The Cauchy problem for the Lame system in infinite domains in R m

Transcription

1 The Cauchy problem for the Lame system in infinite domains in R m O. I. Makhmudov, I. E. Niyozov Samarkand State University, Departament of Mechanics and Mathematics, University Boulbald 5, 734 Samarkand, Uzbekinstan Abstract We consider the problem of analytic continuation of the solution of the multidimensional Lame system in infinite domains through known values of the solution and the corresponding strain tensor on a part of the boundary, i.e,the Cauchy problem. Key words: the Cauchy problem, system Lame, elliptic system, illposed problem, Carleman matrix, regularization, Laplace equation. Introduction We consider the problem of analytic continuation of the solution of the multidimensional Lame system in infinite domains through known values of the solution and the corresponding strain tensor on a part of the boundary, i.e,the Cauchy problem. Let R m, m, be m dimensional real Euclidean space, D a domain in R m with piecewise smooth boundary D, S a part of D with smooth boundary. olimjan@yahoo.com iqbol@samdu.uz

2 Suppose that x = x,..., x m, y = y,..., y m are points in R m and Ux satisfies in domain D the following uniform system of Lame equations: LUx =, were L = µ + λ + µ grad div, is the Laplace operator, λ, µ are the Lame constants, and µ, λ µ. Set Uy = fy, T y, nuy = gy, y S here fy = f y,..., f m y and gy = g y,..., g m y are given continuous vector functions on S, T y, n is the stress operator,i.e., T y, n = µδ ij n + λn iy y j + µn j y y i, i, j =,..., m, where δ ij is the Kronecker delta, and n = n,..., n m is the unit normal vector to the surface S. Given f and g determine the function Uy in D. The Lame system is elliptic. The Cauchy problem for elliptic equations is unstable with respect to small variations of the data, i.e., it is an ill-posed problem. For ill-posed problems, one does not prove the existence theorem; the existence is assumed a priori. Moreover, the solution is assumed to belong to some given subset of the function space, usually a compact one the Tikhonov well-posedness class [5], [3]. The uniqueness of the solution of problem, follows from the general Holmgren theorem. In this paper, we construct an approximate solution of problem, by using Carleman s function. Extending Lavrent ev s idea, Yarmukhamedov constructed the Carleman function for the Cauchy problem for the Laplace and Helmholtz equations [4], [5]. It was proved earlier [], [] that for any Cauchy problem for solutions of elliptic systems, the Carleman matrix exists if the Cauchy data are given on the boundary set of positive measure. Since we deal here with explicit formulas, the construction of the Carleman matrix in elementary and special functions is of considerable interest. Starting from of the Carleman function for the Cauchy problem for the Laplace equation, was constructed Carleman matrix for the Cauchy problem for the system of elasticity theory for m =, 3 in works [3], [6], [7], [8], [9].

3 Since f and g can be given approximately, the solution of problem found only approximately. Suppose that instead of fy and gy we know their approximations f δ y and g δ y with accuracy δ, < δ < in the uniform metric; these approximations do not necessarily belong to the class of existence of the solution. Let us construct a family of functions Ux, f δ, g δ = U σδ x that depend on a parameter σ, and let us prove that under some conditions and for a special choice of the parameter σδ, the family U σδ x converges in the usual sense to the solution Ux of problem, as δ. Following Tikhonov [3] we call U σδ x a regularized solution of the problem. The regularized solution defines a stable method for approximately solving problem,. In the work [] the problem is solved by the polynomial approximation method and yields a solvability criterion of the problem, too. Construction of the matrix of fundamental solution for the system of elasticity of a special form Definition.. The matrix Γy, x = Γ ij y, x m m, is called the matrix of fundamental solutions of system, where Γ ij y, x = i, j =,,..., m, µλ + µ λ + 3µδ ijqy, x λ + µy j x j qy, x, x i qy, x = { mω m π and ω m is the area of unit sphere in R m., y x m m >, ln y x, m =, The matrix Γy, x is symmetric and each of its columns and rows satisfy equation at an arbitrary point x R m, except y = x. Thus, we have LΓy, x =, y x. Developing Lavrent ev s idea concerning the notion of Carleman function of the Cauchy problem for the Laplace equation [5], we introduce the following notion. 3

