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Volume 14, Issue 1 (Jan 2016)

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Parabolic oblique derivative problem with discontinuous coefficients in generalized weighted Morrey spaces

Vagif S. Guliyev
  • Corresponding author
  • Ahi Evran University, Department of Mathematics, 40100, Kirsehir, Turkey and Institute of Mathematics and Mechanics, AZ 1141 Baku, Azerbaijan
  • Email:
/ Mehriban N. Omarova
  • Institute of Mathematics and Mechanics, AZ 1141 Baku, Azerbaijan and Baku State University, Baku, AZ 1148, Azerbaijan
  • Email:
Published Online: 2016-02-13 | DOI: https://doi.org/10.1515/math-2016-0006

Abstract

We obtain the global weighted Morrey-type regularity of the solution of the regular oblique derivative problem for linear uniformly parabolic operators with VMO coefficients. We show that if the right-hand side of the parabolic equation belongs to certain generalized weighted Morrey space Mp,ϕ(Q, w), than the strong solution belongs to the generalized weighted Sobolev- Morrey space W˙2,1p,φ(Q,ω).

Keywords: Generalized weighted Morrey spaces; Uniformly parabolic operators; Regular oblique derivative problem; VMO

MSC 2010: 35K20; 35D35; 35B45; 35R05

1 Introduction

We consider the regular oblique derivative problem in generalized weighted Sobolev- Morrey space W˙2,1p,φ(Q,ω) for linear non-divergence form parabolic equations in a cylinder utaij(x)Diju=f(x) a.e. in Q,u(x',0)=0,        on  Ω,u/l=li(x)Diu=0  on S.

The unique strong solvability of this problem was proved in [38]. In [39] Softova studied the regularity of the solution in the Morrey spaces Lp,ƛ with p ∈ (1, ∞), ƛ ∈ (0, n + 2) and also its Hölder regularity. In [41] Softova extended these studies on generalized Morrey spaces Lp,ϕ with a Morrey function ϕ satisfying the doubling and integral conditions introduced in [27, 31]. The approach associated to the names of Calderón and Zygmund and developed by Chiarenza, Frasca and Longo in [7, 8] consists of obtaining of explicit representation formula for the higher order derivatives of the solution by singular and nonsingular integrals. Further the regularity properties of the solution follows by the continuity properties of these integrals in the corresponding spaces. In [39] and [40] the regularity of the corresponding operators in the Morrey and generalized Morrey spaces is studied, while in [38] we can find the corresponding results obtained in Lp by [9] and [5]. In recent works there have been studied the regularity of the solutions of elliptic and parabolic problems with Dirichlet data on the boundary in generalized Morrey spaces Mp,ϕ with a weight ϕ satisfying (10) with w ≡ 1 (cf. [18, 19]). Precisely, a boundedness in Mp,ϕ was obtained for sub-linear operators generated by singular integrals as the Calderon-Zygmund. More results concerning sub-linear operators in generalized Morrey spaces can be found in [3, 12, 40] see also the references therein.

After studying generalized Morrey spaces in detail, researchers passed to weighted Morrey spaces and generalized weighted Morrey spaces. Recently, Komori and Shirai [23] defined the weighted Morrey spaces and studied the boundedness of some classical operators, such as the Hardy-Littlewood maximal operator or the Calderón-Zygmund operator on these spaces. Also, Guliyev in [13] first introduced the generalized weighted Morrey spaces Mwp,ϕ and studied the boundedness of the sublinear operators and their higher order commutators generated by Calderón-Zygmund operators and Riesz potentials in these spaces (see, also [15, 17]). Note that, Guliyev [13] gave the concept of generalized weighted Morrey space which could be viewed as an extension of both Mp,ϕ and Lp,κ(w).

