where 2 ≤ i ≤ n, ν = ξ | ξ | . In addition, we have ∂$_{ξ}$ i ∂$_{ξ}$ j ( | ξ | . z k ( x, ξ | ξ | , t )) = g$_{ij}$ | ξ | . z k − 1 | ξ | 3 ( n ∑ r =2 g$_{jr}$ξ r ) ( n ∑ r =2 g$_{ir}$ξ r ) . z k − 1 | ξ | 2 n ∑ r =2 g$_{jr}$ξ r ) ( n ∑ s =1 ( ∂$_{ν}$ s . z k ) 1 | ξ | 2 ( ξ s n r =2 g$_{ir}$ξ r ) − ∂$_{ν}$ i . z k ) − ( ∂$_{ν}$ i . z k ) 1 | ξ | $_{2}$( n ∑ r =2 g$_{jr}$ξ $^{r}$) − n ∑ s =1 ( ∂$_{ν}$ s . z k ) ξ $^{s}$g$_{ij}$ | ξ | 2 + 2 | ξ | 4 ( n ∑ r =2 g$_{jr}$ξ r ) ( n ∑ s =1 ( ∂$_{ν}$ s . z k ) ξ s ( n ∑ r =2 g$_{jr}$ξ r )) + n ∑ s =1 1 | ξ | $_{3}$ξ $^{s}$( n ∑ r =2 g$_{ir}$ξ $^{r}$) ( n m =1 ( ∂$_{ν}$ m ∂$_{ν}$ s . z k ) 1 | ξ | $_{2}$ξ m n ∑ r =2 g$_{ir}$ξ r − ∂$_{ν}$ s ∂$_{ν}$ i . z k ) − 1 | ξ | 3 ( n ∑ r =2 g$_{ir}$ξ r ) n ∑ m =1 ξ $^{m}$∂$_{ν}$ m ∂$_{ν}$ j . z k + 1 | ξ | ∂$_{ν}$ j ∂$_{ν}$ i . z $^{k}$, (A.8) where 2 ≤ i, j ≤ n. Since . z k ∈ C $^{5}$(Ω ( d$_{0}$ )) and the set G ′ is bounded and closed, by (A.3)-(A.8), it is easy to see that ∣ | ξ | | α |− $^{1}$∂ α ξ ( | ξ | . z k )∣ ∣ ≤ K$_{2}$, 1 ≤ | α | ≤ 4 , (A.9) in Ω , where K$_{2}$ > 0 is independent of ( x, ξ ) ∈ ( D × G ). In (A.9), K$_{2}$ depends on the norm of the vector function . z ( x, ν, t ) in the space C $^{5}$(Ω ( d$_{0}$ )) and on the Euclidean distance between G ′ and 0 ∈ R n − 1 (0 / ∈ G ′ ) and the Euclidean diameter of G ′ . Moreover, the following inequalities are valid in Ω: for 0 ≤ | α | ≤ 4, where K$_{3}$, K$_{4}$ depend on the same parameters as K$_{2}$ in (A.9). Consequently, by the differentiation of an integral with respect to a parameter, the boundedness of the functions | ξ | . z $^{k}$, ∂$_{ξ}$ j ( | ξ | . z k ) and relations (A.9) - (A.10), we complete the proof of the second assertion of Lemma 1. 21 ∣ | ξ | | α | $^{+1}$∂ α ξ ( 1 | ξ $_{|}$)∣ ∣ ≤ K$_{3}$, ∣ | ξ | | α | ∂ α ξ ( b ( z )) ∣ ≤ K$_{4}$, (A.10)