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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

# 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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• Other articles by this author:
/ Mehriban N. Omarova
• Institute of Mathematics and Mechanics, AZ 1141 Baku, Azerbaijan and Baku State University, Baku, AZ 1148, Azerbaijan
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• Other articles by this author:
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 ${ℝ}_{+}^{n+1}$, and S = ∂Ω × (0, T) stands for the lateral boundary of Q. We consider the problem

${Pu:=ut−aij(x)Diju=fa.e. in Q,𝔍u:=u(x′,0)=0,on Ω,𝔅u:=∂u/∂l=li(x)Diu=0on S,$(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 ${C}_{0}^{\infty }$ functions in the weighted Lebesgue space ${L}_{\omega }^{p}$ 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)

${‖{D}_{y}^{\beta }𝕶‖}_{\infty ;{ℝ}^{n+1}×}{}_{{𝕊}^{n}}\le M\left(\beta \right)<\infty$

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 $u\in {\stackrel{\circ }{W}}_{2,1}^{p}\left(Q,\omega \right)$ 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 $u\in {\stackrel{\circ }{W}}_{2,1}^{p}\left(Q,\omega \right)$. 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 $Θ1≤εΘ2+CεΘ0 for any ε∈(0,2).$

Proof. For functions $u\in {W}_{2,1}^{p,\omega }\left({\mathcal{C}}_{r}\right),p\in \left(1,\infty \right)$ and ωAp we dispose with the following interpolation inequality which may be proved analogously in [7]. $‖Du‖p,ω;Cr≤C(‖u‖p,ω;Cr+‖u‖p,ω;Cr1/2+(‖Dtu‖p,ω;Cr+‖D2u‖p,ω;Cr)1/2).$

Then for any > 0 we have $‖Du‖p,ω;Cr≤C((1+12ϵ)‖u‖p,ω;Cr+ϵ2(‖Dtu‖p,ω;Cr+‖D2u‖p,ω;Cr)).$

Then for any > 0 we have $‖Du‖p,ω;Cr≤C((1+12ϵ)‖u‖p,ω;Cr+ϵ2(‖Dtu‖p,ω;Cr+‖D2u‖p,ω;Cr)).$

Choosing small enough, taking $\delta =\frac{Cϵ}{2}<1$, dividing all terms of $\phi \left(x,r\right)w{\left({\mathcal{C}}_{r}\right)}^{1/p}$ and taking the supremum over 𝓒r we get the desired interpolation inequality in Mp, φ(ω) $Dup,φ,ω;Cr⩽δDtup,φ,ω;Cr+D2up,φ,ω;Cr+Cδup,φ,ω;Cr.$(*)

We can always find some θ0 ∈ (0, 1) such that $Θ1≤2[θ0(1−θ0)r]Dup,φ,ω;Cθ0r≤2[θ0(1−θ0)r]δ(Dtup,φ,ω;Cθ+D2up,φ,ω;Cθ)+Cδup,φ,ω;Cθ0r.$

The assertion follows choosing $\delta =\frac{\epsilon }{2}\left[{\theta }_{0}\left(1-{\theta }_{0}\right)r\right]<{\theta }_{0}r$ for any ∈ (0, 2). □

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

$Θ1≤εΘ2+CεΘ0 for any ε∈(0,2).$

Thus from (23) in [5] becomes

$[θ(1−θ)r]2Dtup,φ,ω;Cr+D2up,φ,ω;Cr≤Θ2≤C(r2fp,φ,ω;Q+Θ0)$

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

$‖Dtu‖p,φ,ω;Cr/2(x0)+‖D2u‖p,φ,ω;Cr/2(x0)≤C(‖f‖p,φ,ω;Q+1r2‖u‖p,φ,ω;Cr(x0)).$

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

$‖u‖W2,1p,φ(Q′,ω)≤C(‖f‖p,φ,ω;Q+‖u‖p,φ,ω;Q″),$(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

$‖u‖W2,1p,φ(CR+,ω)≤C‖f‖p,φ,ω;CR+$(30)

for all R < R0 (see [9, 11] for details). Making a covering $\left\{{\mathcal{C}}_{\alpha }^{+}\right\},\alpha \in \mathcal{A}$ such that $Q\{Q}^{\prime }\subset \underset{\alpha \in \mathcal{A}}{\cup }{\mathcal{C}}_{\alpha }^{+}$, considering a partition of unity subordinated to that covering and applying (30) for each ${\mathcal{C}}_{\alpha }^{+}$ we get

$‖u‖W2,1p,φ(Q\Q′,ω)≤C‖f‖p,φ,ω;Q$(31)

with a constant depending on n, p, [ω]Ap, Λ, T, diamΩ, ||DΓ||∞, Q, ηa, ||a||∞, Q, ${‖l‖}_{Lip\left(\overline{\mathcal{S}}\right)}$, 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.

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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,

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