We call weight a positive measurable function defined on ℝn ×. ℝ+. In [29] Muckenhoupt shows that the maximal inequality holds in weighted Lebesgue spaces Lwq if and only if the weight w satisfies the following integral condition called parabolic Muckenhoupt condition or parabolic Aq-condition. We say that the measurable, nonnegative function w : ℝn → ℝ+ satisfies the parabolic Aq-condition for q ∈ (1, ∞) if supI(1|I|Iw(x,t)dxdt)(1|I|Iw(x,t)1q1dxdt)q1A<(1) for all parabolic cylinders ℐ in ℝn+1. Then w(ℐ) means the weighted measure of ℐ, that is w(I)=Iw(x,t)dxdt.

This measure satisfies strong and reverse doubling property. Precisely, for each ℐ and each measurable subset 𝒜 ⊂ ℐ, there exist positive constants c1 and τ1 ∈ (0, 1) such that 1[w]q(|A||I|)qw(A)w(I)c1(|A||I|)τ1,(2) where c1 and τ1 depend on n and q but not on ℐ and 𝒜.

Throughout this paper the following notations are to be used: x=(x',t)=(x",xn,t)n+1,R+n+1=x'n,t>0 and D+n+1=x"Rn1,xn>0,t>0,Diu=u/xi,Diju=2u/xixj,Dtu=ut=u/t stand for the corresponding derivatives while Du = (D1u, … , Dnu) and D2u=Dijui,j=1n mean the spatial gradient and the Hessian matrix of u. For any measurable function f and A ⊂ ℝn+1 we write fp,A=(A|f(y)|pdy)1/p,fA=1|A|Af(y)dy where |A| is the Lebesgue measure of A. Through all the paper the standard summation convention on repeated upper and lower indexes is adopted. The letter C is used for various constants and may change from one occurrence to another.

2 Definitions and statement of the problem

Let Ω ⊂ ℝn, n ≥ 1 be a bounded C1,1-domain, Q = Ω × (0, T) be a cylinder in +n+1, and S = ∂Ω × (0, T) stands for the lateral boundary of Q. We consider the problem Bu:=utaij(x)Diju=f a.e. in   Q,Fu:=u(x',0)=0,             on    Ω,Bu:=u/l=li(x)Diu=0  on S,(3) under the following conditions:

  • (i)

    The operator 𝔅 is supposed to be uniformly parabolic, i.e. there exists a constant Λ > 0 such that for almost all xQ Λ1|ξ|2aij(x)ξiξjΛ|ξ|2,ξn,aij(x)=aji(x),       i,j=1,,n.(4)

    The symmetry of the coefficient matrix a=aiji,j=1n implies essential boundedness of aij’s and we set a,Q=i,j=1naij,Q.

  • (ii)

    The boundary operator 𝔅 is prescribed in terms of a directional derivative with respect to the unit vector field l(x) = (l1(x), … , ln(x)), xS. We suppose that 𝔅 is a regular oblique derivative operator, i.e., the field l is never tangential to S: l(x)n(x)=li(x)ni(x)>0  on S,liLip(S¯).(5) Here Lip(S) is the class of uniformly Lipschitz continuous functions on S and n(x) stands for the unit outward normal to Ω.

In the following, besides the parabolic metric Q(x) = max(|x′|, |t|1/2) and the defined by it parabolic cylinders Ir(x)=yn+1:|x'y'|<r,|tτ|<r2,   |Ir|=Crn+2. we use the equivalent one ρ(x)=(|x'|2+|x'|4+4t22)12 (see [9]). The balls with respect to this metric are ellipsoids εr(x)=yn+1:|x'y'|2r2+|tτ|2r4<1,|εr|=Crn+2.

Because of the equivalence of the metrics all estimates obtained over ellipsoids hold true also over parabolic cylinders and in the following we shall use this without explicit references.

Definition 2.1: ([20, 37]). LetaLIoc1(n+1), denoted by ηa(R)=supεr,rR1|εr|εr|f(y)fεr|dy,  foreveryR>0 wherer ranges over all ellipsoids inn+1. The Banach space BMO (bounded mean oscillation) consists of functions for which the following norm is finitea*=supR>0ηa(R)<.A function a belongs to VMO (vanishing mean oscillation) with VMO- modulus ηa(R) providedlimR0ηa(R)=0.For any bounded cylinder Q we define BMO(Q) and VMO(Q) taking aL1(Q) and Qr = Q ⋂ ℐr instead ofr in the definition above.

