Regularization of the Cauchy Problem for the System of Elasticity Theory in R m

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1 Regularization of the Cauchy Problem for the System of Elasticity Theory in R m O. I. Makhmudov, I. E. Niyozov Samarkand State University, Departament of Mechanics and Mathematics, University Boulevard 5, Samarkand, Uzbekinstan Abstract In this paper we consider the regularization of the Cauchy problem for a system of second order differential equations with constant coefficients. Key words: the Cauchy problem, Lame system, elliptic system, ill-posed problem, Carleman matrix, regularization, Laplace equation. Introduction As is well known, the Cauchy problem for elliptic equations is ill-posed, the solution of the problem is unique, but unstable (Hadamard s example). For ill-posed problems, one does not prove existence theorems, the existence is assumed a priori. Moreover, the solution is assumed to belong to a given subset of a function space, usually a compact one [4]. The uniqueness of the solution follows from the general Holmgren theorem [9]. Let x = (x,..., x m ) and y = (y,..., y m ) be points in R m, D ρ be a bounded simply connected domain in R m whose boundary consists of a cone surface Σ : α = τy m, α = y y m, τ = tg π ρ, y m > 0, ρ > and a smooth surface S, lying in the cone. In the domain D ρ, consider the system of elasticity theory µ U(x) + (λ + µ)grad divu(x) = 0; olimjan@yahoo.com, makhmudovo@rambler.ru iqbol@samdu.uz, iqboln@rambler.ru

2 here U = (U,..., U m ) is the displacement vector, is the Laplace operator, λ and µ are the Lame constants. For brevity, it is convenient to use matrix-valued notation. Let us introduce the matrix differential operator A( x ) = A ij ( x ) m m, where A ij ( x ) = δ ij µ + (λ + µ). x i x j Then the elliptic system can be written in matrix form A( x )U(x) = 0. () Statement of the problem. Assume the Cauchy data of a solution U are given on S, U(y) = f(y), y S, T ( y, n(y))u(y) = g(y), y S, () where f = (f,..., f m ) and g = (g,..., g m ) are prescribed continuous vector functions on S, T ( y, n(y)) is the strain operator, i.e., T ( y, n(y)) = T ij ( y, n(y)) m m = λn i + µn j + µδ ij y j y i n. m m δ ij is the Kronecker delta, and n(y) = (n (y),..., n m (y)) is the unit normal vector to the surface S at a point y. It is required to determine the function U(y) in D, i.e., find an analytic continuation of the solution of the system of equations in a domain from the values of f and g on a smooth part S of the boundary. On establishing uniqueness in theoretical studies of ill - posed problems, one comes across important questions concerning the derivation of estimates of conditional stability and the construction of regularizing operators. Suppose that, instead of f(y) and g(y), we are given their approximations f δ (y) and g δ (y) with accuracy δ, 0 < δ < (in the metric of C) which do not necessarily belong to the class of solutions. In this paper, we construct a family of functions U(x, f δ, g δ ) = U σδ (x) depending on the parameter σ and prove that under certain conditions and a special choice of the parameter σ(δ) the family U σδ (x) converges in the usual sense to the solution U(x) of problem (),(), as δ 0 Following A.N.Tikhonov, U σδ (x) is called a regularized solution of the problem. A regularized solution determines a stable method of approximate solution of problem [3]. Using results of [4],[4] concerning the Cauchy problem for the Laplace equation, we succeeded in constructing a Carleman matrix in explicit form and on in constructing a

3 regularized solution of the Cauchy problem for the system (). Since we refer to explicit formulas, it follows that the construction of the Carleman matrix in terms of elementary and special functions is of considerable interest. For m =, 3 the problem under consideration coincides with the Cauchy problem for the system of elasticity theory describing statics of an isotropic elastic medium. In these cases problem (),() was studied for special classes of domains in [5], [6], [7], [8]. Further, the Cauchy problem for systems describing steady-state elastic vibrations, for systems of thermoelasticity, and for systems of Navier-Stokes was studied in [5], [], [0], [3]. Earlier, is was proved in [], [] that the Carleman matrix exists in any Cauchy problem for solutions of elliptic systems whenever the Cauchy data are given on a boundary set of positive measure..construction of the matrix of fundamental solution for the system of elasticity of a special form Definition. Γ(y, x) = Γ ij (y, x) m m, is called the matrix of fundamental solutions of system (), where Γ ij (y, x) = µ(λ + µ) ((λ + 3µ)δ ijq(y, x) (λ + µ)(y j x j ) x i q(y, x)), i, j =,..., m, q(y, x) = and ω m is the area of unit sphere in R m. { ( m)ω m π, m > y x m ln y x, m =, The matrix Γ(y, x) is symmetric and its columns and rows satisfy equation () at an arbitrary point x R m, except y = x. Thus, we have A( x )Γ(y, x) = 0, y x. Developing Lavrent ev s idea concerning the notion of Carleman function of the Cauchy problem for the Laplace equation [4], we introduce the following notion. Definition. By a Carleman matrix of problem (),() we mean an (m m) matrix Π(y, x, σ) satisfying the following two conditions: ) Π(y, x, σ) = Γ(y, x) + G(y, x, σ), where σ is a positive numerical parameter and, with respect to the variable y, the matrix G(y, x, σ) satisfies system () everywhere in the domain D. ) The relation holds D\S ( Π(y, x, σ) + T ( y, n)π(y, x, σ) )ds y ε(σ), 3

