Weak Carleman estimates with two large parameters for second order operators and applications to elasticity with residual stress

Transcription

1 Weak Carleman estimates with two large parameters for second order operators and applications to elasticity with residual stress V. Isakov and N. Kim October 25, Introduction We consider the general partial differential operator of second order A = j,k=1 a jk j k + b j j + c in a bounded domain of the space R n with the real-valued coefficients a jk C 1 ( ), b j, c L (). The principal symbol of this operator is A(x; ζ) = a jk (x)ζ j ζ k. (1) We use the following convention and notations. Sums are over repeated indices j, k, l, m = 1,..., n. Let = ( 1,..., n ), D = i, α = (α 1,..., α n ) is a multi-index with integer components, ζ α = ζ α1 1 ζn αn, D α and α are defined similarly. ν is the outward normal to the boundary of a domain. ε = {ψ(x) > ε}. We recall that u (k) () = ( α u 2 ) 1 2 α k is the norm in the Sobolev space H (k) () and 2 = (0) is the L 2 -norm. Let K be a positive constant. A function ψ is called K-pseudo-convex on with respect to A if ψ C 2 ( ), A(x, ψ(x)) 0, x, and j k ψ(x) A A (x; ξ) + ( A A k k A 2 A ) j ψ(x, ξ) K ξ 2 ζ j ζ k ζ k ζ j ζ j ζ k (2) for some positive constant K, for any ξ R n and any point x of provided A(x; ξ) = 0, A ζ j (x, ξ) j ψ(x) = 0. (3) 1

2 We will assume that the coefficients of A admit the following bound a jk 2 () + b j () + c () M. Let C be generic constants (different at different places) depending only on M, on K, on the function ψ, on ε 0, on C 2 ()-norms of the coefficients ρ, λ, µ, r jk of the elasticity system, and on the domain. Any additional dependence will be indicated. We will use the weight function and let σ = γτϕ. Now we will state main results of this paper. ϕ = e γψ (4) Theorem 1.1. Let ψ be K pseudo-convex with respect to A in. Then there are constants C, C 0 (γ) such that σ 3 2 α e 2τϕ α u 2 C e 2τϕ Au 2 (5) for all u C 2 0(), α 1, C < γ, and C 0 (γ) < τ. In [6] this result (for a jk C ) with constants depending on A was stated without proof, in [7] there are proofs for isotropic hyperbolic equations, and in [17] there are proofs with constants depending on A. In [22] it is shown that ψ(x, t) = x a 2 θ 2 t 2 is pseudo-convex with respect to A if the speed of propagation is monotone in a certain direction. According to [23], [25], ψ(x, t) = d 2 (x, a) θ 2 t 2 (d is the distance in the Riemannian metric determined by the elliptic part of A) is pseudo-convex if sectional curvatures are nonpositive. In [1], [7], [15] Carleman estimates with second large parameter under additional assumptions were used to obtain uniqueness of the continuation and controllability results for thermoelasticity systems. Now we state a weak form of Theorem 1.1. Theorem 1.2. Let A be a linear partial differential operator of second order with the principal coefficients in C 2 ( ) and with the coefficients of the first order derivatives in C 1 ( ). Let ψ be a K pseudo-convex C 3 ( )-function with respect to A in. Let Av = f 0 + n j=1 jf j in. Then there is C 2 > 0 such that σe 2τϕ v 2 C e 2τϕ ( 1 σ 2 f provided γ > C, τ > C(γ). fj 2 ) for all v H0 2 () (6) j=1 2

3 As an important application we consider an elasticity system with residual stress R [24]. In Theorems we let x R 3 and (x, t) R 4. The residual stress is modeled by a symmetric second-rank tensor R(x) = (r jk (x)) 3 j,k=1 C 2 () which is divergence free R = 0. Let u(x, t) = (u 1, u 2, u 3 ) : R 3 be the displacement vector in. We introduce the operator of the linear elasticity with the residual stress A R u = ρ 2 t u µ u (λ + µ) (divu) λdivu 2ɛ(u) µ div(( u)r), (7) where ρ C 1 ( ), λ, µ C 2 ( ) are density and Lame parameters depending only on x, ɛ(u) = ( 1 2 ( iu j + j u i )). Let (µ; R) = t 2 µδ jk +r jk jk ρ j k and σ = τγϕ. Theorem 1.3. Let ψ be K- pseudo-convex with respect to (µ; R); (λ+2µ; R) in and let λ 2 () + µ 2 () + r jk 2 () M. Then there are constants C, C 0 (γ) such that σ( u 2 + divu 2 + curlu 2 )e 2τϕ C A R u 2 e 2τϕ (8) for all u H 2 0 (), C < γ, C 0 < τ. A more complicated version of this estimate (with additional spatial derivatives on the right hand) was obtained in [19] when R is small and in [17], [18] without any smallness condition. Let us consider the following Cauchy problem A R u = f in, u = g 0, ν u = g 1 on Γ, (9) where Γ C 3. Let δ = {ψ > δ}. The Carleman estimate of Theorem 1.2 by standard argument ([16], section 3.2) implies the following conditional Hölder stability estimate for (9) in (δ) (and hence uniqueness in (0)). Theorem 1.4. Suppose that all coefficients λ, µ, ρ, R are in C 2 ( ). Let ψ be K pseudo-convex with respect to (µ; R); (λ + 2µ; R) in. Assume that 0 Γ. Then there exist C = C(δ), κ = κ(δ) (0, 1) such that for a solution u H 2 () to (9) one has u (0) ( δ ) + x u (0) ( δ ) C(F + M 1 κ 1 F κ ), (10) where F = f (0) ( 0 ) + g 0 ( 3 2 ) (Γ) + g 1 ( 1 2 ) (Γ), M 1 = u (1) (). Now we state results about identification of residual stress from additional boundary data. In Theorem 1.5 = G ( T, T ) where G is a bounded domain in R 3 with C 8 -boundary and Γ G ( T, T ). Let u(; 1), u(; 2) be solutions to A R u = 0 in, 3

4 u = u 0, t u = u 1 on G {0}, u = g 0 on G ( T, T ), (11) corresponding to sets of coefficients R(; 1) and R(; 2), respectively. We will assume that u 0 H 9 (G), u 1 H 8 (G), g 0 C 9 ( G [ T, T ]). Then our conditions on the coefficients of residual stress systems and on G by known energy estimates (like in [12], [20]) imply that α x β t u () C, when α 2, β 5. (12) We can consider the boundary stress data as measurements (observations). We introduce the norm of the differences of the lateral Cauchy data F c = 4 β=2 β t ν (u(; 2) u(; 1)) ( 1 2 ) (Γ). (13) By examining the equation (7), we can see that since residual stress tensor is divergence free it appears in the equation without first derivatives. It turns out that a single set of Cauchy data is sufficient to recover the symmetric (variable) matrix R. To guarantee the uniqueness, we impose some non-degeneracy condition on the initial data (u 0, u 1 ). Let ( ) 2 M = 1 u u u 0 2u u 0 3u 2 0 1u u u 1 2u u 1 3u 2. (14) 1 Note that M is a 6 6 matrix-valued function. We assume that det M > ε 0 > 0 on. (15) One can check that, for example, u 0 (x) = (x 2 1, x 2 2, x 2 3) and u 1 (x) = (x 2 x 3, x 1 x 3, x 1 x 2 ) satisfy (15) with ε 0 = 2 6. Now we state the Hölder type estimate of determining coefficients in (ε) defined as {ψ > ε}. Theorem 1.5. Assume that ψ is K-pseudo-convex with respect to (µ; R(; 2)), (λ + 2µ; R(; 2)) in. Let the initial data (u 0, u 1 ), satisfy (15). Assume that (0) Γ. Then there exist constants C = C(δ), κ = κ(δ) (0, 1) such that R(; 2) R(; 1) (0) ((δ)) CF κ. (16) A weaker result with C depending on R(; 2) is derived in [18]. If Γ is the whole lateral boundary and T is sufficiently large, then under more restrictive conditions a much stronger (and in a certain sense best possible) Lipschitz stability estimate holds [17], [18]. Carleman estimates were introduced by Carleman in 1939 to demonstrate uniqueness in the Cauchy problem for a system of first order in R 2 with non analytic coefficients. His idea turned out to be very fruitful and until now it dominates the field. In s Carleman type estimates and uniqueness of the continuation theorems have been obtained for wide classes of partial differential equations including general elliptic and parabolic equations of second order and some hyperbolic equations of second order. For accounts on these results we 4