According to [1, 21] having a function aBMO/VMO(Q) it is possible to extend it in the whole ℝn+1 preserving its BMO-norm or VMO-modulus, respectively. In the following we use this property without explicit references.

For this goal we recall some well known properties of the BMO functions.

Lemma 2.2: (John-Nirenberg lemma, [20]). Let aBMO and p ∈ (1, ∞). Then for any ballthere holds (1|B|B|a(y)aB|pdy)1pC(p)a*.

As an immediate consequence of Lemma 2.2 we get the following property.

Corollary 2.3.: Let aBMO then for all 0 < 2r < t holds |aBr+aBt+|Ca*lntr(6) where the constant is independent of a, x, t and r.

As mentioned before, we call weight a positive measurable function defined on ℝn × ℝ+. Given a weight w and a measurable set 𝕊 we denote by w(S)=Sw(x)dx the w-measure of 𝕊. A weight w belongs to the Muckenhoupt class Ap, 1 < p < ∞, if [w]Ap:=supB(1|B|Bw(x)dx)(1|B|Bw(x)p'/pdx)p/p'<,(7) where 1p+1p'=1. Note that, for any ball we have (see [11]) [w]Ap(B)1/p=|B|1wL1(B)1/pw1/pLp'(B)1.(8)

In case p = 1, we say that wA1 if 1|B|Bw(x)dxA essBinfw(x) and [w]A1 is the smallest A for which the above inequality holds. It is an immediate consequence of (7) that whenever wAp than it satisfies the doubling property, precisely w(2Br)C(n,p)w(Br).(9)

The following lemma collects some of the most important properties of the Muckenhoupt weights.

Lemma 2.4: ([11]). We have the following:

  • (1)

    If wAp for some 1 ≤ p < ∞, then for all ƛ > 1 we have w(λB)λnp[w]Apw(B).

  • (2)

    The following equality is valid: A = ⋃1≤p<∞ Ap.

  • (3)

    If wA, then for all ƛ > 1 we havew(λB)2λn[w]Aλnw(B).

  • (4)

    If wAp for some 1 ≤ p ≤ ∞, then there exist C > 0 and δ > 0 such that for any balland a measurable set 𝕊 ⊂ ℬ, 1[w]Ap(|S||B|)w(S)w(B)C(|S||B|)δ.

Lemma 2.5: ([30, Theorem 5]). Let wA. Then the norm of BMO(w) is equivalent to the norm of BMO(ℝn), where BMO(w)={a:a*,w=supxn,r>01w(Br(x))Br(x)|a(y)aBr(x),w|w(y)dy<} and aBr(x),w=1w(Br(x))Br(x)a(y)w(y)dy.

Lemma 2.6: (The John-Nirenberg inequality). Let aBMO,

  • (1)

    there exist constants C1, C2 > 0, such that for all β > 0 |xB:|a(x)aB|>β|C1|B|eC2β/a*,  Bn;

  • (2)

    for all p ∈ (1, ∞)a*=CsupB(1|B|B|a(y)aB|pdy)1/p;

  • (3)

    for all p ∈ [1, ∞) and wAa*=CsupB(1w(B)B|a(y)aB|pw(y)dy)1/p.

Definition 2.7.: Let ϕ(x, r) be weight in ϕ: ℝn × ℝ+ → ℝ+ and ωAp, p ∈ [1, ∞). The generalized weighted Morrey space Mp,ϕ(ℝn, ω) or Mp,ϕ(ω) consists of all functions fLp,ωloc(n) such thatfp,φ;ω=supxn,r>0φ(x,r)1(ω(εr(x))1εr(x)|f(y)|pω(y)dy)1/p<.The space Mp,ϕ(Q, ω) consists of Lωp(Q) functions provided the following norm is finite fp,φ,w;Q=supxn,r>0φ(x,r)1(ω(Qr(x))1Qr(x)|f(y)|pω(y)dy)1/p.The generalized Sobolev-Morrey space consist of all Sobolev functions uW2,1p(Q,ω) with distributional derivatives DtlDxsuMp,φ(Q,ω),02l+|s|2, endowed by the norm uW2,1p,φ(Q,ω)=utp,φ,ω;Q+|s|2Dsup,φ,ω;Q and W˙2,1p,φ(Q,ω)={uW2,1p,φ(Q,ω):u(x)=0,xQ},uW˙2,1p,φ(Q,ω)=uW2,1p,φ(Q,ω) where ∂Q means the parabolic boundary Ω ∪ ﹛Ω × (0, T)﹜.

