Boundary Value Problems

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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

Boundary Value Problems20142014:116

DOI: 10.1186/1687-2770-2014-116

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 $u\in {W}^{2,p}\left(0,1;E\left(A\right),E\right)$ for ${f}_{k}\in {E}_{k}$, $\lambda \in {S}_{\psi }$, with sufficiently large $|\lambda |$ and the following coercive uniform estimate holds:
$\sum _{i=0}^{2}|\lambda {|}^{1-\frac{i}{2}}{\parallel {u}^{\left(i\right)}\parallel }_{{L}^{p}\left(0,1;E\right)}+{\parallel Au\parallel }_{{L}^{p}\left(0,1;E\right)}\le M\sum _{k=1}^{2}\left({\parallel {f}_{k}\parallel }_{{E}_{k}}+|\lambda {|}^{1-{\theta }_{k}}{\parallel {f}_{k}\parallel }_{E}\right).$
(3.7)
Theorem 3.3 Assume Condition 3.2 holds. Then the operator $u\to \left\{\left(L+\lambda \right)u,{L}_{1}u,{L}_{2}u\right\}$ for $\lambda \in {S}_{\psi ,\varkappa }$ and for sufficiently large $\varkappa >0$ is an isomorphism from
Moreover, the following uniform coercive estimate holds:
$\begin{array}{c}\sum _{i=0}^{2}|\lambda {|}^{1-\frac{i}{2}}{\parallel {u}^{\left(i\right)}\parallel }_{{L}^{p}\left(0,1;E\right)}+{\parallel Au\parallel }_{{L}^{p}\left(0,1;E\right)}\hfill \\ \phantom{\rule{1em}{0ex}}\le C\left[{\parallel f\parallel }_{{L}^{,p}\left(0,1;E\right)}+\sum _{k=1}^{2}\left({\parallel {f}_{k}\parallel }_{{E}_{k}}+|\lambda {|}^{1-{\theta }_{k}}{\parallel {f}_{k}\parallel }_{E}\right)\right].\hfill \end{array}$
(3.12)

Authors’ Affiliations

(1)
Department of Mechanical Engineering, Okan University
(2)
Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences
(3)
Okan University

References

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