5 refer to books [10], [16]. While still there are challenges for scalar partial differential operators, in many cases results are quite complete. The situation with systems is quite different. A useful concept of pseudo-convexity is not available for systems and Carleman estimates are at present obtained only in particular cases. A general result of Calderon of 1958 is applicable only to some elliptic systems of first order. Only recently there was a progress for classical isotropic dynamical Maxwell s and elasticity systems [8]. This progress was achieved by using principal diagonalization of these systems and Carleman estimates for scalar hyperbolic equations. An important system of thermoelasticity can not be prinicipally diagonalized, however it has triangular structure which allows to obtain Carleman estimates and uniqueness of the continuation by exploiting Carleman estimates for second order scalar operators with two large parameters [1], [7], [15]. So far Carleman estimates with two large parameters were obtained only for elliptic, parabolic, and isotropic hyperbolic operators of second order [7]. Carleman estimates are also very useful in control theory (controllability and stabilization for initial boundary value problems) and inverse problems [16]. In particular, they were a main tool in the first proof of uniqueness and stability of all three elastic parameters in dynamical Lame system from two sets of boundary data [12]. Until now anisotropic systems can be studied only in very special cases, like small scalar perturbation of classical elasticity (elasticity with residual stress) in [19], [20], [21] where there are Carleman estimates, uniqueness and stability of the continuation and of the identification of elastic coefficients for such system. In the recent papers [17], [18] the Carleman estimates with two large parameters have been obtained for general second order equations (including as a particular case operators of hyperbolic type). Applying these estimates, in [17], [18] we obtained Carleman estimates, uniqueness and stability of the continuation results and of the identification of residual stress R without smallness assumption of R, i.e. globally. We have to assume pseudo-convexity with respect to two scalar operators involving residual stress. Constants in the estimates in [17], [18] depend on partial differential operators. In this paper by improving results of [17], [18], [20], [21] we show that constants in Carleman estimates depend only on some constants in the pseudoconvexity conditions and bounds on the coefficients of differential operators. We give explicit conditions of pseudo-convexity with respect to the Euclidean metrics. Moreover by using the methods and results of [12] we obtain for general scalar operators Carleman estimates with two large parameters in Sobolev spaces of negative order and using the methods of [20], [21] we derive most natural Carleman estimates for elasticity system with residual stress. In [17] such estimates are obtained with additional spatial derivatives. Accordingly, in identification problems we can derive from them stability estimates for elastic coefficients which depend only on a priori bounds on these coefficients and pseudo-convexity constant and not coefficients (as in [17], [18]) to be determined. We stated our basic results in section 1. In section 2 we discuss pseudoconvexity conditions. In section 3 we give proofs of crucial symbols bounds and of fundamental Carleman estimate of Theorem 1.1. In section 4 we prove estimates of Theorem 1.2 in negative norms. A crucial idea of proofs is to localize 5

6 estimates and to freeze coefficients in an appropriate way. This substantially facilitates use of the Fourier analysis. In section 5 we derive from Theorem 1.2 a Carleman estimate for a system of elasticity with residual stress which demonstrates the use of two large parameters. In section 6 we prove Hölder type stability estimate for the lateral Cauchy problem for A R. In section 7 we show uniqueness and stability of identification of 6 functions defining residual stress from one set of special boundary measurements of u. A crucial assumption is that the initial data are in certain sense independent (which excludes practically important zero initial conditions). This assumption enables to use a modification of the method of Bukhgeim and Klibanov of 1981 proposed and used by Imanuvilov and Yamamoto in 2002 for scalar equations. This modification was essential in handling inverse problems for the isotropic elasticity in [12]. We observe that, if the initial data are zero, currently there are no uniqueness of identification results for residual stress, moreover, most likely, these results are not true. Indeed, as known, one can not uniquely identify general (anisotropic) hyperbolic equations from all possible boundary measurements. In addition, even for the isotropic and scalar case while there are complete uniqueness of the identification results obtained by method of boundary control of Belishev (see for example, [16]), stability of identification is very weak (of logarithmic type). This weak stability leads to poor results of numerical solution of inverse problems. Special nonzero initial data and pseudo-convexity assumptions imply much better Hölder and in some cases even best possible Lipschitz stability. Better stability promises a substantial improvement in numerical solution which is crucial for applications. The condition (15) is quite restrictive for many applications. However, recently new measurements (in particular by using interior sources) become available and feasible in important geophysical and medical applications (e.g. [2]). The non degeneration conditions, like (15), are getting important since they explain what positions and intensities of interior sources are needed for stable reconstruction of coefficients of partial differential equations modelling corresponding physical phenomenae. 2 Pseudo-convexity condition It is not obvious and easy to find functions ψ which are pseudo convex with respect to a general anisotropic operator, in particular, to the hyperbolic operator A = t 2 n j,k=1 a jk j k. In the isotropic case explicit and verifiable conditions for ψ(x, t) = x β 2 θ 2 t 2 were found by Isakov in 1980 and their simplifications are given in [16], section 3.4. In general anisotropic case Khaidarov [22] showed that under certain conditions the same ψ is pseudo-convex if the speed of the propagation determined by A is monotone in a certain direction. The most suitable choice is ψ(x, t) = d 2 (x, β) θ 2 t 2 where d is the distance in the Riemannian metrics determined by the spacial part of A. Lasiecka, Triggiani, and Yao [23] showed that this function is indeed pseudo convex when d is convex in the Riemannian metric. Romanov [26] gave a simple independent proof and 6

7 emphasized that negativity of sectional curvatures are sufficient. A disadvantage of this choice of ψ is that in most of inverse problems A and therefore the corresponding Riemannian metric are not known. In addition, the known conditions of pseudo convexity in the anisotropic case are not so easy to verify. For example, conditions in [23], [26] impose restrictions on second partials of a jk. In applications residual stress is relatively small [24]. Motivated by these reasons, we will give simple sufficient condition of pseudo-convexity for R where smallness of R is explicit. Moreover, we derive explicit global sufficient conditions for A when ψ(x, t) = x β 2 θ 2 t 2 is K-pseudo convex. Lemma 2.1. Let the matrix R be symmetric positive at any point of and 2µρθ R + µi R x < 2µ 2 on. (17) Let θ 2 < µ ρ (18) Then the function ψ(x, t) = x 2 θ 2 t 2 d 2 is pseudo-convex with respect to the anisotropic wave operator (µ; R) in Ḡ. A proof which is using standard calculations is given in [17]. Now in the following Lemma for a general anisotropic hyperbolic operator we give the condition of K-pseudo-convexity of ψ(x, t) = x β 2 θ 2 t 2, x, β = (0,..., 0, β n ) R n with β n large enough. Lemma 2.2. Let A = t 2 a jk j k, a jk = a kj, j,k=1 where a jk C 1 satisfies the uniform ellipticity condition a jk ξ j ξ k ε 0 ξ 2, ξ R n, ε 0 > 0. (19) j,k=1 Let Assume that ψ(x, t) = x β 2 θ 2 t 2, β = (0,..., 0, β n ). n 1 ( a nk k a jl 2 a lk k a jn )ξ j ξ l ε 1 ξ 2, ξ R n. (20) j,l=1 k=1 k=1 for some ε 1 > 0. Then there is large β n such that the function ψ is K-pseudo-convex with respect to A in. 7

8 and Proof. Denoting the left side in (2) by H we will have H = H 1 + H 2 where H 2 = j,k=0 H 1 = j,k=0 Using the first equality of (3) we yield H 1 = 8θ 2 j k ψ A ξ j A ξ k ( ( k A ξ j ) A ξ k k A 2 A ξ j ξ k ) j ψ. n j,k=1 a jk ξ j ξ k + 8 ( a jk ξ k ) 2. j=1 k=1 A Using that 0 A, 0 ξ 0, 2 A A ξ j ξ 0, k ξ 0, 2 A ξ 0 ξ k, j, k = 1,..., n are all zeros, and that j ψ = 2(x j β j ) and β j = 0 for j = 1,..., n 1, H 2 = { n ( 2 k a jl ξ l )( 2 a mk ξ m ) j,k=1 l=1 m=1 Hence Let + 4 j,k=1 ( p,q=1 = 8θ 2 n l=1 k a pq ξ p ξ q )( 2a jk ) } 2(x j β j ). j,k=1 H 1 + H 2 a jk ξ j ξ k + 8 ( a jk ξ k ) 2 j=1 k=1 { n 2( k a jl ξ l )( a mk ξ m ) a jk ( n { + 4β n ank ( k=1 H 3 = p,q=1 m=1 l=1 p,q=1 k a pq ξ p ξ q ) } x j k a pq ξ p ξ q ) 2( k a nl ξ l )( a mk ξ m ) }. j,l=1 k=1 m=1 { } ank k a jl 2a lk k a jn ξj ξ l. (21) Then with large β n the positivity of (21) guarantees the positivity of H 1 +H 2, and hence (2). From the second condition of pseudo-convexity (3) 4θ 2 tξ j,k=1 a jk ξ k x j 2 a nk ξ k β n = 0, k=1 8