Theorem 2.8.: (Main result) Let (i) and (ii) hold, aVMO(Q, ω) and uW˙2,1p(Q,ω) , ω), p(1, ∞), ωAp be a strong solution of (3). If fMp,ϕ (Q, ω) with ϕ(x, r) being measurable positive function satisfying r(1+lnsr)ess infφ(x,ς)ω(Qς(x))1ps<ς<ω(Qs(x))1pdssC(10) for each (x, r) ∈ Q × ℝ+, then uW˙2,1p,φ(Q,ω) and uW˙2,1p,φ(Q,ω)Cfp,φ,ω;Q(11) with C=C(n,p,[ω]Ap,Λ,Ω,T,a;Q,ηa) and ηa=i,j=1nηaij.If ϕ(x, r) = r(ƛ−n−2)/p, then Mp,ϕLp,ƛ and the condition (10) holds with a constant depending on n, p and ƛ. If ϕ(x, r) = ω(x, r)1/pr−(n+2)/p with ω : ℝn+1 × ℝ+ → ℝ+ satisfying the conditions k1ω(x0,s)ω(x0,r)k2  x0n+1,rs2rrω(x0,s)sdsk3ω(x0,r)ki>i=1,2,3 than we obtain the spaces Lp,ω studied in [27, 31]. The following results are obtained in [19] and treat continuity in Mp,ϕ(ℝn+1, ω) of certain singular and nonsingular integrals.

Definition 2.9.: A measurable function 𝔎(x; ζ) : ℝn+1 × ℝn+1 \ ﹛0﹜ → ℝ is called variable parabolic Calderon-Zygmund kernel (PCZK) if:

  • i)

    𝔎(x; ·) is a PCZK for a.a. xn+1:

    • a)

      𝔎(x; ·)C(ℝn+1 \ ﹛0﹜),

    • b)

      𝔎(x; μζ) = μ−(n+2) 𝔎(x; ζ) ∀μ > 0,

    • c)

      SnK(x;ξ)dσξ=0,   Sn|K(x;ξ)|dσξ<+.

  • ii)

    DξβK;n+1×SnM(β)< for each multi-index β.

    Consider the singular integrals Kf(x)=P.V.n+1K(x;xy)f(y)dy,[a,f](x)=P.V.n+1K(x;xy)[a(y)a(x)]f(y)dy.(12)

Theorem 2.10.: For any fMp,ϕ(ℝn+1, ω) with (p, ϕ) as in Theorem 2.8 and aBMO there exist constants depending on n, p, ϕ, ω and the kernel such that Kfp,φ,ω;n+1C[ω]Ap1pfp,φ,ω;n+1[a,f]p,φ,ω;n+1C[ω]Ap1pa*fp,φ,ω;n+1.(13)

Corollary 2.11.: Let Q be a cylinder in+n+1,fMp,φ(Q,ω),aBMO(Q) and K(x,ξ):Q×+n+1\0. Then the operators (12) are bounded in Mp,ϕ (Q, ω) with p, ϕ, and ω as in Theorem 2.10. Then Kfp,φ,ω;QC[ω]Ap1pfp,φ,ω;Q,[a,f]p,φ,ω;QC[ω]Ap1pa*fp,φ,ω;Q(14) withC=C(n,p,φ,[ω]Ap,|Ω|,K).

