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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 14, Issue 1

Issues

Corrigendum to: 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
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/ Mehriban N. Omarova
  • Institute of Mathematics and Mechanics, AZ 1141 Baku, Azerbaijan and Baku State University, Baku, AZ 1148, Azerbaijan
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Published Online: 2016-05-19 | DOI: https://doi.org/10.1515/math-2016-0026

We rectify an error in the proof of the global weighted Morrey-type regularity of the solution of the regular oblique derivative problem for linear uniformly parabolic operators with VMO coefficients.

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

{Pu:=utaij(x)Diju=fa.e.  inQ,𝔍u:=u(x,0)=0,on    Ω,𝔅u:=u/l=li(x)Diu=0onS,(3)

under the following conditions (i) and (ii) in [5].

The unique strong solvability of this problem was proved in [9]. In [10] Softova studied the regularity of the solution in the Morrey spaces and in [11] extended these studies on generalized Morrey spaces Mp,φ with a Morrey function φ satisfying Guliyev type conditions considered in [6].

The density of the C0 functions in the weighted Lebesgue space Lωp is proved in [8, Chapter 3, Theorem 3.11].

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

  • i)

    𝕶(x; ·) is a PCZK for a.a. x ∈ ℝn+1:

    • a)

      𝕶(x; ·) ∈ C(ℝn+1 \ {0}),

    • b)

      𝕶(x; (μy′, μ2τ)) = μ–(n + 2) 𝕶(x; y), ∀μ > 0, y = (y′, τ)

    • c)

      𝕊n 𝕶(x; y)y = 0, ∫𝕊n |𝕶(x; y)|y < +∞.

  • ii)

    Dyβ𝕶;n+1×𝕊nM(β)<

for each multi-index β.

Note that, the Theorems 2.10, Theorem 2.13 and Corollaries 2.11, 2.12, 2.14 in [5] the isotropic case are obtained in [4] and treat continuity in Mp,φ(ℝn+1, ω) of certain parabolic singular and parabolic nonsingular integrals.

3 Proof of the main result

Suppose that uW2,1p(Q,ω) is a solution of (3). Note that the solution of (3) exists according to Remark A (see also, [1], [12]).

We are going to show that fMp,φ(Q, ω) implies uW2,1p(Q,ω). For this goal we obtain an a priori estimate of u. Following the method used by Chiarenza, Frasca and Longo in [2] and [3], we prove the results considering two steps: interior estimate and boundary estimate.

In the proof of the interior estimate after inequality (23) in [5] we need the following interpolation inequality.

(Interpolation inequality). There exists a constant C independent of r such that Θ1εΘ2+CεΘ0for  any  ε(0,2).

Proof. For functions uW2,1p,ω(Cr),p(1,) and ωAp we dispose with the following interpolation inequality which may be proved analogously in [7]. Dup,ω;CrC(up,ω;Cr+up,ω;Cr1/2+(Dtup,ω;Cr+D2up,ω;Cr)1/2).

Then for any > 0 we have Dup,ω;CrC((1+12ϵ)up,ω;Cr+ϵ2(Dtup,ω;Cr+D2up,ω;Cr)).

Then for any > 0 we have Dup,ω;CrC((1+12ϵ)up,ω;Cr+ϵ2(Dtup,ω;Cr+D2up,ω;Cr)).

Choosing small enough, taking δ=Cϵ2<1, dividing all terms of φ(x,r)w(Cr)1/p and taking the supremum over 𝓒r we get the desired interpolation inequality in Mp, φ(ω) Dup,φ,ω;CrδDtup,φ,ω;Cr+D2up,φ,ω;Cr+Cδup,φ,ω;Cr.(*)

We can always find some θ0 ∈ (0, 1) such that Θ12[θ0(1θ0)r]Dup,φ,ω;Cθ0r2[θ0(1θ0)r]δ(Dtup,φ,ω;Cθ+D2up,φ,ω;Cθ)+Cδup,φ,ω;Cθ0r.

The assertion follows choosing δ=ε2[θ0(1θ0)r]<θ0r for any ∈ (0, 2). □

The interpolation inequality (see Lemma 3.1) gives that there exists a positive constant C independent of r such that

Θ1εΘ2+CεΘ0foranyε(0,2).

Thus from (23) in [5] becomes

[θ(1θ)r]2Dtup,φ,ω;Cr+D2up,φ,ω;CrΘ2C(r2fp,φ,ω;Q+Θ0)

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

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

Consider cylinders Q′ = Ω′ × (0, T) and Q″ = Ω″ × (0, T) with Q″ ⋐ Ω″ ⋐ 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]Ap, Λ, T, ||DΓ||∞,Q, ηa(r), ||a||∞, Q and dist(Ω′, ∂Ω″).

In the proof of the boundary estimates after equality (27) in [5] applying Theorem 2.10 and Corollary 2.11 in [5], the interpolation inequality (*), taking into account the VMO properties of the coefficients aij’s, it is possible to choose R0 small

uW2,1p,φ(CR+,ω)Cfp,φ,ω;CR+(30)

for all R < R0 (see [9, 11] for details). 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]Ap, Λ, T, diamΩ, ||DΓ||∞, Q, ηa, ||a||∞, Q, lLip(S¯), and ||Dl||∞, 𝓢.

The main estimate (11) in the Theorem 2.8 in [5] follows from (24) and (31).

Acknowledgement

The authors thank to Prof. L. Softova who called their attention to some gaps in the proof of the main result. 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 Presidium Azerbaijan National Academy of Science 2015.

References

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About the article

Published Online: 2016-05-19

Published in Print: 2016-01-01


Citation Information: Open Mathematics, Volume 14, Issue 1, Pages 283–285, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0026.

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

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