9 hence Therefore, H 3 (ξ) = ( j,l=1 k=1 k=1 a nk ξ k = 1 β n 0( ξ ). (22) n 1 a nk k a jl 2 k=1 a lk ( k a jn ))ξ j ξ l + 1 β n 0( ξ 2 ). We have A(x, t; ψ(x, t)) = 4θ 4 t 2 4 n j,k=1 a jk(x)(x j β j )(x k β k ) = a nn (x)β 2 n +... where... denotes the terms bounded by Cβ n. So ψ is not characteristic in for large β n. Lemma follows. Corollary 2.3. Let us assume the monotonicity of the speed of the propagation with respect to A: j,k,l=1 a nk k a jl ξ j ξ l ε 1 ξ 2, ξ R n, (23) for some ε 1 > 0 and that the symmetrization of the matrix ( n 1 k=1 a lk k a jn ) is non positive. Then there is large β n such that the function ψ is K-pseudo-convex with respect to A in. One can give more precise sufficient conditions for (20). For example, by using the (Frobenius) operator norm of a matrix in L 2 (R n ) one of these conditions is 4 n 1 ( j,k=1 k=1 a lk k a jn ) 2 < ε 2 1. This corollary gives more general pseudo-convexity conditions than in [22] where it was assumed that a jn = 0 when j < n. Now we consider in more detail the cases of 2D and 3D. Example 1 : n = 2 From (22), we have ξ 2 = a21 a 22 ξ 1 + β2 1 O( ξ ) so, by routine calculations, 2 a l1 1 a j2 ξ j ξ l = j,l=1 (a 11 a 22 a 2 12) a 22 1 a 12 a 12 1 a 22 a 2 ξ1 2 + β2 1 O(ξ 1) 22 Due to positivity of the matrix (a jk ), a 11 a 22 a 2 12 > 0, a 22 > 0, so the non positivity of the principal term (with respect to large β 2 ) is 1 a 12 a

10 which is therefore a sufficient condition for K-pseudo-convexity of ψ for large β 2, provided we have monotonicity of the speed of the propagation in the x 2 - direction. Using ξ 2 = a21 a 22 ξ 1 + β2 1 O( ξ ), from (22) by routine calculations we have H 3 + β2 1 O( ξ ) = 2 j,k,l=1 a 2k k a jl ξ j ξ l 2 2 a l1 1 a j2 ξ j ξ l = j,l=1 ( (a 21 1 a 11 + a 22 2 a 11 ) (a (a 11 a 22 a 2 12)) a 22 1 a 12 a 12 1 a 22 a 2 12( 2a 12 2a 22 ) a 2 a 12 a 12( 1a 12 + ) 2a 12 ) ξ 22 a 22 a a Since 12 2 a 22 0 implies that 2a12 a 12 2a22 a 22 0, and 1 a 12 a a 12 a 12 = a 2 22 a ( 2a 12 2a 22 ) + 1 a 22 a 12 a 22 a 2 (a 12 1 a 22 + a 22 2 a 22 ), 22 we have some new sufficient conditions for positivity of H 3 : 0 < a 12 1 a 11 + a 22 2 a 11, 1 a 12 a 22 0, 2 a 12 a 22 0, a 12 1 a 22 + a 22 2 a These conditions mean positivity or negativity of some derivatives or conormal derivatives in the x 2 direction. Example 2: n = 3 By using ξ 3 = a31 a 33 ξ 1 a32 a 33 ξ from (22), by standard calculations we can obtain H 3 + β3 1 O( ξ ) = 3 j,k,l=1 a 3k k a jl ξ j ξ l 2 2 k=1 j,l=1 3 a lk k a j3 ξ j ξ l where = A 1 ξ A 2 ξ 1 ξ 2 + A 3 ξ 2 2 A 1 = a 13 1 a 11 + a 23 2 a 11 + a 33 3 a 11 a2 13 a 2 (a 13 1 a 33 + a 23 2 a 33 + a 33 3 a 33 ) 33 2a 33 (a 11 1 a 13 a 33 + a 23 2 a 13 a 33 + a 13 3 a 13 a 33 ), A 2 = (a 13 1 a 12 + a 23 2 a 12 + a 33 3 a 12 ) a 33 (a 21 1 a 13 a 33 + a 22 2 a 13 a 33 + a 23 3 a 13 a 33 ) a 23 a 23 a 23 a 33 (a a a 13 3 ) a 13a 23 (a 13 1 a 33 +a 23 2 a 33 +a 33 3 a 33 ), a 33 a 33 a a 2 33

11 A 3 = a 13 1 a 22 + a 23 2 a 22 + a 33 3 a 22 a2 23 a 2 (a 13 1 a 33 + a 23 2 a 33 + a 33 3 a 33 ) 33 2a 33 (a 12 1 a 23 a 33 + a 22 2 a 23 a 33 + a 23 3 a 23 a 33 ). The positivity of H 3 follows from the inequalities 0 < A 1, A 2 2 < A 1 A 3. While the formulae for A 1, A 2, A 3 contain several simple and meaningful blocks so far we could not find positivity conditions which have geometrical or physical interpretations, but we hope that such conditions exist and can be found in near future. 3 Proof of strong Carleman estimates for scalar operators. In the following ζ(ϕ)(x) = ξ + iτ ϕ(x). quadratic form We will introduce the differential F(x, τ, D, D)v v = A(x, D + iτ ϕ(x))v 2 A(x, D iτ ϕ(x))v 2. (24) This differential quadratic form is of order (3, 2), since the coefficients of the principal part of A are real valued. By Lemma in Hörmander s book [10] there exists differential quadratic form G(x, τ, D, D) of order (2, 1) such that G(x, D, D)v v = F(x, D, D)v v (25) and its symbol where G(x, τ, ξ, ξ) = x k η k F(x, τ, ζ, ζ), ζ = ξ + iη, at η = 0 F(x, τ, ζ, ζ) = A(x, ζ + iτ ϕ)a(x, ζ iτ ϕ) A(x, ζ iτ ϕ)a(x, ζ + iτ ϕ) Lemma 3.1. We have G(x, τ, ξ, ξ) = 2τ A A j k ϕ + 2I k A A + 2I A ( ζ j ζ k ζ k where A, k A,... are taken at (x, ζ(ϕ)(x)). 2 A iτ ζ k x k 2 A j k ϕ ) ζ j ζ k Proof is given in [17], Lemma 3.1. Using these formulae, from Lemma 2.1 by standard calculations we yield: (26) τ 1 G(x, τ, ξ, ξ) = G 1 (x, τ, ξ, ξ) + G 2 (x, τ, ξ, ξ) + G 3 (x, τ, ξ, ξ) + G 4 (x, τ, ξ, ξ) (27) 11