Corollary 2.12.: Let aVMO and (p, ϕ) be as in Theorem 2.8. Then for any ϵ > 0 there exists a positive number r0 = r0(ϵ, ηa) such that for anyr(x0) with a radius r(0, r0) and all fMp,ϕ(ℰr(x0), ω) [a,f]p,φ,ω;εr(x0)Cεfp,φ,ω;εr(x0)(15) where C is independent of ϵ, f, r and x0.For any x'+n and any fixed t > 0 define the generalized reflection T(x)=(T'(x),t),  T(x)=x'2xnan(x',t)ann(x',t)(16) where an(x) is the last row of the coefficients matrix a(x) of (3). The function 𝒯′(x) maps +n into n and the kernel 𝔎(x; 𝒯(x) − y) = 𝒦(x; 𝒯′(x) − y′, t − τ) is a nonsingular one for any x,yD+n+1. Taking x = (ʺ, −xn, t) there exist positive constants k1 and k2 such that k1ρ(x˜y)ρ(T(x)y)k2ρ(x˜y)(17)For anyfMp,φ(D+n+1,ω) with a norm fp,φ,ω;D+n+1=supxD+n+1,r>0φ(x,r)1(ω(εr(x))1εr(x)|f(y)|pω(y)dy)1/p and aBMO(D+n+1,ω) defines the nonsingular integral operators K˜f(x)=D+n+1K(x;T(x)y)f(y)dy˜[a,f](x)=D+n+1K(x;T(x)y)[a(x)a(y)f(y)dy.](18)

Theorem 2.13.: Let aBMO(D+n+1),ωAp andfMp,φ(D+n+1,ω) with (p, ϕ) as in Theorem 2.8. Then the operators K˜f and ˜[a,f] are continuous in Mp,φ(D+n+1,ω) and     K˜fp,φ,ω;D+n+1Cfp,φ,ω;D+n+1,˜[a,f]p,φ,ω;D+n+1C[ω]Ap1pa*fp,φ,ω;D+n+1.(19) with a constant independend of a and f.

Corollary 2.14.: Let a2 ϵ > 0 there exists a positive number r0 = r0(ϵ, ηa) such that for anyεr+(x0)=εr(x0)D+n+1 with a radius r(0, r0) and center x0 = (ʺ, 0, 0) and for all fMp,φ(εr+(x0),ω) holds ˜[a,f]p,φ,ω;εr+(x0)Cεfp,φ,ω;εr+(x0),(20) where C is independent of ϵ, f, r and x0.

3 Proof of the main result

As it follows by [39], the problem (3) is uniquely solvable in W˙2,1p(Q,ω) .

We are going to show that fMp,ϕ(Q, ω) implies uW˙2,1p,φ(Q,ω). For this goal we obtain an a priori estimate of u. Following the method used by Chiarenza, Frasca and Longo in [7] and [8], we prove the results considering two steps.

Interior estimate. For any x0+n+1 consider the parabolic semi-cylinders 𝒞r(x0) = ℬr(x′0) × (t0r2, t0). Let ν𝒞0(𝒞r) and suppose that ν(x, t) = 0 for t ≤ 0. According to [[5], Theorem 1.4] for any xsupp ν the following representation formula for the second derivatives of ν holds true

Dijυ(x)=P.V.n+1Γij(x;xy)[ahk(y)ahk(x)]Dhkυ(y)dy+P.V.n+1Γij(x;xy)Bυ(y)dy+Bυ(x)SnΓj(x;y)vidσy,(21)

where ν(ν1, … , νn+1) is the outward normal to 𝕊n. Here Γ(x; ζ) is the fundamental solution of the operator 𝔅 and Γ(x; ζ) = ∂2Γ(x; ζ)/∂ζiζj.

