator satisfying(22)and(24). ThenT¯is bounded inMp,φD+n+1,p1and


with a constant c independent of f.

Lemma 4.2.Letp1,aBMOD+n+1andT¯asatisfy(23)and(24). Suppose that, for allfLp,locD+n+1,




with a constant c independent of a,f,x0 and r.

Theorem 4.2.Letp1,aBMOD+n+1,letφx0rbe a weight function satisfying(20)andT¯abe a sublinear operator satisfying(7), (8). Then sublinear operatorT¯ais bounded inMp,φD+n+1and


constant c independent of a and f.


5. Singular and non-singular integrals in generalized Morrey spaces

We apply the above results to Calderon-Zygmund-type operators with parabolic kernel. Since these operators are sublinear and bounded in LpRn+1, their continuity in Mp,φ follows immediately. We are called a parabolic Calderon-Zygmund kernel if the following a measurable function Kxξ:Rn+1×Rn+1\0R.

  1. Kx is a parabolic Calderon-Zygmund kernel for a.e. xRn+1:


    1b.Kxμξμ2s=μn2Kxξ for all μ>0, ξ=ξs,


  2. DξβKLRn+1×SnMβ< for every multi-index β.



which means the singular integrals


are sublinear and bounded in LpRn+1 according to the results in [1, 13].

Theorem 5.1.LetfMp,φRn+1m then there exist constantscdepending onn,pand the kernel such that


Corollary 5.1.For any cylinderQinR+n+1,fMp,φQ,aBMOQandKxξ:Q×R+n+1\0R.Then the operators(30)are bounded inMp,φQand


constant c independent of a and f.

We define the extensions


and then the singular integral satisfying inequalities




Corollary 5.2.LetaVMO. Then for anyε>0there exists a positive numberr0=r0εηasuch that for anyErx0with a radiusr0r0and allfMp,φErx0


where c is independent of E,f,r, and x0.

For the proof of corollary see [12].

For any xR+n and any fixed t>0, define the generalized reflexion


where αm,βm,aαβnx is the last row of the coefficients matrix ax=aαβx of (1). The function τx maps R+n into Rn, and the kernel Kxτxy=Kxτxytτ is non-singular for any x,yD+n+1. Taking x¯D+n+1, there exists positive constants K1 and K2 such that


Let fMp,φD+n+1, aBMOD+n+1 define the non-singular integral operators


Since Kxτxy is still homogeneous and satisfies 1b, we have


Hence, the operators (36) are sublinear and bounded in LpD+n+1,p1. From section 4 the following results are obtained.

Theorem 5.2.LetaBMOD+n+1andfMp,φD+n+1withpφas in(8)Then the non-singular operators are continuous inMp,φD+n+1and


constant C independent of a and f.

Corollary 5.3.For anyaVMO. Then there exists a positive numberr0=r0εφasuch that for anyErx0with a radiusr0r0and allfMp,φEr+x0


where C is independent of E,f,r and x0, ε>0.


6. Proof of the first main result

Now using boundedness of singular integral of Calderon-Zygmund operators in generalized Morrey spaces we will get interval estimates for solutions of problem (1), (2) with coefficients from VMO spaces.

Let Ω to be open bounded domain in Rn,n3 and we suppose that its boundary is sufficiently smoothness.

Let coefficients aαβx, α,βm are symmetric and satisfying to the condition uniform ellipticity, essential boundedness of the coefficient aαβxLQ and regularity aαβxVMOQ. Let fMp,φQ,pφ as in (8) Since Mp,φQ is a proper subset of LpQ,(1) and (2) is uniquely solvable and the solution ux belongs at least to Wp2m,1Q. Our aim is to show that this solution also belong to W0p,φ2m,1Q. For this we need an a priori estimate of u, which we prove in two steps. Before we give interior estimate. For any x0R+n+1 define the parabolic semi-cylinders Crx0=Brx0×t0r2t0. Let ϑC0Cr and suppose that ϑxt=0, for t0. According to [1, 7, 16], for any xsuppϑ the following representation formula for the higher derivatives of ϑ holds true if uW0p2mQ


where ν=ν1νn+1 is the outward normal to Sn. Here, Γxξ is the fundamental solution of the operator L. Γxt can be represented in form


for a.e. xRn+1 and ξRn\0, where Aαβn×n is inverse matrix for aαβn×n. Since any function ϑWp2m,1Q can be approximated by C0 functions, the representation formula (39) still holds for any ϑWp2m,1Crx0. The properties of the fundamental solution (see [7, 17]) imply that DαΓxy are variable Calderon-Zygmund kernels in the sense of our definition above. By notation above, we can write


