1 Correction to: Computational Optimization and Applications https://doi.org/10.1007/s10589-022-00415-6


The original version of this article unfortunately contained an error in Eq. 74 caused by a mistake during the production process. The right hand side of this equation is missing, and the article was published with incomplete equation.


Now, the correct and complete equation of (74) is presented here:

$$\begin{aligned}&\int _{J\times \mathbb {R}^{n+1}} \bigl (\partial _t (\rho (\alpha ) \delta u) +{\text{ div }}(\rho (\alpha ) (\delta u\otimes u+u\otimes \delta u))+\delta c\bigr)^\top \varphi \,d(t,x,y)\nonumber \\&+\int _{J\times \mathbb {R}^{n+1}} S(\delta u,\delta q;\mu (\alpha )):\nabla \varphi \,d(t,x,y)\nonumber \\&+\int _{J}\int _{\mathbb {R}^{n+1}} (\rho _2-\rho _1) u^\top \bigl (\partial _t\varphi +u\cdot \nabla \varphi \bigr ) d\delta \alpha (t)(x,y)\,dt\nonumber \\&-\int _{J}\int _{\mathbb {R}^{n+1}} \left[ S(u,q;\mu (\alpha ))\right] :\nabla \varphi \,d\delta \alpha (t)(x,y) dt= \\&=\lim _{\varepsilon \searrow 0}-\int _{J\times \mathbb {R}^{n+1}} \sigma \left( \frac{\delta \nu _{\varepsilon }^\top }{|\nu _{\varepsilon }|}- \frac{\delta \nu _{\varepsilon }^\top \nu _{\varepsilon }\nu _{\varepsilon }^\top }{|\nu _{\varepsilon }|^3} \right) \left( D\varphi -{\text{ div }}(\varphi )I \right) \nabla \alpha \,d(t,x,y) \nonumber \\&\qquad -\int _{J\times \mathbb {R}^{n+1}} \sigma \frac{\nu _{\varepsilon }^\top }{|\nu _{\varepsilon }|} \left( D\varphi -{\text{ div }}(\varphi )I \right) \nabla d\delta \alpha (t)(x,y)\qquad \forall \,\varphi \in C_c^2(J\times \mathbb {R}^{n+1};\mathbb {R}^{n+1}),\end{aligned}$$
(74)

The original article has been corrected.