Because of density arguments the representation formula (21) still holds for any νW2, 1p(𝒞r(x0), ω). The properties of the fundamental solution (cf. [5, 25, 38]) imply Γij are Calderon-Zygmund kernels in the sense of Definition 2.9. We denote by 𝒦ij and Cij the singular integrals defined in (12) with kernels 𝔎(x; x − y) = Γij (x; x − y). Then we can write that

Dijυ(x)=ij[ahk,Dhkυ](x)+Kij(Bυ)(x)+Bυ(x)SnΓj(x;y)vidσy.(22)

Because of Corollaries 2.11 and 2.12 and the equivalence of the metrics we get

D2υp,φ,ω;Cr(x0)C(εD2υp,φ,ω;Cr(x0)+Bup,φ,ω;Cr(x0))

for some r small enough. Moving the norm of D2ν on the left-hand side we get

D2υp,φ,ω;Cr(x0)CBυp,φ,ω;Cr(x0)

with a constant depending on n, p, ηa(r), ║a║∞,Q and ║DΓ║∞,Q. Define a cut-off function ϕ(x) = ϕ1(x′)ϕ2(t), with ϕ1C0(ℬr(x′0)), ϕ2C0(ℝ) such that

ϕ1(x')=1x'Bθr(x0')0x'Bθ'r(x0'),ϕ2(t)=1  t(t0(θr)2,t0]0t<t0(θ'r)2

with θ(0, 1), θ′ = θ(3 − θ)/2 > θ and |Dsϕ| ≤ C ∈ [θ(1 − θ)r]−s, s = 0, 1, 2, |ϕt| ∼ |D2ϕ|. For any solution uW2,1p(Q, ω) of (3) define ν(x) = ϕ(x)u(x)W2,1p(𝒞r, ω). Then we get

D2up,φ,ω;Cθr(x0)D2υp,φ,ω;Cθ',r(x0)CBυp,φ,ω;Cθ'r(x0)C(fp,φ,ω;Cθ'r(x0)+Dup,φ,ω;Cθ'r(x0)θ(1θ)r+up,φ,ω;Cθ'r(x0)[θ(1θ)r]2).

By the choice of θ′ it holds θ(1 − θ) ≤ 2θ′(1 − θ′) which leads to

[θ(1θ)r]2D2up,φ,ω;Cθr(x0)C(r2fp,φ,ω;Q+θ(1θ)rDup,φ,ω;Cθr(x0)+up,φ,ω;Cθr(x0)).

Introducing the semi-norms

Θs=sup0<θ<1[θ(1θ)r]sD2up,φ,ω;Cθr(x0)s=0,1,2

and taking the supremo with respect to θ and θ′ we get

Θ2C(r2fp,φ,ω;Q+Θ1+Θ0).(23)

The interpolation inequality [26, Lemma 4.2] gives that there exists a positive constant C independent of r such that

Θ1εΘ2+CεΘ0 for anyε(0,2).

Thus (23) becomes

[θ(1θ)r]2D2up,φ,ω;Cθr(x0)Θ2C(r2fp,φ,ω;Q+Θ0)

for each θ(0, 1). Taking θ = 1/2 we get the Caccioppoli-type estimate

D2up,φ,ω;Cr/2(x0)C(fp,φ,ω;Q+1r2up,φ,ω;Cr(x0)).

To estimate ut we exploit the parabolic structure of the equation and the boundedness of the coefficients

utp,φ,ω;Cr/2(x0)a,QD2up,φ,ω;Cr/2(x0)+fp,φ,ω;Cr/2(x0)C(fp,φ,ω;Q+1r2up,φ,ω;Cr(x0)).

Consider cylinders Q′ = Ω′ × (0, T) and = Ωʺ × (0, T) with Ω′ ⋐ Ωʺ ⋐ Ω, by standard covering procedure and partition of the unity we get

uW2,1p,φ(Q',ω)C(fp,φ,ω;Q+up,φ,ω;Q").(24)

where C depends on n,p,[ω]Ap12Λ,TDΓ;Q,ηa(r),a,Q and dist(Ω′, ∂Ωʺ).

Boundary estimates. For any fixed R > 0 and x0 = (ʺ, 0, 0) define the semi-cylinders

CR+(x0)=CR(x0)D+n+1.