12 where G 1 (x, τ, ξ, ξ) = 8γϕ a jm a kl (ξ m ξ l + σ 2 m ψ l ψ) j k ψ, G 2 (x, τ, ξ, ξ) = 4γϕ a lk k a jm (σ 2 j ψ m ψ l ψ + 2ξ m ξ l j ψ ξ j ξ m l ψ) G 3 (x, τ, ξ, ξ) = 4γϕ(2 a km j a lj k ψξ l ξ m a jk ( m a lm l ψ + a lm l m ψ)(ξ j ξ k σ 2 j ψ k ψ) G 4 (x, τ, ξ, ξ) = 4γ 2 ϕ((2 a jm ξ m j ψ) 2 + 2σ 2 ( a jm j ψ m ψ) 2 ( a lm (ξ l ξ m σ 2 l ψ m ψ))( a jk j ψ k ψ)). Observe that the terms of τ 1 G with highest powers of γ are collected in G 4. Proof of Theorem 1.1. First, make the substitution u = e τϕ v. Obviously, D k (e τϕ v) = e τϕ (D k + iτ k ϕ)v. Hence a jk D j D k (e τϕ v) = a jk e τϕ (D j + iτ j ϕ)(d k + iτ k ϕ)v Accordingly, the bound (5) is transformed into σ 3 2 α α v 2 C A(, D + iτ ϕ)v 2 (28) Lemma 3.2. Under conditions of Theorem 1.1 for any ε 0 there is C such that γϕ(x)(2k ε 0 ) ζ(ϕ)(x) 2 τ 1 2 A(x, ζ(ϕ)(x)) 2 G(x, τ, ξ, ξ) + γϕ(x)cγ ζ(ϕ)(x) 2 (29) for all C < γ, ξ R n, and x. Proof: By homogeneity reasons we can assume ζ(ϕ) (x) = 1. We will also assume that γ 1. To derive (29) we will use pseudo-convexity of ψ and consider 4 possible cases. Case 1): σ = 0, A(x, ξ) = 0, A ζ j (x, ξ) j ψ(x) = 0. Then and from (27) we yield σ = 0, a jk ξ j ξ k = 0, a jk ξ j k ψ = 0, τ 1 G(x, 0, ξ, ξ) = 2γϕ j k ψ2a jm ξ m 2a kl ξ l + 4γϕ a lk k a jm (2ξ l ξ m j ψ ξ j ξ m l ψ) = 2γϕ j k ψ A ζ j A ζ k + 2γϕ (( k A ζ j ) A ζ k ( k A) 2 A ζ j ζ k ) j ψ(x, ξ) 12

13 by pseudo-convexity of ψ (2). Case 2): 2γϕK (30) σ < δ, γa(x, ξ) < δ, A ζ j (x, ξ) j ψ(x) < δ < 1 (31) where δ is a (small) positive number to be chosen later. Using (27) as in the case 1), bounding the terms with σ 2 by Cγϕδ 2 and dropping the first two and the last (positive) terms in G 4 we obtain τ 1 G(x, τ, ξ, ξ) 2γϕ j k ψ2a jm ξ m 2a kl ξ l Cγϕδ 2 +4γϕ a lk k a jm (2ξ l ξ m j ψ ξ j ξ m l ψ)+ 4γϕ(2 (a km k ψξ m j a lj ξ l a jk ξ j ξ k ( m a lm l ψ + a lm l m ψ)) 4γϕ γa jk ξ j ξ k a lm l ψ m ψ γϕ(2 j k ψ2a jm ξ m 2a kl ξ l + 4 a lk k a jm (2ξ l ξ m j ψ ξ j ξ m l ψ) Cδ) γϕ(2k ε(δ)) where ε(δ) 0 as δ 0. Indeed, let us assume the opposite. Then there are ε 0 > 0 and sequencies ξ(p), x(p), σ(p), A(p), p = 1, 2,... with ξ(p) 2 + σ(p) 2 ψ(x(p)) 2 = 1 and A(p) with coefficients bounded by M in C 2 () such that σ(p) < p 1, γ(p)a(p)(x(p), ξ(p)) < p 1, A(p) (x(p), ξ(p)) j ψ(x(p)) < p 1 ζ j but 2 j k ψ2a jm (p)(x(p))ξ m (p)2a kl (p)(x(p))ξ l (p)+ 4 a lk (p) k a jm (p)(x(p))(2ξ l (p)ξ m (p) j ψ(x(p)) ξ j (p)ξ m (p) l ψ(x(p)) Cp 1 2K ε 0. Using compactness and subtracting subsequencies, we can assume that x(p) x, ξ(p) ξ in R n and a jk (p) a jk in C 1 (). Passing to the limits and using that γ(p) 1 we arrive at the case 1) and at the inequality 2 j k ψ(x)2a jm (x)ξ m 2a kl (x)ξ l + 4 a lk k a jm (x)(2ξ l ξ m j ψ(x) ξ j ξ m l ψ(x) which contradicts (30). 2K ε 0. 13

14 From now on we will fix δ such that ε(δ) < ε 0 and denote it by δ 0. Observe that we can choose δ 0 to be dependent on the same parameters as C. Case 3): γτϕ(x) > δ 0, γa(x, ζ(ϕ)(x)) < δ 0. Using (27) as above we yield τ 1 G(x, τ, ξ, ξ) Cγϕ(x) + 8C 1 γ 2 ϕδ 2 0 2γϕ(x)K when we choose γ > C 2. Case 4): γa(x, ζ(ϕ)(x)) > δ 0 From (27) we have τ 1 G(x, τ, ξ, ξ) + γϕ(x)c 1 γa(x, ζ(ϕ)(x)) 2 Cγϕ(x) Cγ 2 ϕ A(x, ζ(ϕ)(x)) + γϕc 1 γa(x, ζ(ϕ)(x)) 2 Cγϕ(x) Cγϕ(x) γa(x, ζ(ϕ)(x)) + γϕc 1 γa(x, ζ(ϕ)(x)) 2 Cγϕ(x)+Cγϕ(x) γa(x, ζ(ϕ)(x)) ( C 1 2C γa(x, ζ(ϕ)(x)) 1)+γϕ(x)C 1 2 γa(x, ζ(ϕ)(x)) 2 Cγϕ(x) + Cγϕ(x) γa(x, ζ(ϕ)(x)) ( C 1δ 0 2C 1) + γϕ(x)c 1 2 δ2 0 Kγϕ(x) when C 1 > 2C δ 0 + C+K. δ0 2 The proof is complete. We fix x 0, introduce the norm v 1 = ( ˆv(ξ) 2 1 ξ 2 + τ 2 γ 2 ϕ 2 (x 0 ) ψ(x 0 ) 2 dξ) 2 (32) and observe that v 1 Cτ 1 v 2. (33) Lemma 3.3. There is a function ε(δ; γ) 0 as δ 0 and γ is fixed and C(γ) such that τ 1 (G(x 0, τ, D, D) G(, τ, D, D))v v ε(δ; γ) τ 2 2 α α v 2, (34) for all v C 2 0(B(x 0 ; δ)). α 1 A(x 0, D + iτ ϕ(x 0 ))v A(, D + iτ ϕ)v 2 1 (ε(δ; γ) + C(γ)τ 1 ) (γτϕ(x 0 )) 2 2 α α v 2 (35) α 1 Proof with constants depending on A is given in [17], Lemma 3.3. It is easy to observe that these constants depend only on Lipschitz constants of a jk, so this proof is valid in our situation. Now we continue the proof of Theorem

15 By using Parseval s identity, (τ 2 ϕ(x 0 ) 2 ) m α α v 2 dx (2π) n ζ 2m (ϕ)(x 0 ) ˆv(ξ) 2 dξ Hence multiplying the inequality (29) by ˆv(ξ) 2, v C0( 2 ε ), and integrating over R n we yield C 1 γϕ(x 0 ) (γτϕ(x 0 )) 2 2 α α v 2 α 1 τ 1 G(x 0, τ, D, D)v v + γϕ(x 0 )γ 2 A(x0, ζ(ϕ)(x 0 )) 2 ζ(ϕ)(x 0 ) 2 ˆv(ξ) 2 dξ τ 1 G(x 0, τ, D, D)v v + γϕ(x 0 )γ 2 A(x 0, D + iτ ϕ(x 0 ))v 2 1 τ 1 G(x, τ, D, D)v v + ε(δ; γ) α 1 τ 2 2 α γϕ(x 0 )γ 2 A(, D+iτ ϕ)v 2 1+(ε(δ; γ)+c(γ)τ 2 ) α 1 α v 2 + τ 3 2 α α v 2 (36) for v C 2 0( ε B(x 0, δ)). Here we used Lemma 3.3 and the elementary inequality a 2 2b 2 + 2(b a) 2. Choosing δ > 0 small and τ large enough so that (2C) 1 γϕ(x 0 )(γτϕ(x 0 )) 2 2 α > (ε(δ; γ) + C(γ)τ 2 )τ 2 2 α we absorb the second and fourth term on the right side of the inequality (36) to arrive at the inequality (γτϕ(x 0 )) 3 2 α α v 2 C( α 1 G(, τ, D, D)v v + τγϕ(x 0 )γ 2 A(, D + iτ ϕ)v 2 1) As above by choosing large τ > C(γ) one can replace ϕ(x 0 ) on the left side of this inequality by ϕ. Using (24), (25) and the property (33) of the norm 1 we conclude that (γτϕ) 3 2 α α v 2 α 1 C A(, D + iτ ϕ)v C(γ)τ 1 A(, D + iτ ϕ)v 2 2 for v C0(B(x 2 0 ; δ)). Choosing τ > C(γ) we eliminate the second term on the right side. Now the bound (28) follows by partition of the unity argument. Since our choice of δ 0 depends on γ we give this argument in some detail. The balls B(x 0 ; δ 0 ) form an open covering of the compact set. Hence we can find a finite subcovering B(x 0j ; δ 0 ) and a special partition of the unity 15