4 where ε(σ) 0, as σ, uniformly in 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 a vector U = (U,..., U m ). i= Definition 3. A vector function U(y) = (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 show the formula of Somilian-Bettis [] below. Theorem. Any regular solution U(x) of equation () in the domain D is represented by the formula U(x) = (Γ(y, x){t ( y, n)u(y)} {T ( y, n)γ(y, x)} U(y))ds y, x D. (3) D Here A is conjugate to A. Suppose that a Carleman matrix Π(y, x, σ) of the problem (),() exists. Then for the regular functions v(y) and u(y) the following holds i,j= [v(y){a( y )U(y)} {A( y )v(y)} U(y)]dy = = [v(y){t ( y, n)u(y)} {T ( y, n)v(y)}u(y)]ds y. Substituting v(y) = G(y, x, σ) and u(y) = U(y), a regular solution system (), into the above equality, we get [G(y, x, σ){a( y )U(y)} {A( y )G(y, x, σ)} U(y)]dy = 0. (4) Adding (3) and (4) gives the following theorem. Theorem. Any regular solution U(x) of equation () in the domain D ρ is represented by the formula U(x) = (Π(y, x, σ){t ( y, n)u(y)} {T ( y, n)π(y, x, σ)} U(y))ds y, x D ρ, (5) where Π(y, x, σ) is a Carleman matrix. Suppose that K(ω), ω = u + iv (u, v are real), is an entire function taking real values on the real axis and satisfying the conditions K(u) 0, sup v p K (p) (ω) = M(p, u) <, p = 0,..., m, u R. v 4

5 Let s = α = (y x ) (y m x m ). For α > 0, we define the function Φ(y, x) by the following relations: if m =, then πk(x )Φ(y, x) = if m = n +, n, then C m K(x m )Φ(y, x) = n s n 0 Im[ K(i u + α + y ) i udu ] u + α + y x u + α, (6) where C m = ( ) n n (m )πω m (n )!, if m = n, n, then where C m = ( ) n (n )!(m )ω m. 0 Im[ K(i u + α + y m ) i du ] u + α + y m x m u + α ; (7) C m K(x m )Φ(y, x) = n s n Im K(αi + y m) α(α + y m x m ), (8) Lemma. The function Φ(y, x) can be expressed as Φ(y, x) = π ln r + g (y, x), m =, r = y x, Φ(y, x) = r m ω m (m ) + g m(y, x), m 3, r = y x, where g m (y, x), m is a function defined for all values of y, x and harmonic in the variable y in all of R m. With the help of function Φ(y, x) we construct a matrix: Π(y, x) = Π ij (y, x) m m = λ + 3µ µ(λ + µ) δ ijφ(y, x) λ + µ µ(λ + µ) (y j x j ) Φ(y, x) y i, m m i, j =,,..., m. (9). The solution of problems (), () in domain D ρ I. Let x 0 = (0,..., 0, x m ) D ρ. We adopt the notation β = τy m α 0, = τx m α 0, α 0 = x x m, r = x y, s = α = (y x ) (y m x m ), w = iτ u + α + β, w 0 = iτα + β. We now construct Carleman s matrix for the problem (), () for the domain D ρ. The Carleman matrix is explicitly expressed by Mittag-Löffler s a entire function. It is defined by series [] E ρ (w) = n=0 Γ w n ( ), ρ > 0, E (w) = exp w, + n ρ 5