The operators Dα,βB and Dα,βC are defined by (30) with Kxxy=Dα,βΓxxy. Due to (30) and (31) and the equivalence of the metrics we obtain for E>0 there exists r0E such that for any r<r0E


for some r small enough. From (41) we get that


Define a cut-off function ψx=ψ1xψ2t, with ψ1C0Brx0, ψ2C0R such that


with θ01, θ'=θ3θ/2>0 and DαψCθ1θrα, α2m, ψtDαψ. For any solution uWp2m,1Q of (1) and (2) define ϑx=φxuxWp2m,1Cr. Hence,


As so,


We introduce


the above inequality becomes


Now we use following interpolation inequality (see [5])


where there exists a positive constant C independent of r. Thus (42) becomes


Taking θ=12 we obtain the Caccioppoli-type estimate


We get the boundedness of the coefficients


Let Q=Ω×0T and Q=Ω×0T the cylinders with ΩΩΩ. By the standard covering procedure and partition of the unity we obtain that


where C depends on n,p,Λ,T,DΓLQ,ηα,aLQ and distΩΩ. Now we give boundary estimates. For any fixed x0rRn+1×R+ define the semi-cylinders


with Sr+=x0t:x0x<r,0<t<r2. For any solution uWp2m,1Cr+x0 with suppuCr+x0, the following boundary representation formula holds (see [1, 7, 16]).




Here B¯ij and C¯ij are non-singular operators defined by (36) with a kernel Kxτxy=Dα,βΓxτxy. Applying the estimates (37) and (38) and having in mind that the components of the vector τxxn are bounded, we obtain that


Taking r small enough we can move the norm of u on the left-hand side, obtaining that


with a constant C depending on n,p,Λ,T,ηα,aLQ. By covering the boundary with small cylinders, using a partition of the unit subordinated by that covering and local flattening of ∂Ω we get that


Using (43) and (44), we obtain (5).


7. The higher order elliptic equations in unbounded domains

Now we are consider boundary value the Dirichlet problem for higher order nondivergence uniformly elliptic equations with coefficients in modified Morrey spaces in unbounded domains Ω


where the coefficients matrix ax=aα,βijxi,j=1n satisfies


for a.e. xΩ, ξRn, aα,β=aβ,α,ξ=ξαα=mRN, N–number different multiindeks with length equal to m.

Under these assumptions we prove that the maximal operator M are bounded from the modified Morrey space L˜p,λRn to L˜q,λRn if and only if,


For xRn and t>0, let Bxt denote the open ball centered at x of radius t and Bxt=Rn\Bxt. One of the most important variants of the Hardy-Littlewood maximal function is the so-called fractional maximal function defined by the formula


where Bxt is the Lebesgue measure of the ball B(x,t). The fractional maximal function Mαf coincides for α=0 with the Hardy-Littlewood maximal function MfM0f.

Let 1p<,0λn,t1=min1t. We denote by L˜p,λRn the modified Morrey space, as the set of locally integrable functions fx,xRn, with the finite norm


Note that


and if λ<0 or λ>n, then Lp,λRn=L˜p,λRn=θ, where θ is the set of all functions equivalent to 0 on Rn. WL˜p,λRn-the modified weak Morrey space as the set of locally integrable functions fx,xRn with finite norm


Note that


We study the L˜p,λ-boundedness of the maximal operator M.

The classical result by Hardy-Littlewood-Sobolev states that if 1<p<q<, then the Riesz potential Iα is bounded from LpRn to LqRn if and only if α=n1p1q and for p=1<q<, Iα is bounded from L1Rn to WLqRn if and only if α=n11q. D.R. Adams studied the boundedness of the Iα in Morrey spaces and proved the follows statement.

Theorem(Adams) Let 0<α<n and 0λ<nα, 1p<nλα.

  1. If 1<p<nλα, then condition 1p1q=αnλ is necessary and sufficient for the boundedness of the operator Iα from Lp,λRn to Lq,λRn.

  2. If p=1, then condition 11q=αnλ is necessary and sufficient for the boundedness of the operator Iα from L1,λRn to WLq,λRn.

If α=npnq, then λ=0 and the statement of Theorem reduced to the aforementioned result by Hardy-Littlewood-Sobolev Theorem also implies the boundedness of the fractional maximal operator Mα.

In this section we study the fractional maximal integral and the Riesz potential in the modified Morrey space. In the case p=1 we prove that the operator Iα is bounded from L˜1,λRn to WL˜q,λRn if and only if, αn11qαnλ. In the case 1<p<nλα we prove that the operator Iα is bounded from L˜p,λRn to L˜q,λRn if and only if, αn1p1qαnλ.

Theorem 7.1.IffL˜p,λRn,1<p<,0λ<n,thenMfL˜p,λRnand


where Cp,λ depends only on p,λ and n.

Proof. We use Fefferman-Stein inequality and get


Later from some estimates for MχBxt we have the following inequalities


Theorem 7.2.(see [18]) Let0<α<n,0λ<nαand1p<nλα.