16 χ j (; γ) subordinated to this subcovering. In particular, χ j C 2 0(B(x 0j ; δ 0 )), 0 χ j 1, and χ 2 j = 1 on. By Leibniz formula α (χ j v) = χ j α v + ( α χ j )v, A(, D + iτ ϕ)(χ j v) = χ j A(, D + iτ ϕ)v + a β τ 1 β β v β 1 with a β C(γ). Hence applying the Carleman estimate (28) to χ j v and using the elementary inequality a + b a2 b 2 we obtain 1 2 α 1 σ 3 2 α χ j α v 2 σ 3 ( α (χ j ))v) 2 C χ j A(, D + iτ ϕ)v C(γ) τ 2 2 β β v 2 2. β 1 Summing up over j = 1,..., J and using that χ 2 j = 1 we yield 1 σ 3 2 α α v 2 σ ( α (χ j ))v) 2 2 α 1,j J C A(, D + iτ ϕ)v C(γ) τ 2 2 β β v 2 2. β 1 Since the highest powers of τ are in the first term of the left side, choosing C(γ) < τ we absorb by this term the second term on the left side and the second term on the right side. The proof is complete. 4 Proof of Carleman estimates in negative norms A proof of Theorem 1.2 will be based on Theorem 1.1 and some additional lemmas. By basic differentiation rules, A ϕ (D) = A(D + iτ ϕ) = A(D) + τ(a 1 (D) + a 0 ) τ 2 A( ϕ), (37) where A 1 is the first order differential operator with the C 1 -coefficients depending on γ and a 0 is some function in L depending on γ. We will use the notation < ξ >= ( ξ 2 + 1) 1 2 and the pseudo-differential operator Λ s τ f = F 1 (< ξ > +τ) s Ff, where F is the Fourier transform and ξ = (ξ 1,..., ξ n ). Let be a bounded domain in R n with a smooth boundary such that. We can extend all coefficients of the operator A ϕ onto R n preserving the regularity in such a way that they have support in. 16

17 Lemma 4.1. There exists a constant C(γ) such that Λ 1 τ A ϕ u A ϕ Λ 1 τ u (0) ( ) C(γ) u (0) () for all u H0 2 (). Proof: Due to (37) it suffices to show that Λτ 1 a α u a α Λ 1 τ u (0) ( ) C(γ) u (0) (), for all α 2, (38) for a C 2 ( ), a 2 ( ) < M, α = (α 1,..., α n ), that τ Λ 1 τ a β u a β Λ 1 τ u (0) ( ) C(γ) u (0) (), for all β 1, (39) for α = (α 1,..., α n )that τ 2 Λ 1 τ au aλ 1 τ u (0) ( ) C(γ) u (0) (). (40) for a C 1 ( ), α = (α 1,..., α n ), and that τ Λ 1 τ a 0 u a 0 Λ 1 τ u (0) ( ) C(γ) u (0) (). (41) Let α j > 0 and β j = 1 while other components of β be zero. We introduce u 1 = Λ 1 τ α β u. Using also that Λ τ = Λ 0 + σ, we have Λ 1 τ a α u a α Λ 1 τ u = Λ 1 τ (aλ τ Λ τ a) j u 1 = Λ 1 τ (aλ 0 Λ 0 a) j u 1 = Λ 1 τ (a j Λ 0 j (Λ 0 a) + Λ 0 j a)u 1 = Λ 1 τ ( j (aλ 0 Λ 0 a) + (Λ 0 j a j aλ 0 ))u 1 From the Parseval identity, u 1 (0) (R n ) C u (0) (). By known (e.g. Coifman and Meyer ([4])) estimates of commutators of pseudo-differential operators and of multiplication operators (aλ 0 Λ 0 a)u 1 (0) ( ) C(γ) u 1 (0) (R n ). A similar estimate is valid when we replace a by j a. Using that, as above, Λ 1 0 j is a bounded operator in L 2 we complete the proof of the bound (38). Proofs of (39), (40) are similar. The bound (41) is obvious. Indeed, τλ 1 τ (a 0 u) τa 0 Λ 1 τ u (0) τλ 1 τ (a 0 u) (0) + τa 0 Λ 1 τ u (0) C u (0) since τλ 1 τ v (0) v (0). In Lemmas 4.2, 4.3 x, y denote elements of R n. Lemma 4.2. Let K(x, y; τ) be the Schwartz kernel of the pseudo-differential operator Λ 1 τ with τ > 1. Then x α K(x, y; τ) C(d 1 )τ 2 x y 2n 2 provided α 2 and 0 < d 1 x y. 17

18 A proof can be found in [12], Lemma 3.4. This proof is valid in our case when we choose l = n + 1 and replace n + 1 by n. Let σ = supσ and σ = infσ over. Lemma 4.3. Let ψ be K pseudo-convex with respect to A on. Then for any x 0 there are δ(γ) and C such that (σ 3 Λ 1 σ v 2 + σ v 2 ) C Λ 1 σ A ϕv 2 R n R n for all v H 2 0 (B(x 0 ; δ)) provided τ > C. Proof: We can assume that x 0 = 0 and we let B(δ) = B(x 0 ; δ). For continuity reasons, ψ is K/2 pseudo-convex in. By Theorem 1.1 there exists C such that the following Carleman estimate holds 1 α =0 σ 3 2 α α v 0 2 C A ϕ v 0 2 for all v 0 H0 2 () provided C < γ, C(γ) < τ. Let χ C0 (), χ = 1 on B(2δ). Using this Carleman type estimate for v 0 = χλ 1 σ v, we obtain (σ 3 χ 2 Λ 1 σ v 2 + σ χ α (Λ 1 σ v) + α χλ 1 σ v 2 ) C A ϕ (χλ 1 σ v) 2 (42) C ( A ϕ (Λ 1 σ v) 2 + C(γ)(τ 2 Λ 1 σ v 2 + α (Λ 1 σ v) 2 )). where we used (37), the Leibniz formulas A(χw) = χaw + A 1 (; χ)w + A(χ)w, A 1 (χw; ϕ) = χa 1 (w; ϕ) + A 1 (χ; ϕ)w, and the triangle inequality. Due to the Parseval identity α Λ 1 σ v 2 α Λ 1 σ v 2 R n v 2 = R n Similarly, B(2δ) τ 2 v 2 when α = 1. (43) Λ 1 σ v 2 v 2. (44) Using these inequalities and recalling that χ = 1 on B(2δ) we derive from the bound (42) that (σ 3 Λ 1 σ v 2 + σ α (Λ 1 σ v) 2 ) C(γ) v 2 18

19 C ( A ϕ (Λ 1 σ v) 2 + C(γ) v 2 ). (45) The Parseval identity and the definition of Λ τ yield σv 2 σ v 2 = σ σ Rn R < ξ > 2 +σ 2 ˆv(ξ) 2 dξ + σ ξ 2 < ξ > 2 +σ 2 ˆv(ξ) 2 dξ = n (σ + (σ ) 3 ) Λ 1 σ v 2 + σ α (Λ 1 σ v) 2 R n R n 2( σ ) 3 σ R n \ σ 3 Λ 1 σ v 2 + σ σ ((σ ) 3 Λ 1 σ v 2 + σ σ α (Λ 1 σ v) 2 + α (Λ 1 σ v) 2 ). Since ψ C 2, using (4) we will choose δ(γ) so that 1 2 < σ σ. Choosing τ > C(γ) and using Lemma 4.1 we will have from (45) (σ 3 Λ 1 σ v 2 + σ v 2 ) R n C Λ 1 σ A ϕv 2 + C(γ) R n \B(2δ) (τ 3 Λ 1 σ v 2 + τ α Λ 1 σ v 2 ) (46) when τ > C(γ). Now, by using Lemma 4.2 we will eliminate the last integral in this bound and complete the proof. Since supp v B(δ), x α Λτ 1 v(x) v(y) x α K(x, y; τ) dy C(γ)τ 2 x y 2n 2 v(y) dy B(δ) by Lemma 4.2, provided x R n \ B(2δ). When y B(δ), B(δ) x y 1 2 x y x y δ x 1 4 y δ x x 4 C(γ). Hence by using the Schwarz inequality α Λ 1 τ v (x) C(γ)τ 2 (1 + x ) 2n 2 ( B(δ) v 2 ) 1 2 for all α 1. provided x R n \ B(2δ). Using this estimate we conclude that the last integral in (46) is less than C(γ) B(δ) v 2. So choosing τ > C(γ) we will bound this integral by the last integral in the left side of (46) and eliminate this integral. This completes the proof of Lemma