4 Definition.. By the Carleman matrix of problem, we mean an m m matrix Πy, x, σ satisfying the following two conditions: Πy, x, σ = Γy, x + Gy, x, σ, where σ is a positive numerical parameter and, with respect to the variable y, the matrix Gy, x, σ satisfies system every where in the domain D. The following relation holds: Πy, x, σ + T y, nπy, x, σ ds y εσ, D\S where εσ, as σ uniformly x on compact subsets of D ; here and elsewhere Π denotes the Euclidean norm of the matrix Π = Π ij, i.e., Π = m Π ij, in particular U = m Ui for the vector U = U,..., U m. i,j= i= Definition.3. A vector function Uy = U y,..., U m y is said to be regular in D, if it is continuous together with its partial derivatives of second order in D and those of first order in D = D D. In the theory of partial differential equations, an important role is played by representations of solutions of these equations as functions of potential type. As an example of such representations, we present Somilian-Betti formula [4] below. Theorem.4. Any regular solution Ux of equation in the domain D is specified by the formula Ux = Γy, x{t y, nuy} Uy{T y, nγy, x}ds y, x D. D Since the Carleman matrix differs from the matrix of fundamental solutions by a solution of the transposed system, it follows that Somilian-Betti formula remains valid if the fundamental solution is replaced by the Carleman matrix. Thus, we obtain the following assertion. Theorem.5. Any regular solution Ux of equation in the domain D is specified by the formula Ux = Πy, x, σ{t y, nuy} Uy{T y, nπy, x, σ}ds y, x D, D where Πy, x, σ the Carleman matrix. 3 4

5 Suppose that Kω, ω = u + iv u, v are real, is an entire function taking real values the real axis and satisfying the conditions Let Ku, sup v p K p ω = Mp, u <, p =,..., m, u R. v s = α = y x y m x m. For α >, we define the function φy, x by the following relations: if m =, then πkx φy, x = if m = n +, n, then C m Kx m φy, x = n s n Im[ Ki u + α + y i udu ] u + α + y x u + α, Im[ Ki u + α + y m i du ] u + α + y m x m u + α, 4 where C m = n n m πω m n!, if m = n, n, then C m Kx m φy, x = n s n Im Kαi + y m αα + y m x m, 5 where C m = n n!m ω m. With the help of function φy, x we construct a matrix Πy, x = Π ij y, x m m = λ + 3µ µλ + µ δ ijφy, x λ + µ µλ + µ y j x j φy, x y i, m m i, j =,,..., m. 6 Now suppose D is an infinite domain in R m, m, lying in the shall: < y m < h, h = π, ρ >. Moreover, its boundary consists of a smooth ρ surface S lying in the half-space y m > and of the hyperplane D\S : y m =. Suppose that for the some b > the boundary area satisfies the conditions exp{ b exp ρ y }ds y <, < ρ < ρ, y = y,..., y m R m. D 7 5

6 For σ in 4-6, put where Kω = expσω bchiρ ω h b chiρ ω h, Kx m = expσx m + b cos ρ x m h b cos ρ x m h, ω = y m + iv, ρ, ρ, ρ, < x m < h, b >, b > b cos ρ h. Take φy, x, σ = φy, x and Πy, x, σ = Πy, x. In [4], the following assertion was proved. Lemma.6. The function φy, x, σ can be expressed as φy, x, σ = π ln r + g y, x, σ, m =, r = y x, r m φy, x, σ = ω m m + g my, x, σ, m 3, r = y x, where g m y, x, σ, m is a function defined for all values y, x and harmonic in the variable y in all of R m. Using in Lemma.6, we obtain. Lemma.7. The matrix Πy, x, σ is the Carleman matrix for problem,. Proof. By 4-6 and Lemma.6, we have where Πy, x, σ = Γy, x + Gy, x, σ, Gy, x, σ = G kj y, x, σ m m = λ + 3µ µλ + µ δ kjg m y, x, σ λ + µ µλ + µ y j x j g m y, x, σ y i. m m 6

