1 Correction to: Bull. Malays. Math. Sci. Soc. https://doi.org/10.1007/s40840-022-01356-3

In this article Equation 4.22 was incorrect. The Equation should have appeared as shown below. The original article has been corrected.

$$\begin{aligned} \Big \vert \int _{\Omega } \nabla \mathbf {R}^{\varepsilon } \nabla \mathbf {w}^{\varepsilon } \Big \vert&\le \vert \vert \nabla \mathbf {R}^{\varepsilon } \vert \vert _{L^{2}(\Omega )} \vert \vert \nabla \mathbf {w}^{\varepsilon } \vert \vert _{L^{2}(\Omega )} \le C \vert \vert \nabla \mathbf {R}^{\varepsilon } \vert \vert _{L^{2}(\Omega )} \vert \vert r^{\varepsilon } \vert \vert _{L^{2}(\Omega )}, \nonumber \\ \varepsilon \Big \vert \int _{\Omega } 2h \frac{\partial \mathbf {R}^{\varepsilon }}{\partial z} \frac{\partial \mathbf {w}^{\varepsilon }}{\partial z} \Big \vert&\le C \varepsilon \vert \vert \nabla \mathbf {R}^{\varepsilon } \vert \vert _{L^{2}(\Omega )} \vert \vert \nabla \mathbf {w}^{\varepsilon } \vert \vert _{L^{2}(\Omega )}\le C \varepsilon \vert \vert \nabla \mathbf {R}^{\varepsilon } \vert \vert _{L^{2}(\Omega )}\vert \vert r^{\varepsilon } \vert \vert _{L^{2}(\Omega )}, \nonumber \\ \varepsilon \Big \vert \int _{\Omega } 2h' \frac{\partial \mathbf {R}^{\varepsilon }}{\partial x}\mathbf {w}^{\varepsilon } \Big \vert&\le C \varepsilon \vert \vert \nabla \mathbf {R}^{\varepsilon } \vert \vert _{L^{2}(\Omega )} \vert \vert \nabla \mathbf {w}^{\varepsilon } \vert \vert _{L^{2}(\Omega )}\le C \varepsilon \vert \vert \nabla \mathbf {R}^{\varepsilon } \vert \vert _{L^{2}(\Omega )}\vert \vert r^{\varepsilon } \vert \vert _{L^{2}(\Omega )}, \nonumber \\ \varepsilon \Big \vert \int _{\Omega } 2h'z \frac{\partial \mathbf {R}^{\varepsilon }}{\partial x} \frac{\partial \mathbf {w}^{\varepsilon }}{\partial z} \Big \vert&\le C \varepsilon \vert \vert \nabla \mathbf {R}^{\varepsilon } \vert \vert _{L^{2}(\Omega )} \vert \vert \nabla \mathbf {w}^{\varepsilon } \vert \vert _{L^{2}(\Omega )}\le C \varepsilon \vert \vert \nabla \mathbf {R}^{\varepsilon } \vert \vert _{L^{2}(\Omega )}\vert \vert r^{\varepsilon } \vert \vert _{L^{2}(\Omega )}, \nonumber \\ \varepsilon \Big \vert \int _{\Omega } zh'' \frac{\partial \mathbf {R}^{\varepsilon }}{\partial z}\mathbf {w}^{\varepsilon } \Big \vert&\le C \varepsilon \vert \vert \nabla \mathbf {R}^{\varepsilon } \vert \vert _{L^{2}(\Omega )} \vert \vert \nabla \mathbf {w}^{\varepsilon } \vert \vert _{L^{2}(\Omega )}\le C \varepsilon \vert \vert \nabla \mathbf {R}^{\varepsilon } \vert \vert _{L^{2}(\Omega )}\vert \vert r^{\varepsilon } \vert \vert _{L^{2}(\Omega )}, \nonumber \\ k^2 \Big \vert \int _{\Omega } \mathbf {R}^{\varepsilon } \mathbf {w}^{\varepsilon } \Big \vert&\le C \vert \vert \nabla \mathbf {R}^{\varepsilon } \vert \vert _{L^{2}(\Omega )} \vert \vert \nabla \mathbf {w}^{\varepsilon } \vert \vert _{L^{2}(\Omega )} \le C \vert \vert \nabla \mathbf {R}^{\varepsilon } \vert \vert _{L^{2}(\Omega )}\vert \vert r^{\varepsilon } \vert \vert _{L^{2}(\Omega )}, \nonumber \\ M^2 \Big \vert \int _{\Omega } R_x^{\varepsilon }\mathbf {e}_{1}\mathbf {w}^{\varepsilon } \Big \vert&\le C \vert \vert \nabla \mathbf {R}^{\varepsilon } \vert \vert _{L^{2}(\Omega )} \vert \vert \nabla \mathbf {w}^{\varepsilon } \vert \vert _{L^{2}(\Omega )} \le C \vert \vert \nabla \mathbf {R}^{\varepsilon } \vert \vert _{L^{2}(\Omega )}\vert \vert r^{\varepsilon } \vert \vert _{L^{2}(\Omega )}, \\ \varepsilon \Big \vert \int _{\Omega }zh'r^{\varepsilon } \frac{\partial \mathbf {w}^{\varepsilon }}{\partial z}\mathbf {e}_{1} \Big \vert&\le C \varepsilon \vert \vert r^{\varepsilon } \vert \vert _{L^{2}(\Omega )} \vert \vert \nabla \mathbf {w}^{\varepsilon } \vert \vert _{L^{2}(\Omega )} \nonumber \\&\le C \varepsilon \vert \vert r^{\varepsilon } \vert \vert _{L^{2}(\Omega )}^{2}, \nonumber \\ \varepsilon \Big \vert \int _{\Omega } h'r^{\varepsilon } \mathbf {w}^{\varepsilon } \mathbf {e}_{1} \Big \vert&\le C \varepsilon \vert \vert r^{\varepsilon } \vert \vert _{L^{2}(\Omega )} \vert \vert \nabla \mathbf {w}^{\varepsilon } \vert \vert _{L^{2}(\Omega )} \nonumber \\&\le C \varepsilon \vert \vert r^{\varepsilon } \vert \vert _{L^{2}(\Omega )}^2, \nonumber \\ \varepsilon \Big \vert \int _{\Omega } h'r^{\varepsilon } \frac{\partial \mathbf {w}^{\varepsilon }}{\partial z} \mathbf {e}_{2} \Big \vert&\le C \varepsilon \vert \vert r^{\varepsilon } \vert \vert _{L^{2}(\Omega )} \vert \vert \nabla \mathbf {w}^{\varepsilon } \vert \vert _{L^{2}(\Omega )}, \nonumber \\&\le C \varepsilon \vert \vert r^{\varepsilon } \vert \vert _{L^{2}(\Omega )}^2, \nonumber \\ \Big \vert \int _{\Omega } \mathbf {E}^{\varepsilon } \mathbf {w}^{\varepsilon } \Big \vert&\le C\vert \vert \mathbf {E} \vert \vert _{L^{2}(\Omega )} \vert \vert \nabla \mathbf {w} \vert \vert _{L^{2}(\Omega )} \le C \varepsilon ^2 \vert \vert r^{\varepsilon } \vert \vert _{L^{2}(\Omega )}. \nonumber \end{aligned}$$
(4.22)