20 Proof of Theorem 1.2 We first assume that suppv B(x 0, δ). Using the substitution v = e τϕ w and the identity Av = e τϕ A ϕ w we reduce (6) to the bound σ w 2 C ( f 0 2 B(x 0,δ) σ 2 provided A ϕ w = f 0 + n j=1 jf j, with f 0 = e τϕ (f 0 Using Lemma 4.3 we will have B(x 0,δ) n + f j) 2 when τ > C(γ), j=1 τ j ϕf j ), f j = e τϕ f j. j=1 σ w 2 C ( Λ 1 σ f B(x 0,δ) C( (σ 2 f R n j=1 f j 2 ) j=1 Λ 1 σ jf j 2 ) by the Parseval identity. Using the definition of f j we complete the proof when suppu B(x 0, δ(x 0 )). Now we will complete the proof by using a special partition of the unity argument. Due to compactness of we can find a finite covering of by balls B(x(k), δ(γ)), k = 1,..., K. Let χ(; k) be the special C - partition of the unity subordinated to this covering, i.e. suppχ(; k) B(x(k); δ) and K k=1 χ2 (; k) = 1 on. By the Leibniz formula A(χ(; k)v) = χ(; k)av + j (b j v) + cv = χ(; k)f 0 + j (χ(; k)f j ) j=1 j=1 ( j χ(; k))f j + j=1 j (b j v) + cv. where b j, c depend on γ. Applying Theorem 1.2 to χ(; k)v we obtain σe 2τϕ χ 2 (; k)v 2 j=1 j=1 j=1 C ( f 2 n 0 σ 2 + fj 2 f j + C(γ) σ 2 + C(γ)v2 )e 2τϕ. Summing over k = 1,..., K and choosing τ > C(γ) we will absorb the terms containing v in the right side by the left side and complete the proof. 20

21 5 Proof of Carleman estimates for elasticity system Lemma 5.1. Let ψ > 0 on. Then there are constants C, C 0 (γ) such that γ σ 4 2 α e 2τϕ α v 2 C σe 2τϕ v 2 (47) for all v C 2 0(), α 2, C < γ, and C 0 (γ) < τ. A proof is given in [7], [17]. Lemma 5.2. There exists a constant C(γ) such that Λ 1 σ ( α curl) ϕ v ( α curl) ϕ Λ 1 σ v (0)( ) C(γ) v (0) () for all v H 2 0 (), α 1. A proof is similar to the proof of Lemma 4.1. Lemma 5.3. Let ψ > 0 on. Then there are C, C 0 (γ) such that γ (σ 2 u 2 + α u 2 )e 2τϕ C σ( curlu 2 + divu 2 )e 2τϕ for all u H 1 0 () provided C < γ, C 0 (γ) < τ. Proof: Let x 0. We first consider u supported in B(x 0 ; δ). Using the standard substitution u = e τϕ v as above, this Lemma follows from the bound γ B(δ) (σ 4 Λ 1 σ v 2 + σ 2 v 2 + α v 2 ) C σ( (curl) ϕ v 2 + (div) ϕ v 2 ) B(δ) Since v 0 = curlcurlv 0 divv 0, by Lemma 5.1 there exists C, C(γ) such that the following Carleman estimate holds γ σ 4 2 α α v 0 2 C σ ( ( α curl) ϕ v ( α div) ϕ v 0 2 ) α 2 B(δ) for all v 0 H 2 0 () provided C < γ, C(γ) < τ. 21

22 Let χ C0 (), χ = 1 on B(2δ). Applying this Carleman type estimate to v 0 = χλ 1 σ v, we obtain γ (σ 4 χ 2 Λ 1 σ v 2 + σ 2 χ α (Λ 1 σ v) + α χλ 1 σ v 2 )+ α =2 C χ α (Λ 1 σ v) + 2 α χ α α (Λ 1 σ v) + ( α χ)λ 1 σ v 2 σ( ( α curl) ϕ (Λ 1 σ v) 2 + ( α div) ϕ (Λ 1 σ v) 2 )+ C(γ)(τ 3 Λ 1 σ v 2 + τ α (Λ 1 σ v) 2 ), (48) where we used that due to (37), for any second order partial differential operator A with constant coefficients, A ϕ (, D)(χΛ 1 σ v) = = χa(, D + iτ ϕ)λ 1 σ v + (τa 0 + A 1 (, D))Λ 1 σ v where A 1, A 0 are first, and zero order operators with bounded coefficients depending on γ and A, applied this relations to components of curl ϕ, div ϕ, and used the triangle inequality. Recalling that χ = 1 on B(2δ), we derive from inequalities (43), (44), and the bound (48) that γ σ 4 2 α α Λ 1 σ v 2 C(γ) v 2 B(2δ) α 2 C (σ ( ( α curl) ϕ (Λ 1 σ v) 2 + ( α div) ϕ (Λ 1 σ v) 2 ) + C(γ)τ v 2 ). (49) The Parseval identity and the definition of Λ τ yield σ 2 v 2 (σ ) 2 v 2 = (σ ) 2 σ R3 R < ξ > 2 +σ 2 ˆv(ξ) 2 dξ + (σ ) 2 ξ 2 < ξ > 2 +σ 2 ˆv(ξ) 2 dξ 3 2(σ ) 4 Λ 1 σ v 2 + (σ ) 2 α (Λ 1 σ v) 2 R 3 R 3 2( σ ) 4 σ B(2δ) σ 4 Λ 1 σ v 2 + ( σ σ ) 2 B(2δ) σ 2 α (Λ 1 σ v)

23 R 3 \B(2δ) (σ ) 4 Λ 1 σ v 2 + (σ ) 2 α Λ 1 σ v 2 ). Since ψ C 2, using (4) we will choose δ(γ) so that 1 2 < σ σ. Similarly, α v 2 = ξ 2 ˆv(ξ) 2 dξ = R 3 8 R3 ((σ ) 2 ξ 2 R3 ξ 4 + 1) < ξ > 2 +σ 2 ˆv(ξ) 2 dξ + < ξ > 2 +σ 2 ˆv(ξ) 2 dξ 2 (σ ) 4 2 α α Λ 1 σ v 2 B(2δ) 1 α 2 R 3 1 α 2 σ 4 2 α α Λ 1 σ v 2 + R 3 \B(2δ) 1 α 2 (σ ) 4 2 α α Λ 1 σ v 2, where we used that σ 2σ on. Choosing τ > C(γ) and using Lemma 5.2 we will have from (49) γ (σ 4 Λ 1 σ v 2 + σ 2 v 2 + α v 2 ) R 3 C σ C(γ) ( α Λ 1 σ (curl) ϕv 2 + α Λ 1 R 3 \B(2δ) α 2 when τ > C(γ). From (43), (44) σ ( α Λ 1 σ (curl) ϕv 2 + α Λ 1 σ (div) ϕv 2 ) σ (div) ϕv 2 )+ τ 4 2 α α Λ 1 σ v 2 (50) B(δ) σ( (curl) ϕ v 2 + (div) ϕ v 2 ). Now, by using Lemma 4.3 we will eliminate the last integral in the bound (50). Since suppv B(δ), x α Λ 1 σ v(x) v(y) α x K(x, y; σ ) dy C(γ)τ 2 x y 8 v(y) dy B(δ) by Lemma 4.2, provided x R 3 \ B(2δ). When y B(δ), as in the proof of Lemma 4.3, x y 1+ x C(γ). Hence by using the Schwarz inequality α Λ 1 σ v (x) C(γ)τ 2 (1 + x ) 8 ( v 2 ) 1 2 for all α 1. provided x R 3 \ B(2δ). Using this estimate we conclude that the last integral in (50) is less than C(γ) B(δ) v 2. So choosing τ > C(γ) we will bound this B(δ) 23

