Erratum to: ‘Abstract elliptic operators appearing in atmospheric dispersion’ by Veli B Shakhmurov and Aida Sahmurova published in the journal of Boundary Value Problems, 2014, V. 2014: 43

Veli B Shakhmurov; Aida Sahmurova

doi:10.1186/1687-2770-2014-116

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Erratum to: ‘Abstract elliptic operators appearing in atmospheric dispersion’ by Veli B Shakhmurov and Aida Sahmurova published in the journal of Boundary Value Problems, 2014, V. 2014: 43

Veli B Shakhmurov^{1, 2}Email author and
Aida Sahmurova³

Boundary Value Problems20142014:116

https://doi.org/10.1186/1687-2770-2014-116

Received: 1 May 2014

Accepted: 2 May 2014

Published: 14 May 2014

The original article was published in Boundary Value Problems 2014 2014:43

Correction

Errata of paper [1]. In Theorems 3.2 and 3.3 it should say $m = 0$ , i.e., these theorems should read as follows.

Theorem 3.2 Let Condition 3.2 hold. Then problem (3.5)-(3.6) has a unique solution

u \in W^{2, p} (0, 1; E (A), E)

for

f_{k} \in E_{k}

,

λ \in S_{ψ}

, with sufficiently large

| λ |

and the following coercive uniform estimate holds:

\sum_{i = 0}^{2} | λ |^{1 - \frac{i}{2}} {∥ u^{(i)} ∥}_{L^{p} (0, 1; E)} + {∥ A u ∥}_{L^{p} (0, 1; E)} \leq M \sum_{k = 1}^{2} ({∥ f_{k} ∥}_{E_{k}} + | λ |^{1 - θ_{k}} {∥ f_{k} ∥}_{E}) .

(3.7)

Theorem 3.3 Assume Condition 3.2 holds. Then the operator

u \to {(L + λ) u, L_{1} u, L_{2} u}

for

λ \in S_{ψ, ϰ}

and for sufficiently large

ϰ > 0

is an isomorphism from

W^{2, p} (0, 1; E (A), E) onto L^{p} (0, 1; E) \times E_{1} \times E_{2} .

Moreover, the following uniform coercive estimate holds:

\begin{matrix} \sum_{i = 0}^{2} | λ |^{1 - \frac{i}{2}} {∥ u^{(i)} ∥}_{L^{p} (0, 1; E)} + {∥ A u ∥}_{L^{p} (0, 1; E)} \\ \leq C [{∥ f ∥}_{L^{, p} (0, 1; E)} + \sum_{k = 1}^{2} ({∥ f_{k} ∥}_{E_{k}} + | λ |^{1 - θ_{k}} {∥ f_{k} ∥}_{E})] . \end{matrix}

(3.12)

Notes

Authors’ Affiliations

(1)

Department of Mechanical Engineering, Okan University

(2)

Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences

(3)

Okan University

References

Shakhmurov VB, Sahmurova A: Abstract elliptic operators appearing in atmospheric dispersion. Bound. Value Probl. 2014., 2014: Article ID 43Google Scholar

Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.