6 where Γ( ) is the Euler function. Denote by = (, θ), 0 < θ < π, ρ > the contour in the complex plane w, running ρ in the direction nondecreasing argw and consisting of the following part s. )ray argw = θ, w, )arc θ argw θ of the circle w =, 3)ray argw = θ, w. The contour divides complex plane into two parts: D and D + lying on the left and on the right from, respectively. Suppose that π < θ < π, ρ >. Then the formula holds ρ ρ where Ψ ρ (w) = E ρ (w) = exp w ρ + Ψ ρ (w), w D + E ρ (w) = Ψ ρ (w), E ρ(w) = Ψ ρ(w), w D, (0) ρ πi ReΨ ρ (w) = Ψ ρ(w) + Ψ ρ (w) ImΨ ρ (w) = Ψ ρ(w) Ψ ρ (w) i exp ζ ρ ζ w dζ, Ψ ρ(w) = ρ πi ImΨ ρ(w) Imw = ρ πi = ρ πi = ρimw πi exp ζ ρ dζ. () (ζ w) exp ζ ρ (ζ Rew) (ζ w)(ζ w) dζ, exp ζ ρ (ζ Rew) (ζ w) (ζ w) dζ. exp ζ ρ dζ, () (ζ w)(ζ w) In what follows, we take θ = π ρ + ε, ρ >, ε > 0. It is clear that if π ρ +ε argw π, then w D and E ρ (w) = Ψ ρ (w). Set E k,q (w) = ρ πi ζ q exp ζ ρ dζ, k =,,..., q = 0,,,.... (ζ w) k (ζ w) k If π ρ + ε argw π, then the inequalities are valid where M, M, M 3 are constants. and E ρ (w) M + w, E ρ(w) M + w, E k,q (w) M 3, k =,,..., (3) + w k Suppose that in formula (0) θ = π ρ + ε < π ρ, ρ >. Then E ρ(w) = Ψ ρ (w), cos ρθ < 0 ζ q exp(cos ρθ ζ q ) dζ <, q = 0,,,.... (4) 6

7 In this case for sufficiently large w (w D +, w D ), we have Now from (0) and for large w we obtain From this it follows that From (),(5) and for large w, we obtain or min ζ ζ w = w sin ε, min ζ ζ w = w sin ε. (5) ζ w = w + ζ w(ζ w), ζ w = w + ζ w(ζ w), (6) E ρ(w) Γ ( ) ρ w ρ π sin ε w ζ exp [cos ρθ ζ ρ ] dζ const w, ( Γ ) ρ = ρ πi E ρ (w) M + w. exp (ζ ρ ) dζ. (ζ w) = w ζ w (ζ w) + ζ w (ζ w) E ρ(w) Γ ( ) ρ w const w 3 E ρ(w) = M + w. For k =,,... we have from (6) [ ] [ ] ( ) k (ζ w) k (ζ w) = ζ k ( ) k ζ k = k w k w k (ζ w) k w k w k (ζ w) k = w k k w k+ ζ w From this for large w, (4) and (5) we get E k,q(w) Γ ( ) ρ w k const or E k,q(w) = w k+ M 3, k =,, w k 7

8 then Therefore, since (ζ w)(ζ w) = ζ ζ(y m x m ) + u + α + (y m x m ), α = s, n s n (ζ w)(ζ w) = ( )n (n )! (ζ w) n (ζ w) n. Now from () we obtain d n ds ReE ρ(w) = ( )n (n )!ρ n πi Then from () we have d n ImE ρ (w) ds n u + α = ( )n (n )!ρ πi Now for σ > 0, we set in formulas (6)-(9) Then, for ρ > we obtain where ϕ σ (y, x) is defined as follows: if m =, then if m = n +, n, then d n ds ReE ρ(w) n const r + w d n ImE ρ (w) const r ds n u + α + w. (ζ (y m x m )) exp ζ ρ (ζ w) n (ζ w) n dζ, exp ζ ρ (ζ w) n (ζ w) n dζ, K(w) = E ρ (σ ρ w), K(xm ) = E ρ (σ ρ ). (7) Φ(y, x) = Φ σ (y, x) = ϕ σ(y, x) c m E ρ (σ ρ ), y x, ϕ σ (y, x) = ϕ σ (y, x) = dn ds n 0 0 E ρ (σ ρ w) Im i udu u + α + y x u + α ; E ρ (σ ρ w) Im i udu u + α + y m x m u + α, y x; if m = n, n, then ρ (σ ρ w ) E ϕ σ (y, x) = dn Im, y x. dsn α(iα + y m x m We now define the matrix Π(y, x, σ) by formula (9) for Φ(y, x) = Φ σ (y, x). In the work [4] there is proved 8

