parameter ν = ξ/ | ξ | and work only with the parameter ξ. Lemma 4 . In the new coordinate system, the functions are bounded on the set G for | ξ | ≤ 1 / 2. Moreover, the functions tend to zero in L$_{2}$ ( R 1 ξ ) as $_{ξ}$′ → 0. First, let us show that if up to the second order derivatives of the functions . z $^{i}$( x, ξ, t ) with respect to ξ are bounded, then it is possible to prove that the functions ∂$_{ξ}$ i ∂$_{ξ}$ 1 u are bounded for 0 < | ξ | ≤ 1 / 2 and 1 ≤ i ≤ n . Since supp b ⊂ D and the solution of Problem (3.2)-(3.3) has property (3.4), the integral in (4.2) can be considered on the finite interval [ 0 , d 0 | ξ | ] , where d$_{0}$ is the diameter of D in the metric g = ( g$_{ij}$ ( x )). Hence we can write (4.2) as Moreover, since the metric g = ( g$_{ij}$ ) is written down in the semi-geodesic coordi- nates, the equality . z k ( x, ξ $^{1}$, 0 , t ) = 0 ( ξ 1 = 0) is satisfied for each k (2 ≤ k ≤ n ) (see the proof of Lemma 1 in Appendix) and thus ∂$_{ξ}$ 1 z k ( x, ξ $^{1}$, 0 , t ) = 0 . Then, for each fixed x ∈ D and 0 < | ξ $^{1}$|  ≤ 1, by Taylor’s formula with respect to the variable $_{ξ}$′ = ( ξ $^{2}$, ..., ξ $^{n}$) (e.g., [13], p. 285), we obtain the equalities ∂$_{ξ}$ i ∂$_{ξ}$ 1 u (0 , ξ ) , ∂$_{ξ}$ i ∂$_{ξ}$ 1 ∂$_{x}$ j u (0 , ξ ), 1 ≤ i, j ≤ n , u (0 , ξ 1 $_{, ξ}$′$_{)}$$_{,}$ ∂$_{ξ}$ i u (0 , ξ 1 $_{, ξ}$′$_{)}$$_{,}$ ∂$_{ξ}$ i ∂$_{x}$ j u (0 , ξ 1 $_{, ξ}$′$_{)}$$_{,}$ ∂$_{ξ}$ i ∂$_{ξ}$ 1 u (0 , ξ 1 $_{, ξ}$′$_{)}$$_{,}$ ∂$_{x}$ j ∂$_{ξ}$ i ∂$_{ξ}$ 1 u (0 , ξ 1 $_{, ξ}$′$_{)}$$_{,}$$_{1}$ ≤ i, j ≤ n, I$_{ij}$ ( x, ξ ) = ∫ d$_{0}$ | ξ | 0 b ( z ( x, ξ, t )) . z $^{i}$( x, ξ, t ) . z j ( x, ξ, t ) dt . (5.3) . z k ( x, ξ, t ) = n ∑ i =2 ξ $^{i}$∂$_{ξ}$ i . z k ( x, ξ 1 $_{, ξ}$′$_{θ}$$_{1}$$_{, t}$ ) , (5.4) ∂$_{ξ}$ 1 . z k ( x, ξ, t ) = n ∑ i =2 ξ $^{i}$∂$_{ξ}$ i ∂$_{ξ}$ 1 . z k ( x, ξ 1 $_{, ξ}$′$_{θ}$$_{2}$$_{, t}$ ) , (5.5) 11 Proof.