7 Prove that L y Gy, x, σ =. Indeed, since y g m y, x, σ =, y = m and for the j column G j y, x, σ, k= y k divg j y, x, σ = µλ + µ g m y, x, σ, y j then for the k-th component of the vector L y G j y, x, σ we obtain m L y ki G ij y, x, σ i= [ λ + 3µ = µ y µλ + µ δ ijg m y, x, σ λ + µ µλ + µ y j x j ] g m y, x, σ y k +λ + µ y k divg j y, x, σ = λ + µ µλ + µ y j g m y, x, σ + λ + µ µλ + µ y j g m y, x, σ = Hence each column of the matrix Gy, x, σ satisfies Lame system in the variable y every where in R m. We now check the second property of the Carleman matrix. We estimate [ Πy, x, σ + T y, nπy, x, σ ]ds y. D\S To estimate the integral, we have to estimate q φy, x, σ y q, q = q... ym qm q m, q =,,. From the construction of φy, x, σ for we obtain Kω = expσω bchiρ ω h b chiρ ω h, Im Kiθ + y m iθ + y m x m = θ + y m x m θ cos σ + y m x m sin σθ 8 exp σy m b cos ρ ym h chθρ b cos ρ ym h chθρ, 7

8 where θ = u + α, by m = n +, n and θ = α for m = n, n. Then for m = n + and s = α C m φy, x, σ = exp σy m Kx m n s n exp b cos ρ y m h chρ u + s u + s + y m x m y m x m sin σ u + s u + α cos σ u + s exp b cos ρ y m h chρ du u + s u + s. Since the function chρ u + s growths by u, and π < ρ h then function < ρ y m h < ρ h < π, ρ < ρ, π < ρ y m h < π, u + s exp b cos ρ y m h chρ u + s b cos ρ y m h chρ u + s, is decreasing for u. By the mean value theorem for improper integrals [], for some ξ, we have C m φy, x, σ = exp σy m Kx m exp n s n b cos ρ y m h chρ s b cos ρ y m h chρ s 9 ξ s ym x m sin σ u + s u + α cos σ u + s du. u + s + y m x m Hence φy, x, σ Cm C ρσ n s n exp b cos ρ y m h chρ s b cos ρ y m h chρ s exp σy m x m + b cos ρ x m h + b cos ρ x m h, 8

9 q φy, x, σ y q... ym qm C m C ρσ n +k s n k b cos ρ y m h chρ s b cos ρ y m h chρ s exp σy m x m + b cos ρ y m h + b cos ρ y m h, where q = q + + q m, q =,. Then Πy, x, σ + T y, nπy, x, σ Cρ s n σ n+ exp b cos ρ y m h chρ s b cos ρ y m h chρ s exp σy m x m + b cos ρ x m h + b cos ρ x m h. From this inequality by condition 7, we obtain [ Πy, x, σ + T y, nπy, x, σ ]ds y D\S [ Πy, x, σ + T y, nπy, x, σ ]ds y D\S Cρσ n+ exp σx m, x m >. For m = n estimate easily follows from 8. The lemma proved. The formula of Somilian-Betti in infinite domains and its applications I. Let D be a domain in R m, m, with piecewise smooth boundary D. We denote by AD the space for all solutions of system in D if D infinite domain, then the regularity is required in finite points of D. If D is a bounded domain and Ux AD, then the following formula 9

10 3 holds: [Πy, x, σ{t y, nuy} Uy{T y, nπy, x, σ}]ds y D = { Ux, x D,, x D. Let Ux AD, where D is an infinite domain. Write D R = D {x R m : x < R}, DR = D\D R, R >. If for each fixed x D we have lim [Πy, x, σ{t y, nuy} Uy{T y, nπy, x, σ}]ds y =, R DR then [Πy, x, σ{t y, nuy} Uy{T y, nπy, x, σ}]ds y D = { Ux, x D, x D. Indeed, by fixed x D x < R we have [Πy, x, σ{t y, nuy} Uy{T y, nπy, x, σ}]ds y D = [Πy, x, σ{t y, nuy} Uy{T y, nπy, x, σ}]ds y D R + [Πy, x, σ{t y, nuy} Uy{T y, nπy, x, σ}]ds y = Ux + [Πy, x, σ{t y, nuy} Uy{T y, nπy, x, σ}]ds y. Letting R, we obtain.