9 Lemma. The function Φ σ (y, x) can be expressed as Φ σ (y, x) = π ln r + g (y, x, σ), m =, r = y x, Φ σ (y, x) = r m ω m (m ) + g m(y, x, σ), m 3, r = y x, where g m (y, x, σ), m is a function defined for all y, x and harmonic in the variable y in all of R m. Using Lemma, we obtain. Theorem 3. The matrix Π(y, x, σ) given by (7)-(9) is a Carleman matrix for problem (), (). We first consider some properties of function Φ σ (y, x) I. Let m = n +, n, x D ρ, y x, σ σ 0 > 0, then ) for β α the following inequality holds: Φ σ (y, x) C (ρ) σm r m exp( σρ ), Φ σ n C (ρ) σm r m exp( σρ ), y, Φ σ x i n C 3(ρ) σm+ r m exp( σρ ), i =,..., m, (8) ) for β > α the following inequalities hold: Φ σ (y, x) C 4 (ρ) σm r m exp( σρ + σreω0), ρ Φ σ n C 5(ρ) σm r m exp( σρ + σreω0), ρ y, Φ σ x i n C 6(ρ) σm+ r m exp( σρ + σreω0), ρ i =,..., m. (9) II. Let m = n, n, x D ρ, x y, σ σ 0 > 0, then ) for β α the following inequalities hold: Φ σ (y, x) C (ρ) σm 3 r m exp( σρ ), Φ σ n C (ρ) σm r m exp( σρ ), y, Φ σ x i n C 3 (ρ) σm+ r m exp( σρ ), y, i =,..., m, (0) ) for β > α the following inequalities hold: Φ σ (y, x) C 4 (ρ) σm r m exp( σρ + σreω ρ 0), 9

10 Φ σ n C 5 (ρ) σm r m exp( σρ + σreω0), ρ y, Φ σ x i n C 6 (ρ) σm+ r m exp( σρ + σreω0), ρ y, i =,..., m. () III. Let m =, x D ρ, x y, σ σ 0 > 0, then ) if β α, then ) if β > α, then Φ σ (y, x) C 7 (ρ)e (σ + r ρ )ln, r Φ σ y i C 8(ρ) E ρ (σ ρ ), () r Φ σ (y, x) C 7 (ρ)e (σ + r ρ )(ln ) exp(σreω r 0), ρ Φ σ y i C 8 (ρ)eρ (σ ρ ) exp(σreωρ 0). (3) Here all coefficients C i (ρ) and C i (ρ), i =,..., 8, depend on ρ. where Proof Theorem 3. From the definition of Π(y, x, σ) and Lemma, we have Π(y, x, σ) = Γ(y, x) + G(y, x, σ), G(y, x, σ) = G kj (y, x, σ) m m = = λ + 3µ µ(λ + µ) δ kjg m (y, x, σ) λ + µ µ(λ + µ) (y j x j ) g m (y, x, σ) y i. m m Prove that A( y )G(y, x, σ) = 0. Indeed, since y g m (y, x, σ) = 0, y = m jth column G j (y, x, σ) : divg j (y, x, σ) = µ(λ + µ) g m (y, x, σ), y j then for the kth components of A( y )G j (y, x, σ) we obtain k= y k and for the m λ + 3µ A( y ) ki G ij (y, x, σ) = µ y [ µ(λ + µ) δ kjg m (y, x, σ) λ + µ µ(λ + µ) (y j x j ) g m (y, x, σ)]+ y k i= +(λ + µ) divg j (y, x, σ) = λ + µ y k µ(λ + µ) y j g m (y, x, σ) + λ + µ µ(λ + µ) y j g m (y, x, σ) = 0 Therefore, each column of the matrix G(y, x, σ) satisfies to system () in the variable y everywhere on R m. The second condition of Carleman s matrix follows from inequalities (8)-(3). theorem proved. 0 The

