KOMABA, TOKYO, JAPANA CARLEMAN ESTIMATE FOR THE LINEAR SHALLOW SHELL EQUATION AND AN INVERSE SOURCE PROBLEM

We consider an elastic bi-dimensional body whose reference 
 configuration is a shallow shell. We establish a Carleman estimate 
 for the linear shallow shell equations and apply it to prove a 
 conditional stability for an inverse problem of determining external 
 source terms by observations of displacement in a neighbourhood of 
 the boundary over a time interval.

this paper we consider the case of an elastic body that occupies a two-dimensional domain slightly curved and we establish a Carleman estimate. A Carleman estimate yields observability inequalities by an argument in Cheng, Isakov, Yamamoto and Zhou [4], Kazemi and Klibanov [14]. From a geometrical point of view, this is of great interest because a general shell can be approximated through a juxtaposition of shallow shells. From a theoretical point of view, the approach retained here which is based on Carleman estimates, is powerful, while by the multipliers technique, we were not able to obtain the controllability of a Koiter shell by a boundary action without a "shallowness" restriction (Miara and Valente [23]). In the static case, let us now briefly recall the equilibrium equations of a shallow shell with middle surface S, thickness 2ε and curvature εθ (this expression of the curvature has been rigorously justified in Ciarlet and Miara [5]). More precisely, let Ω ⊂ R 2 be a bounded connected domain with Lipschitz boundary ∂Ω, and let a point in Ω be denoted by x = (x 1 , x 2 ), let θ: Ω → R, θ ∈ C 3 (Ω). Let ∂ j = ∂ ∂x j and Then the middle surface of the shell is therefore given by the set S = {(x 1 , x 2 , εθ(x 1 , x 2 )); (x 1 , x 2 ) ∈ Ω} and the shell with thickness 2ε occupies the domain {(x 1 , x 2 , εθ(x 1 , x 2 )) + x 3 a(x 1 , x 2 ); (x 1 , x 2 ) ∈ Ω, −ε ≤ x 3 ≤ ε}, where a(x 1 , x 2 ) is the unit outward normal vector to the middle surface S at the point (x 1 , x 2 ). Hence, if we denote by Θ ε the mapping then the reference configuration of the shell is Θ ε Ω×(−ε, ε) . Subjected to applied volume forces F = (F 1 , F 2 , F 3 ) ∈ {L 2 (Ω)} 3 (for simplicity, no surface forces are taken into account in this presentation), a shell which is clamped on its lateral surface, undergoes a scaled Kirchhoff-Love displacement field of the form u 1 − 3 . For the precise meaning of the scaling, see Ciarlet and Miara [5]. In this section, Latin exponents and indices take their values in the set {1, 2, 3}, Greek indices take their values in the set {1, 2}, the Einstein convention for repeated exponents and indices is used. These notations are used especially for the compatibility with the notations in Ciarlet and Miara [5], Miara and Valente [23]. [24]. Throughout this paper, bold face letters represent vectors.
The three-dimensional vector-valued function u = (u 1 , u 2 , u 3 ) : Ω −→ R 3 describes the displacement of the middle surface of the shell and solves a boundary value problem for shallow shell equations: For a Saint Venant-Kirchhoff isotropic material with Lamé coefficients λ and µ, the constitutive law reads: (1. 2) The outline of the contents of the paper is as follows: In the next section we rewrite the shallow shell equations in a more appropriate form to deal with the evolution problem, in section 3 we establish a Carleman estimate for the evolution problem and finally in section 4 we solve the inverse problem. §2. The evolution problem of the shallow shell equation.
Let Ω ⊂ R 2 be a bounded domain with smooth boundary ∂Ω, x = (x 1 , x 2 ), ∂x j ∂x k , and let θ: Ω → R be given and sufficiently smooth, Then, considering the force of inertia in (1.1) and (1.2), we can describe an evolution problem for shallow shell equation: and n θ jk ∂ jk θ, and (∇ u) T is the transpose matrix of ∇ u.
Henceforth we set and In this section, we establish a Carleman estimate for the shallow shell equation.
We assume that ρ = ρ(x), λ = λ(x) and µ = µ(x) are in C 2 (Ω) and positive in Ω. We set t 0 = T /2, γ and ν are positive constants, First we present Lemma 1. We assume that ρ, µ and λ are in C 2 (Ω) and positive on Ω and that Let ν > 0 be arbitrarily fixed in (3.1). Then there exists a number γ 0 > 0 such that for arbitrary γ ≥ γ 0 , we can choose s 0 ≥ 0 satisfying: there exists a constant C > 0 such that hand side of what is finite. The constants s 0 and C continuously depend on T , We note that if λ and µ are positive constants, then condition (3.2) is automatically satisfied.
Proof of Lemma 1. Except for the term 2 j,k, =1 |∂ j ∂ k ∂ v| 2 e 2sϕ , by Yuan and Yamamoto [28], all the terms on the left hand side is proved to be estimated by the right hand side. We have to estimate and so Similarly we have Lemma 1 yields Here the a priori estimate for the Dirichlet problem for implies Since in terms of Lemma 1. Hence by (3.3) and Lemma 1, we have Hence the a priori estimate for the Dirichlet problem for ∆ yields In terms of (3.4), we obtain Thus the proof of Lemma 1 is completed.
Inverse source problem. We assume that Let an observation subdomain ω ⊂ Ω satisfy ∂Ω ⊂ ∂ω and T > 0 be suitably The condition ∂ω ⊃ ∂Ω means that ω ⊂ Ω is a neighbourhood of ∂Ω. We can relax the condition ∂ω ⊃ ∂Ω, but we cannot choose an arbitrary subdomain ω, because the equation in u is hyperbolic, so that we need some geometric condition on ω (e.g., [10]). This condition is related with the pseudo-convexity which is necessary for proving a Carleman estimate (e.g., Hörmander [8]).
We are ready to state the main result for the inverse source problem.
Relying on this approach of Carleman estimates which have been proved successful, we are now considering the case of more general geometries to solve the inverse problems for elastic bodies whose equilibrium equations are of Koiter shells type (Li, Miara and Yamamoto [21]) and relax the condition of the shallowness introduced in [23].