11 II. Suppose that an infinite domain D lies in a shall: < y m < h, h = π ρ, ρ >, moreover, the boundary area satisfies condition 7. Let Ux AD satisfy a growth condition Uy + T y, nuy C expexp ρ y, ρ < ρ, y = y, y m D. In formula 4, 6 we take Kω = exp bchiρ ω h b chiρ ω h, where ω = i u + α + y m, h < ρ < ρ < ρ, < x m < h, b >, b > b cos ρ, α = y x. In order to obtain the formula for the domain D, we prove that lim [Πy, x, σ{t y, nuy} Uy{T y, nπy, x, σ}]ds y =. R Indeed, by 6 and for fixed x D we get [Πy, x, σ{t y, nuy} Uy{T y, nπy, x, σ}]ds y D R Πy, x, σ + T y, nπy, x, σ 3 Uy + T y, nuy ds y Cs n σ exp b cos ρ y m h chρ s b cos ρ y m h chρ s exp σy m x m b cos ρ x m h b cos ρ x m h expexp ρ y ds y C exp σx m σ

12 exp σh b cos ρ x m h b cos ρ x m h exp y n b cos ρy m h chρ y b cos ρ y m h chρ y + exp ρ y ds y. From this inequality, 3 easily follows. III. Suppose that D is aninfinite simply connected domain in R m, with smooth boundary D, lying in half-spaces y m >, defined by the equation y m = fy, y R m and satisfying the conditions grad fy C <. Let Kω = ω + x m k exp εω + ρ, ω = i u + α + y m, ε >, y m >, x n >, ρ,, k N. Denote by B ρ D = {Uy : Uy AD, Uy + T y, nuy exp y ρ, ρ > }. Let ρ < ρ <, then for Ux B ρ D condition 3 holds and hence formula is valid. Indeed from 4, 5, 6 for Kω = ω + x m k exp εω + ρ we find Im Kiθ+ym iθ+y m x m, where θ = u + α for m = n +, n and θ = α for m = n, n. Im Kiθ + y m iθ + y m x m = Im iθ + y m + x m k exp εiθ + y m + ρ iθ + y m x m = θ k + y m + x m ρ exp ε θ + y m + cos ρ ϕ [θ coskψ + ε θ + y m + ρ sin ρ ϕ y m x m sinkψ + ε θ + y m + ρ sin ρ ϕ], 4 where ϕ = arctan θ y m +, ψ = arctan θ y m + x m.

13 Now for m = n +, θ = u + α, s = α, we define C m φy, x n s n u ρ exp ε + s + y m + cos ρ ϕ = Kx m [s + u + y m + ] k u + s cos kψ + εu + s + y m + ρ sin ρ ϕ y m x m sin kψ + εu + s + y m + ρ sin ρ ϕ du u + s. Since π < arg ω < π, then π < ϕ < π π and < πρ < ρ ϕ < π therefore cos ρ ϕ δ >. Hence, the integral on the right hand side converges uniformly. Moreover, since function exp εu + α y m + ρ is -decreasing for u, and the integral converges by the Dirichlet criterion, then the mean value theorem is appeicable [], i.e., ξ, that C m φy, x n = Kx m s exp ε ξ + s + y n m + ρ ξ cos ρ ϕ [cos kψ + εu + s + y s + u + y m + k m + ρ y m x m kψ u + s sin + εu + s + y m + ρ sin ρ ϕ ]du. sin ρ ϕ Then q φy, x y q... ym qm Cεn Kx m exp ε ξ + s + y m + ρ cos ρ ϕ where C = const, q =,,, q q m = q. Then by fixed x D [Πy, x, σ{t y, nuy} Uy{T y, nπy, x, σ}]ds y 5 3