11 For fixed x D ρ we denote by S the part of S, where β α. It x = x 0 = (0,..., 0, x m ) D ρ, then S = S. In the point (0,..., 0) D ρ, suppose that Let U σ (y) = U U (0) = (0), n y m Φ σ (0, x) n = Φ σ(0, x) y m. S [Π(y, x, σ){t ( y, n)u(y)} {T ( y, n)π(y, x, σ)} U(y)]ds y, x D ρ. (4) Theorem 4. Let U(x) be a regular solution of system () in D ρ, such that Then, U(y) + T ( y, n)u(y) M, y Σ. (5) ) if m = n +, n, and for the x D ρ, σ σ 0 > 0, the following estimate is valid: U(x) U σ (x) MC (x)σ m+ exp( σ ρ ). ) In case m = n, n, x D ρ, σ σ 0 > 0, the following estimate is valid where C k (ρ) is a constant depending on ρ. Proof. From formula (5) U(x) = + U(x) U σ (x) MC (x)σ m exp( σ ρ ), C k (x) = C k (ρ) ds y, k =,, rm S [Π(y, x, σ){t ( y, n)u(y)} {T ( y, n)π(y, x, σ)} U(y)]ds y + \S [Π(y, x, σ){t ( y, n)u(y)} {T ( y, n)π(y, x, σ)} U(y)]ds y, x D ρ, therefore, (4) implies U(x) U σ (x) \S [Π(y, x, σ){t ( y, n)u(y)} {T ( y, n)π(y, x, σ)} U(y)]ds y \S [ Π(y, x, σ) + T ( y, n)π(y, x, σ) ] [ T ( y, n)π(y, x, σ) + U(y) ] ds y. Therefore for β α we obtain from inequalities (8)-(3), and condition (5) for m = n +, n U(x) U σ (x) MC (ρ)σ m+ exp( σ ρ ) ds y r m,

12 and for m = n, n we obtain U(x) U σ (x) MC (ρ)σ m exp( σ ρ ) ds y r m. The theorem proved. Now we write out a result that allows us to calculate U(x) approximately if, instead of U(y) and T ( y, n)u(y) their continuous approximations f δ (y) and g δ (y) are given on the surface S: max S U(y) f δ (y) + max T ( y, n)u(y) g δ (y) δ, 0 < δ <. (6) S We define a function U σδ (x) by setting U σδ (x) = [Π(y, x, σ)g δ (y) {T ( y, n)π(y, x, σ)} f δ (y)]ds y, x D ρ, (7) s where σ = ln M, R ρ δ Rρ = max y S Reωρ 0. Then the following theorem holds. Then, Theorem 5. Let U(x) be a regular solution of system () in D ρ such that U(y) + T ( y, n)u(y) M, y. ) if m = n +, n, the following estimate is valid: U(x) U σδ (x) C (x)δ ( R )ρ ( ln M δ ) if m = n, n, the following estimate is valid: where U(x) U σδ (x) C (x)δ ( R )ρ ( ln M δ C k (x) = C k (ρ) ds y, k =,. rm ) m+, Proof. From formula (5) and (7) we have U(x) U σδ (x) = [Π(y, x, σ){t ( y, n)u(y)} {T ( y, n)π(y, x, σ)} U(y)]ds y + \S + [Π(y, x, σ){t ( y, n)u(y) g δ (y)} + {T ( y, n)π(y, x, σ)} (U(y) f δ (y))]ds y = I + I. S By Theorem 4 for m = n +, n, I = MC (ρ)σ m+ exp( σ ρ ds y ) r, m ) m,

13 and for the m = n, n I = MC (ρ)σ m exp( σ ρ ) ds y r m. Now consider I : I = S ( Π(y, x, σ) + T ( y, n)π(y, x, σ) ) ( T ( y, n)u(y) g δ (y) + U(y) f δ (y) ) ds y. By Lemma and condition (8) we obtain for m = n +, n I = C (ρ)σ m+ δ exp( σ ρ + σrew0) ρ ds y r m and for m = n, n, I = C (ρ)σ m δ exp( σ ρ + σrew0) ρ ds y r m. Therefore, from we obtain the desired result. Corollary. The limit relation hold uniformly on any compact set from D ρ. σ = R ρ lnm δ, Rρ = max y S Reωρ 0. lim U σ(x) = U(x), lim U σδ (x) = U(x) σ δ 0 References [] M.M. Dzharbashyan. Integral Transformations and Representations of Functions in a Complex Domain. [in Russian], Nauka, Moscow 966. [] T.I. Ishankulov and I.E. Niyozov. Regularization of solution of the Cauchy problem for Navier-Stokes. Uzb.Math.J., No., 35-4 (997). [3] V.D.Kupradze, T.V.Burchuladze, T.G.Gegeliya, ot.ab. Three-Dimensional Problems of the Mathematical Theory of Elasticity, etc. [in Russian], Nauka, Moscow, 976. [4] M.M.Lavrent ev. Some Ill-Posed Problems of Mathematical Physics [in Russian], Computer Center of the Siberian Division of the Russian Academy of Sciences, Novosibirck (96) 9p. 3

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