Without lost of generality we can take x0 = (0, 0, 0). Define ℬR+ = ﹛|x′| < R, xn > 0﹜, SR+ = ﹛|ʺ| < R, xn = 0, t(0, R2)﹜ and consider the problem

Bu:=utaij(x)Diju=f(x)   a.e. in CR+,u:=u(x',0)=0,            on   BR+,Bu:=li(x)Diu=0       on   SR+.(25)

Let uW2, 1p (CR+, ω) with u = 0 for t ≤ 0 and xn ≤ 0, then the following representation formula holds (see [26, 38])

Diju(x)=Iij(x)Jij(x)+Hij(x),

where

Iij(x)=P.V.CR+Γij(x;xy)F(x;y)dy+f(x)SnΓj(x;y)vidσy,  i,j=1,,n;Jij(x)=CR+Γij(x;T(x)y)F(x;y)dy;Jin(x)=Jni(x)=CR+Γil(x;T(x)y)(T(x)xn)lF(x;y)dy,i,j=1,,n1Jnn(x)=CR+Γls(x;T(x)y)(T(x)xn)l(T(x)xn)sF(x;y)dy;F(x;y)dy=f(y)+[ahk(y)ahk(x)]Dhku(y),Hij(x)=(Gij*2g)(x)+g(x",t)SnGj(x;y",xn,τ)nidσ(y",τ),    i,j=1,,n,T(x)xn=(2an1(x)ann(x),,2ann1(x)ann(x),1).

Here the kernel G = Γ 𝒬, is a byproduct of the fundamental solution and a bounded regular function 𝒬. Hence its derivatives Gij behave as Γij and the convolution that appears in Hij is defined as follows

(Gij*2g)(x)=P.V.SR+Gij(x;x"y",xn,tτ)g(y",0,τ)dy"dτ,g(x",0,t)=[(lk(0)lk(x",0,t))Dkulk(0)(Γk*F)]|xn=0(x",0,t),(Γk*F)(x)=CR+Γk(x;xy)F(x;y)dy.

Here Iij are a sum of singular integrals and bounded surface integrals hence the estimates obtained in Corollaries 2.11 and 2.12 hold true. On the nonsingular integrals Jij we apply the estimates obtained in Theorem 2.13 and Corollary 2.14 that give

Iijp,φ,ω;CR++Jijp,φ,ω;CR+C(fp,φ,ω;CR++ηa(R)D2up,φ,ω;CR+)(26)

for all i, j = 1, … , n. To estimate the norm of Hij we suppose that the vector field l is extended in 𝒞R+ preserving its Lipschitz regularity and the norm. This automatically leads to extension of the function g in 𝒞R+ that is

g(x)=(lk(0)lk(x))Dku(x)lk(0)(Γk*F)(x).(27)

Applying the estimates for the heat potentials [[25], Chapter 4] and the trace theorems in Lp [[2], Theorems 7.48, 7.53] (see also [[38], Theorem 1]) we get

CR+|(Gij*2g)(y)|pw(y)dyC(CR+|g(y)|pw(y)dy+CR+|Dg(y)|pw(y)dy).

Taking a parabolic cylinder ℐr(x) centered in some point x𝒞R+ we have

CR+Ir(x)|Gij*2g(y)|pw(y)dyCω(Ir(x))φ(x,r)p(φ(x,r)pω(Ir(x))CR+Ir(x)|g(y)|pw(y)dy+φ(x,r)pω(Ir(x))CR+Ir(x)|Dg(y)|pw(y)dy)Cω(Ir(x))φ(x,r)p(gp,φ,ω;CR+p+Dgp,φ,ω;CR+p).

Moving φ(x,r)pω(Ir(x)) on the left-hand side and taking the supremo with respect to (x, r)𝒞R+ × ℝ+ we get

Gij*2gp,φ,ω;CR+pC(gp,φ,ω;CR+p+Dgp,φ,ω;CR+p).

An immediate consequence of (27) is the estimate

gp,φ,ω;CR+[lk(0)lk()]Dkup,φ,ω;CR++lk(0)(Γk*F)p,φ,ω;CR+CRlLip(S¯)Dup,φ,ω;CR++Γk*fp,φ,ω;CR++Γk*[ahk()ahk(x)]Dhkup,φ,ω;CR+.