24 integral by the last integral in the left side of (50) and eliminate this integral. This completes the proof of Lemma 5.3 when suppu B(x 0 ; δ). Now we will complete the proof by using a special partition of the unity argument. Due to compactness of we can find a finite covering of by balls B(x(k), δ(γ)(k)), k = 1,..., K. Let χ(; k) be the special C - partition of the unity subordinated to this covering, i.e. suppχ(; k) B(x(k); δ) and K k=1 χ2 (; k) = 1 on. By the Leibniz formula curl(χ(; k)u) = χ(; k)curlu + A 01 u, div(χ(; k)u) = χ(; k)divu + A 02 u, where A 0j are bounded matrix-functions depending on γ. Applying Lemma 5.3 to χ(; k)u and using the elementary inequality 1 2 a 2 b 2 a + b 2 we obtain γ (σ 2 χ 2 ((; k) u 2 + χ 2 (; k) α u 2 C(γ) u 2 )e 2τϕ C σ(( curlu 2 + divu 2 ) + C(γ) u 2 )e 2τϕ. Summing over k = 1,..., K and choosing τ > C(γ) we will absorb the terms containing C(γ) u by the first term on the left side and complete the proof. Proof of Theorem 1.3: To use Carleman estimates for scalar equations we will extend this system to a new principally triangular system. By using the standard substitution (u, v = divu, w = curlu) the system A R u = f can be reduced [19], Proposition 2.1, to a new system where the leading part is a special lower triangular matrix differential operator with the wave operators in the diagonal: (µ; R)u = f ρ + A 1;1(u, v), (λ + 2µ; R)v = div f ρ + jk ( r jk ρ ) j k u + A 2;1 (u, v, w), (51) (µ; R)w = curl f ρ + jk ( r jk ρ ) j k u + A 3;1 (u, v, w), where A j;1 are first order differential operators. Appyling Theorem 1.2 to each of seven scalar differential operators forming the extended system (51) and summing up seven Carleman estimates, we get σ( u 2 + v 2 + w 2 )e 2τϕ C f 2 e 2τϕ + C j=1 3 j u 2 e 2τϕ + 24

25 C ( u 2 + v 2 + w 2 )e 2τϕ By choosing σ > 2C we can absorb the third integral in the right side by the left side arriving at the inequality σ( u 2 + v 2 + w 2 )e 2τϕ C f 2 e 2τϕ + C j=1 3 j u 2 e 2τϕ. (52) To eliminate the first order derivatives in the right side we need second large parameter γ. By Lemma 5.3 γ 3 j,k=1 j u 2 e 2τϕ C σ( curlu 2 + divu 2 )e 2τϕ C f 2 e 2τϕ + C j=1 3 j u 2 e 2τϕ. where we used (52). Choosing γ > 2C we can see that the first order derivatives term on the right side is absorbed by the left side. This yields γ 3 j j u 2 e 2τϕ C f 2 e 2τϕ. So using again (52) we complete the proof of (8). 6 Proofs of stability estimates in the Cauchy problem In this section we will prove Theorem 1.4. Proof of Theorem 1.4 By extension theorems for Sobolev spaces we can find u H 2 () so that and Let u = g 0, ν u = g 1 on Γ The function v solves the Cauchy problem u (2) () CF. (53) v = u u. (54) A R v = f A R u in, v = 0, ν v = 0 on Γ. (55) 25

26 To apply Carleman estimates of Theorem 1.3 we need zero Cauchy data on the whole boundary. To achieve it we introduce a cut-off function χ C ( ) so that χ = 1 on δ, χ = 0 on \ 0. By Leibniz formula 2 A R (χv) = χa R v + A 1 v, where A 1 is a matrix linear partial differential operator of order 1 with bounded coefficients depending on χ. Moreover A 1 = 0 on δ. Using the Cauchy data 2 (55) we conclude that v H0 2 (), hence by Carleman estimate of Theorem 1.3 we have ( v + div(χv) 2 + curl(χv) 2 )e 2τϕ C ( f 2 + (A R u ) 2 + A 1 v 2 )e 2τϕ for C < γ, C 0 < τ. Shrinking integration domain on the left side to 3δ (where 4 χ = 1) and splitting integration domain of A 1 v 2 into δ and its complement we yield 2 ( v 2 + div(v) 2 + curl(v) 2 )e 2τϕ 3δ4 C ( f 2 + (A R u ) 2 )e 2τϕ + C A 1 v 2 e 2τϕ CF 2 e 2τΦ + C v 2 (1) ()e2τφ2 \ δ2 where we used definition (10) of F and bound (53) and let Φ = supϕ over and Φ 2 = supϕ over \ δ. Letting Φ 1 = infϕ over 3δ and replacing ϕ on 2 4 the left side of the preceding inequality by Φ 1 we yield ( v 2 (0) ( ) + divv 2 3δ 4 (0) ( ) + curlv 2 3δ 4 (0) ( 3δ 4 ))e 2τΦ1 CF 2 e 2τΦ + C v 2 (1) ()e2τφ2. (56) Observe that Φ 2 < Φ 1. Using interior type Schauder estimates for the elliptic operator curl, div, (56) yields v 2 (0) ( δ) + x v 2 (0) ( δ) CF 2 e 2τ(Φ Φ1) + C v 2 (1) ()e2τ(φ2 Φ1). (57) If v (1) ()F 1 < C, then v (1) () CF. Otherwise we let τ = (Φ + Φ 1 Φ 2 ) 1 log( v (1) ()F 1 ). Then the bound (57) implies that with κ = v (1) ( δ ) C v (1) () 1 κ F κ Φ1 Φ2 Φ+Φ 1 Φ 2. Combining the both cases we yield v (0) ( δ ) + x v (0) ( δ C(F + v (1) () 1 κ F κ ). Using that u = v + u, the triangle inequality, (53) and the elementary inequality (a + b) κ a κ + b κ we obtain (10) and complete the proof. 26

27 7 Proof of Hölder stability for the inverse problem In this section we prove Theorem 1.5. Let u(; 1) and u(; 2) satisfy (11) corresponding to R(; 1) and R(; 2), respectively. Denote u = u(; 2) u(; 1) and F = R(; 2) R(; 1) = (f jk ), j, k = 1,..., 3. By subtracting equations (11) for u(; 1) from the equations for u(; 2) we yield A R(;2) u = A(; u(; 1))F on, where A(; u(; 1))F = and 3 j,k=1 f jk j k u(; 1) (58) u = t u = 0 on G {0}, u = 0 on Γ. (59) Differentiating (58) in t and using time-independence of the coefficients of the system, we get A R(;2) U = A(; U(; 1))F on, (60) where t 2 u U = t 3 t 2 u(; 1) u and U(; 1) = 3 t 4 t u(; 1). u t 4 u(; 1) By extension theorems for Sobolev spaces there exists U H 2 () such that U = 0, ν U = ν U on Γ, (61) and U (2) () C ν U ( 1 2 ) (Γ) CF c, (62) due to (13). We now introduce V = U U. Then and A R(;2) V = A(; u(; 1))F A R(;2) U on (63) V = ν (V) = 0 on Γ. (64) To use the Carleman estimate (8) we need zero Cauchy data on (0). To create such data we introduce a cut-off function χ C 2 (R 4 ) such that 0 χ 1, χ = 1 on ( δ 2 ) and χ = 0 on \ (0). By the Leibniz formula A R(;2) (χv) = χa R(;2) (V) + A 1 V = χaf χa R(;2) U + A 1 V. Here ( and below) A 1 denotes a first order matrix differential operator with coefficients uniformly bounded by C(δ). By the choice of χ, A 1 V = 0 on ( δ 2 ). Because of (64) the function χv H0 2 (), so we can apply to it the Carleman estimate (8) with fixed γ to get τ χv 2 e 2τϕ C(δ) ( F 2 + A R(;2) (U ) 2 )e 2τϕ + C A 1 V 2 e 2τϕ \( δ 2 ) 27