  1. If 1<p<nλα, then condition αn1p1qαnλ is necessary and sufficient for the boundedness of the Riesz potential operator Iα from L˜p,λRn to L˜q,λRn.

  2. If p=1<nλα, then condition αn11qαnλ is necessary and sufficient for the boundedness of the operator Iα from L˜1,λRn to L˜q,λRn.

Recall that, for 0<α<n


where νn is the volume of the unit ball in Rn. From [7] for unbounded domains ΩRn we have following result.

Theorem 7.3.LetΩRnbe an unbounded domains with noncompact boundary∂Ω, and0<α<n,0λ<nαand1<p<nλα. Also let satisfies conditionsαn1p1qαnλ,fL˜q,λΩ, functionUxis a solution of problem(45). Then there is exist constantCwhich dependent only atn,λ,p,q,Ωsuch that


where W˜p,λ2m-is correspondingly modified Sobolev-Morrey space.

The proved Theorem 7.3 consequence from methods of [7] and Theorem 7.1 and 7.2.


Additional classifications

Mathematics Subject Classifications (2010): 35 J25, 35B45,42B20, 47B38


  1. 1. M. Bramanti and M. C. Cerutti, Wp1,2 solvability for the Cauchy–Dirichlet problem for parabolic equations with VMO coefficients, Commun. PDEs 18 (1993), 1735–1763
  2. 2. F. Chiarenza and M. Frasca, Morrey spaces and Hardy–Littlewood maximal function, Rend. Mat. 7 (1987), 27–279
  3. 3. E. Nakai, Hardy–Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces, Math. Nachr. 166 (1994) 95–103,
  4. 4. L. G. Softova, Singular integrals and commutators in generalized Morrey spaces, Acta Math. Sinica 22 (2006), 757–766
  5. 5. L. G. Softova, Morrey-type regularity of solutions to parabolic problems with discontinuous data, Manuscr. Math. 136(3) (2011), 365–382
  6. 6. L. G. Softova, The Dirichlet problem for elliptic equations with VMO coefficients in generalized Morrey spaces, in Advances in harmonic analysis and operator theory, the Stefan Samko anniversary volume, Operator Theory: Advances and Applications, Volume 229, pp. 371–386 (Springer, 2013)
  7. 7. V.S. Guliyev, T.S. Gadjiev, Sh. Galandarova: Dirichlet boundary value problems for uniformly elliptic equations in modified local generalized Sobolev–Morrey spaces, Electron. J. Qual. Theory Differ. Equ. 2017, No. 71, 1–17
  8. 8. S.-S. Byun, D.K. Palagachev, L. Wang, Parabolic systems with measurable coefficients in Reifenberg domains, Int. Math. Res. Not. IMRN 2013(13) (2013) 3053–3086
  9. 9. S.-S. Byun, D.K. Palagachev, L.G. Softova, Global gradient estimate in weighted Lebesgue spaces for parabolic operators, Annales Academiae Scientiarum Fennicae Mathematica 41, (2016), 67–83,
  10. 10. S.-S. Byun, L.G. Softova, Gradient estimates in generalized Morrey spaces for parabolic operators, Math. Nachr. 288, (14–15), 1602–1614 (2015). DOI 10.1002/mana.201400113
  11. 11. V.S. Guliyev, L.G. Softova, Global regularity in generalized Morrey spaces of solutions to nondivergence elliptic equations with VMO coefficients, Potential Anal. 38 (2013) 843–862,
  12. 12. V.S. Guliyev, L.G. Softova, Generalized Morrey regularity for parabolic equations with discontinuousdata, Proc. Edinb. Math. Soc. 58 (2014) 219–229,
  13. 13. E. B. Fabes and N. Rivi‘ere, Singular integrals with mixed homogeneity, Studia Math. 27 (1996), 19–38
  14. 14. D. Sarason, On functions of vanishing mean oscillation, Trans. Am. Math. Soc. 207 (1975), 391–405
  15. 15. P. Acquistapace, On BMO regularity for linear elliptic systems, Annali Mat. Pura Appl. 161 (1992), 231–270
  16. 16. H.-Ch. Grunau, G. Sweers, Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions, Math. Ann. 307(1997), No. 4, 589–626. MR1464133;
  17. 17. E. Nakai, The Campanato, Morrey and Hölder spaces on spaces of homogeneous type, Studia Math. 176(1) (2006), 1–19
  18. 18. V.S. Guliyev, J.J. Hasanov: Necessary and sufficient conditions for the boundedness of B-Riesz potential in the B-Morrey spaces Journal of Mathematical Analysis and Applications Volume 347, Issue 1, 1 November 2008, Pages 113–122

Written By

Tair Gadjiev and Konul Suleymanova

Submitted: October 20th, 2020 Reviewed: February 23rd, 2021 Published: March 22nd, 2021