14 [ Πy, x, σ + T y, nπy, x, σ ] [ Uy + T y, nuy ] ds y Cεn+ Kx m n ξ + s + y m + ρ q q= exp ε ξ + s + y m + ρ cos ρ ϕ + y ρ ds y = Cεn+ exp εx m + ρ n ξ + s + y x m k m + ρ q q= exp y ρ ε ξ + s + y m + ρ Cεn+ exp εx m + ρ x m k Since y ρ cos ρ ϕ ds y n ξ + α + y m + ρ q q= exp y ρ ε α + y m + ρ m= cos ρ ϕ ds y. ρ y j x j + y m +, y, then for < ρ < ρ < j= and for some values c and ε >. we have exp y ρ ε α + y m + ρ cos ρ ϕ c exp ε y ρ 6 Hence, 3 is valid. We now prove condition 3 and, hence formula for m = n. By formula 5 and 4 we obtain n φy, x = [C m Kx m ] s Im Kαi + y m n αiα + y m x m n = [C m Kx m ] s + ym + x s n m exp ε s + y m + ρ cos ρ ϕ k [cos kψ + εs + y m + ρ sin ρ ϕ y m x m sin kψ + εs + y m + ρ s sin ρ ϕ ]. 4

15 Since cos ρ ϕ >, ρ < and k is a positive integer, then q φy, x y q... ym qm C[Kx m] exp ε s + y m + ρ cos ρ ϕ, q =,,, q = q q m. Hence, by 6, [Πy, x, σ{t y, nuy} Uy{T y, nπy, x, σ}]ds y D R [ Πy, x, σ + T y, nπy, x, σ ] [ Uy + T y, nuy ] ds y C[Kx m ] exp y ρ ε α + y m + ρ cos ρ ϕ ds y C[Kx m ] exp ε y ρ ds y. From this inequality we obtain 3 and. Suppose that D is a domain in half-space y m >, with boundary stretched to infinity and given by the equation y m = fy, where grad fy c <, y R m and f y <. Theorem.. Let Ux B ρ D satisfy the boundary condition Uy + y ds T y, nuy m y <, ds + y m y <. 7 D If ρ <, then the formula holds. Moreover D φy, x = π lny x + y + x y x + y x, m =, φy, x = m ω m [ r + ] m r m m 3, r = y x + y m x m, r = y x + y m + x m and Πy, x is determined by formula 6. 5

16 Proof. From 5 and 6 for k = we obtain an asymptotic estimate Πy, x = y m, y, y D T y, nπy, x = y m+, y, y D 8 Then from condition 7 it follows that the integral on the left-hand side of 3 uniformly converges relative to parameter ε. Assume ε = in formula for Kω. Then for m = we have φy, x = x π = x π = x y π udu Im[ ] i u + α + y + x i u + α + y x u + α Im[ it + y + x α it + y x ]dt Im[ t + y + x tdt t + y x = 4π ln [ y x + y + x ] π lnr = π lny x + y + x y x + y x. If m = n +, n, then φy, x = x m C m d n ds n = C m d n ds n = π = d n C m ds n m rω n Im i u + s + y m + x m i du u + s + y m x m u + s [ u + r + s r r r m r m, ] du r = y x, r = s + y m + x m. 6

17 If m = n, n, then φy, x = x m C m d n ds n = d n C m ds n = The theorem is proved. y m= Im iα + y m + x m αiα + y m x m [ r + r ] r n + m rω n r n r m r m... n + C m Theorem.. Let Ux B ρ D, where D is half-space y m >, and Uy + y ds T y, n Uy m y <, ds + y m y <. 9 y m= If ρ <, then for x m > the formula holds Ux = [Πy, x{t y, nuy} Uy{T y, nπy, x}]ds y, x D. y m= Proof. We set. Kω = ω + x m k exp εω + ρ, ω = i u + α + y m, ε >, y m, < ρ <, k N. We construct the function φy, x and matrix Πy, x by formula 4-6. Let D R = {x D : x < R}, D R = {x D : x = R}, DR = D\D R. Then by x D [Πy, x{t y, nuy} Uy{T y, nπy, x}]ds y = D D R [Πy, x{t y, nuy} Uy{T y, nπy, x}]ds y 7