28 C( F 2 e 2τϕ + Fc 2 e 2τΦ + C(δ)e 2τδ1 ) (65) where Φ = supϕ over and δ 1 = e γδ 2. To get the last inequality we used the bounds (62). On the other hand, from (7), (58), (59) we have ρ 2 t u = f jk j k u(; 1), ρ 3 t u = f jk t j k u(; 1) on G {0}. From now on we will consider the symmetric matrix-function F as a vector function with components (f 11, f 12, f 13, f 22, f 23, f 33 ). Using the definition (14) of M we obtain ρ( 2 t u, 3 t u) = MF on G {0}, and from the condition (15) we have F = M 1 (ρ( 2 t u, 3 t u)), on G {0}. Hence, by using (15), Since χ(, T ) = 0, G F 2 C ( β t u(, 0) 2 ). (66) β=2,3 T χ β t u(x, 0) 2 e 2τϕ(x,0) dx = t ( χ β t u(x, t) 2 e 2τϕ(x,t) )dx)dt 0 G 2χ 2 ( β+1 t u β t u + τ t ϕ β t u 2 )e 2τϕ + 2 β \( δ 2 ) t u 2 χ t χ e 2τϕ where β = 2, 3. The right side does not exceed C( τ χu 2 e 2τϕ + C(δ) U 2 e 2τϕ ) \( δ 2 ) C( τ χv 2 e 2τϕ + C(δ) V 2 e 2τϕ + τ U 2 e 2τϕ ) \( δ 2 ) because U = V + U. Using that χ = 1 on G( δ 2 ), ϕ < δ 1 on \ ( δ 2 ) and on G \ G( δ 2 ), and ϕ < Φ on from these inequalities, from (62) and from (65) we yield β t u 2 (, 0)e 2τϕ(,0) C( F 2 e 2τϕ + C(δ)e 2τδ1 + τe 2τΦ Fc 2 ). (67) G First we got this bound with G( δ 2 ) instead of G on the left side and then added to both sides of the inequality the integral over G \ G( δ 2 ) which is bounded by 28

29 C(δ)e 2τδ1 due to the bound (12) and the inequality ϕ < δ 1 on G \ G( δ 2 ). From (66) and (67) we obtain F 2 e 2τϕ(,0) C( F 2 e 2τϕ + τe 2τΦ Fc 2 + C(δ)e 2τδ1 ). (68) G To eliminate the integral in the right side of (68) we observe that F 2 (x)e 2τϕ(x,t) dxdt = G T F 2 (x)e 2τϕ(x,0) ( e 2τ(ϕ(x,t) ϕ(x,0)) dt)dx. T Due to our choice of function ϕ we have ϕ(x, t) ϕ(x, 0) < 0 when t 0. Hence by the Lebesgue Theorem the inner integral (with respect to t) converges to 0 as τ goes to infinity. By reasons of continuity of ϕ, this convergence is uniform with respect to x G. Choosing τ > C we therefore can absorb the integral over ( δ 2 ) in the right side of (68) by the left side arriving at the inequality F 2 e 2τϕ(,0) C(e 2τΦ Fc 2 + C(δ)e 2τδ1 ). (δ) Letting δ 2 = e γδ ϕ on (δ) and dividing the both parts by e 2τδ2 we yield F 2 C(τe 2τ(Φ δ2) Fc 2 + e 2τ(δ2 δ1) ) C(δ)(e 2τΦ Fc 2 + e 2τ(δ2 δ1) ) (69) (δ) since τe 2τδ2 < C(δ). To prove (16) it suffices to assume that F c < 1 C. Then τ = logfc Φ+δ 2 δ 1 > C and we can use this τ in (69). Due to the choice of τ, e 2τ(δ2 δ1) = e 2τΦ F 2 c = F 2 δ 2 δ 1 Φ+δ 2 δ 1 c and from (69) we obtain (16) with κ = The proof of Theorem 1.6 is now complete. 8 Conclusion δ2 δ1 Φ+δ 2 δ 1. We observe that it is not clear at the moment how to include in Carleman estimates of Theorem 1.3 time derivatives of u. Then one can obtain proofs of Lipschitz stability in the lateral Cauchy problem and for identification of residual stress in most natural norms. By using additional spatial derivatives such Lipschitz estimates are derived in [17], [18], [20]. The next realistic goal is to apply Carleman estimates of Theorem 1.2 to obtain Carleman estimates, uniqueness of the continuation and coefficients identification results for the important system of transversally isotropic elasticity, where currently there are no analytic results. 29

30 Problems of the continuation, control, and identification of coefficients for equations and systems of equations describing elastic plates and shells are not well understood. We expect that the developed theory can be extended onto Schrödinger type equations and therefore onto systems for plates or shells. Acknowledgements This work was supported in part by the Emylou Keith and Betty Dutcher Distinguished Professorship and the NSF grant DMS A part of this research was conducted during the Mini Special Semester on Inverse Problems, May 18-July 15, 2009, organized by RICAM, Austrian Academy of Sciences. References [1] P. Albano, and D. Tataru, Carleman estimates and boundary observability for a coupled parabolic-hyperbolic system, Electr. J. Diff. Equat., 2000 (2000), [2] H. Ammari, Y. Capdeboscq, H. Kang, A. Kozhemyak, Mathematical Models and Reconstruction Methods in Magneto-Acoustic Imaging, Eur. J. Appl. Math., 20 (2009), [3] A.L. Bukhgeim, Introduction to the Theory of Inverse Problems, VSP, Utrecht, [4] R. Coifman, Y. Meyer, Commutateurs d integrales singulieres et operateurs multilineaires, Ann. Inst. Fourier Grenoble, 28 (1978), [5] G. Duvaut, J.L. Lions, Inequalities in Mechanics and Physics, Springer- Verlag, [6] M. Eller, Carleman Estimates with a Second Large Parameter, J. Math. Anal. Appl., 249 (2000), [7] M. Eller, V. Isakov, Carleman estimates with two large parameters and applications, Contemp. Math., AMS, 268 (2000), [8] M. Eller, V. Isakov, G., Nakamura, and D. Tataru, Uniqueness and stability in the Cauchy problem for Maxwell s and the elasticity system, in Nonlinear Partial Differential Equations, Vol.16, College de France Seminar,Elsevier-Gauthier Villars Series in Applied Mathematics, Ed. P.G. Ciarlet, P.L.Lions, 7, [9] M. E. Gurtin, The Linear Theory of Elasticity, Springer-Verlag, Berlin, 1984, in Mechanics of Solids, Vol.II, edited by Truesdell, C. [10] L. Hörmander, Linear Partial Differential Operators, Springer-Verlag,

31 [11] O. Yu. Imanuvilov and M. Yamamoto, Global uniqueness and stability in determining coefficients of wave equations, Comm. in Part. Diff. Equations, 26 (2001), [12] O. Imanuvilov, V. Isakov, and M. Yamamoto, An inverse problem for the dynamical Lamé system with two sets of boundary data, Comm. Pure Appl. Math., 56 (2003), [13] V. Isakov, A nonhyperbolic Cauchy problem for b c and its applications to elasticity theory, Comm. Pure and Applied Math., 39 (1986), [14] V. Isakov, Carleman type estimates in an anisotropic case and applications, J. Differential Equations, 105, (1993), [15] V. Isakov, On the uniqueness of the continuation for a thermoelasticity system, SIAM J. Math. Anal., 33 (2001), [16] V. Isakov, Inverse Problems for Partial Differential Equations, Springer- Verlag, New York, [17] V. Isakov, N. Kim, Carleman estimates with second large parameter for second order operators, Some application of Sobolev spaces to PDEs, International Math. Ser., Springer-Verlag, 10 (2008), [18] V. Isakov, N. Kim, Carleman estimates with two large parameters for second order operators and applications to elasticity with residual stress, Applicationes Mathematicae, 3 (2008), [19] V. Isakov, G. Nakamura, J.-N. Wang, Uniqueness and stability in the Cauchy problem for the elasticity system with residual stress, Contemp. Math., AMS, 333 (2003), [20] V. Isakov, J.-N. Wang, M. Yamamoto, Uniqueness and stability of determining the residual stress by one measurement, Comm. Part. Diff. Equat., 23 (2007), [21] V. Isakov, J.-N. Wang, M. Yamamoto, An inverse problem for a dynamical Lame system with residual stress, SIAM J. Math. Anal., 39 (2007), [22] A. Khaidarov, Carleman estimates and inverse problems for second order hyperbolic equations, Math. USSR Sbornik, 58 (1987), [23] I. Lasiecka, R. Triggiani, P.F. Yao, Inverse/observability estimates for second order hyperbolic equations with variable coefficients, J. Math. Anal. Appl., 235 (2000), [24] C.-S. Man, Hartig s law and linear elasticity with initial stress, Inverse Problems, 14 (1998),