content_image/295268.jpg content_image/295270.jpg content_image/295275.jpg
where 2 ≤ i ≤ n, ν = ξ | ξ | . In addition, we have 

<img src='content_image/295268.jpg'>

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 

<img src='content_image/295270.jpg'>

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: 

<img src='content_image/295275.jpg'>

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. 