18 + [Πy, x{t y, nuy} Uy{T y, nπy, x}]ds y = Ux + [Πy, x{t y, nuy} Uy{T y, nπy, x}]ds y. But the last integral converges uniformly to zero when R, by condition 8 and 9. The theorem is proved. 3 Regularization of the solution of the Cauchy problem for the multidimensional Lame system in infinite domains Suppose that D R m lies in a shall < y m < h, h = π, ρ >, and ρ its boundary consists of hyperplane y m = and of some smooth surface S, with given equation y m = fy, y R m, where < fy h and grad fy c. Suppose that Ux AD and Then formula, is valid, where Uy + T y, nuy M, y D. Kω = ω x m + h exp σω, ω = i u + α + y m, Kx m = h exp σx m, < x m < h. We denote U σ x = [Πy, x, σ{t y, nuy} Uy{T y, nπy, x, σ}]ds y, x D. S Theorem 3.. Let Ux AD satisfy the boundary condition. Then Ux U σ x MCρ, xσ n+ exp σx m, x D, where Cρ, x = Cρ y m= ds y r m+. 8

19 In the sequel we denote by Cρ any constant depending on p. It may different in diverse applications. Proof. From formula, we have Ux U σ x [Πy, x, σ{t y, nuy} Uy{T y, nπy, x, σ}] ds y y m= [Πy, x, σ + T y, nπy, x, σ ] [ Uy + T y, nuy ] ds y y m= M y m= [Πy, x, σ + T y, nπy, x, σ ] ds y. We estimate I = [Πy, x, σ + T y, nπy, x, σ ] ds y. y m= From the formula 4, 5 and 6: Im Kiθ + y m exp σy m = iθ + y m x m u + r r + 4hy m x m + 4h + u [ y m x m y m x m + h θ sin σθ θy m x m + h cos σθ]. where θ = u + α for m = n+, n and θ = α for m = n, n. Let for m = n +, n, C m φy, x, σ = h exp σy m x m dn ds n u + s + y m x m s + y m x m + u + 4hy m x m + 4h [ y m x m y m x m + h s u sin σ u + s u + sy m x m + h cos σ ] du u + s u + s. 9

20 Here the mean value theorem is applicable, is since and bounded, therefore ξ such that s+y m x m +u decreasing and C m φy, x, σ h expσy m σx m dn ds n ξ s + y m x m + ξ [ y m x m y m x m + h s u u + s + y m x m + h y m x m + h u + s + y m x m cos σ u + s ] du = n n!hexpσy m σx m dn s + y m x m + ξ n ds n ξ y m x m y m x m + h s u u + s + y m x m + h sin σ u + s u + s Cσn expσy m σx m r m sin σ u + s u + s y m x m + h u + s + y m x m cos σ u + sdu Hence, we have q φy, x, σ y q... ym qm Cσn+ expσy m σx m, q = q r m q m, q =,,. Πy, x, σ + T y, nπy, x, σ Cσn+ expσy m σx m r m+. Then I MCρ, xσ n+ exp σx m. For m = n the Theorem is proved similarly. Let us formulate a result that allows us to approximately solve the problem, if Uy and T y, nuy on S are replaced by their continuous approximations f δ y and g δ y, respectively, i.e., max S Uy f δ y + max T y, nuy g δ y < δ, < δ <. S

21 We define the functions U σδ x by U σδ x = [Πy, x, σg δ y f δ y{t y, nπy, x, σ}] ds y, x D. S The following theorem is valid. Theorem 3.. Let Ux AD satisfy the boundary condition. Then we have the estimate ] n+ Ux U σδ x M xm xm M n Cρ, xδ n [ln, x D, δ where σ = h ln M, Cρ, x = Cρ δ D ds y r m+. Proof. From formula,, and by assumption we obtains Ux U σδ x [Πy, x, σ{t y, nuy g δ y} y m= + Uy f δ y{t y, nπy, x, σ}] ds y + [Πy, x, σ{t y, nuy g δ y} S + Uy f δ y{t y, nπy, x, σ}] ds y MCρ, xσ n+ exp σx m +δ [ Πy, x, σ + T y, nπy, x, σ ] ds y S MCρ, xσ n+ exp σx m C ρσ n+ expσy m σx m +δ ds r m+ y S Cρ, xσ n+ M exp σx m + δ exphσ σx m = Cρ, xσ n+ exp σx m M + δ exp hσ.