The convolution Γk * f is a Riesz potential. On the other hand

|(Γk*f)(x)|CCR+|f(y)|ρ(xy)n+1dyCRCR+|f(y)|ρ(xy)n+2dyCCR+|f(y)|ρ(xy)n+2dy

with a constant depending on T and diam Ω. Apply [[16], Theorem 4.8, [13], Theorem 3.1] that gives

Γk*fp,φ,ω,CR+Cfp,φ,ω,CR+.

Analogously

|Γk*[ahk()ahk(x)]Dhku()|CCR+|ahk(y)ahk(x)||Dhku(y)|ρ(xy)n+2dy

with a constant depending on diam Ω and T. The kernel ρ(x − y)−(n+2) is a nonnegative singular one and applying again the results for sub-linear integrals [[13], Theorem 3.7] we get

|Γk*[ahk()ahk(x)]Dhku()|ρ,ω,φ,CR+Ca*D2uρ,ω,φ,CR+.

Hence

gρ,ω,φ,CR+C(RlLip(S¯)Dup,φ,ω;CR++fp,φ,ω;CR++Rηa(R)D2up,φ,ω;CR+).(28)

Further, the Rademacher theorem asserts existence almost everywhere of the derivatives DhlkL, thus

Dhg(x)=Dhlk(x)Dku(x)+[lk(0)lk(x)]Dkhulk(0)(Γkh*F)(x).

The Mp,ϕ(ω) norm of the last term is estimated as above and

Dgp,φ,ω;CR+C(Dl:SDup,φ,ω;CR++RlLip(S¯)D2up,φ,ω;CR++fp,φ,ω;CR++ηa(R)D2up,φ,ω;CR+).(29)

Finally unifying (26), (28) and (29) we get

D2up,φ,ω;CR+C(fp,φ,ω;Q+(1+R)Dup,φ,ω;CR++(R+ηa(R)+Rηa(R))D2up,φ,ω;CR+)

with a constant depending on known quantities and ║lLip(S) and ║Dl∞;𝒮. Direct calculations lead to an interpolation inequality in Mp,ϕ(ω) analogous to [[25], Lemma 3.3] (cf. [41])

Dup,φ,ω;CR+δD2up,φ,ω;CR++Cδup,φ,ω;CR+,δ(0,R).

Taking 0<δ=RR+1<R we get

D2up,φ,ω;CR+C(fp,φ,ω;Q+RD2up,φ,ω;CR++CRup,φ,ω;CR++(R+ηa(R)+Rηa(R))D2up,φ,ω;CR+).

Choosing R small enough and moving the terms containing the norm of D2u on the left-hand side we get

D2up,φ,ω;CR+C(fp,φ,ω;CR++1Rup,φ,ω;CR+).

Because of the parabolic structure of the equation analogous estimate holds also for ut. Further the Jensen inequality applied to gives

up,φ,ω;CR+CR2utp,φ,ω;CR+C(R2fp,φ,ω;CR++Rup,φ,ω;CR+).

Choosing R smaller, if necessary, we get up,φ,ω;CR+Cfp,φ,ω;CR+ and therefore

uW2,1p,φ(CR+,ω)Cfp,φ,ω;CR+Cfp,φ,ω;CR+.(30)

Making a covering ﹛Cα+﹜, α𝒜 such that Q\QαACα+, considering a partition of unity subordinated to that covering and applying (30) for each 𝒞α+ we get

uW2,1p,φ(Q\Q',ω)Cfp,φ,ω;Q(31)

with a constant depending on n,p,[ω]Ap12,Λ,T,diamΩ, and ║Dl∞,𝒮. The estimate (11) follows from (24) and (31).

Acknowledgement

The authors are thankful to the referee for very valuable comments. The research of V.S. Guliyev and M.N. Omarova is partially supported by the grant of Science Development Foundation under the President of the Republic of Azerbaijan, Grant EIF-2013-9(15)-46/10/1 and by the grant of Presidium Azerbaijan National Academy of Science 2015.

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About the article

Received: 2015-11-15

Accepted: 2016-01-14

Published Online: 2016-02-13

Published in Print: 2016-01-01



Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0006. Export Citation

© 2016 Guliyev and Omarova, published by De Gruyter Open.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

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