22 Now, for the choice of the parameter M = δ exp hσ, we derive Ux U σδ x Cρ, xm xm n h n+ ln M n+ δ xm n. δ The theorem is proved. Corollary 3.3. The limit relations lim U σx = Ux, lim U σδ x = Ux σ δ hold uniformly on each compact subset of D. By slightly modifying the techniques, it is possible to generalize obtained results. Concerning the surface S let s assume that it is smooth and its area satisfies the growth condition exp{ b chρ y }ds y <, < ρ < ρ. 3 S Let Ux AD satisfy a growth condition Uy + T y, nuy C exp a cos ρ y m h exp ρ y, y D. 4 Suppose that given f δ y, g δ y in S, satisfies following inequality: f δ y + g δ y exp a cos ρ y m h exp ρ y, y S, 5 Uy f δ y + T y, nuy g δ y δ exp a cos ρ y m h exp ρ y, y S, where < δ <, < ρ < ρ, a. Put Kω = exp σω bchiρ ω h b chiρ ω h, where ω = i u + α + y m, < x m < h, b, b b cos ρ h + ε, ε >, α = y x, b = a exp ρ x, σ = h lnδ.

23 Under these conditions, we define U σδ x = [Πy, x, σg δ y f δ y{t y, nπy, x, σ}]ds y, x D. S Theorem 3.4. Let Ux AD satisfy conditions 4,5 and surface S satisfy condition 3, then Ux U σδ x 6 Cxδ xm h ln exp a cos ρ x m h δ exp ρ x, x D, where Cx = Cρ D r m exp b chρ α [ ] b exp cos ρ h exp ρ y x exp ρ y x ds y. Proof. By the assumptions of the theorem Ux U σδ x [ Πy, x, σ + T y, nπy, x, σ ] S [Uy f δ y + T y, nuy g δ y ]ds y + [ Πy, x, σ + T y, nπy, x, σ ] y m= [ Uy + T y, nπy, x, σ ]ds y 7 δ exp a cos ρ y m h exp ρ y S [ Πy, x, σ + T y, nπy, x, σ ]ds y h +C exp a cos ρ exp ρ y y m= [ Πy, x, σ + T y, nπy, x, σ ]ds y. 3

24 From 9 we obtain Πy, x, σ + T y, nπy, x, σ Cρ r m σn+ expσy m x m b exp cos ρ y m h exp ρ α b cos ρ y m h chρ α exp b cos ρ x m h. Then from 7 Ux U σδ x Cσ m+ δ exp a cos ρ y m h exp ρ y S exp b cos ρ x m h r exp σy m m x m exp b cos ρ y m h exp ρ α b chρ α ds y [ ] +Cσ n+ h exp a cos ρ exp ρ y y m= exp σx m + b cos ρ x m h exp b cos ρ h exp ρ α exp b chρ α r ds m y Cσ n+ δ exp σh exp σx m [ b exp cos ρ y m h exp ρ y x S b cos ρ y m h ] exp ρ α exp b cos ρ y m h chρ α [ exp b cos ρ x m h ] r ds m y + Cσ n+ exp σx m 4

25 [ b exp cos ρ h exp ρ y x b cos ρ ] h exp ρ y x [ exp b cos ρ x m h ] b chρ α r ds m y σ n+ δ exp σh + exp σx m exp b cos ρ x m h y m= D [ b exp cos ρ h exp ρ y x exp ρ y x ] exp[ b chρ α] r ds y. m If we now take σ = h ln δ and b = a exp ρ x, then we obtain 6. The theorem is proved. Corollary 3.5. The limit relation lim U σδx = Ux δ holds uniformly on each compact subset of D. In conclusion, the authors wishes to thank Professor M.M.Lavrent ev and Sh.Ya.Yarmukhamedov for posing the problem and for discussion in the course of solution. References [] L.A.Aizenberg and N.N.Tarkhanov. An abstract Carleman formula. Dokl.Acad.Nauk SSSR [Soviet Math.Dokl.], 98, No.6, [] G.M. Fichtenholz. Course of differential and integral calculus.[in Russian]. Moscow.969. Vol. p. 6. [3] T.I.Ishankulov and O.I.Makhmudov. The Cauchy problem for a system of thermoplasticity equations in a space. Math. zhametki. [Math.Notes], 64, No.,

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