EXISTENCE OF SOLUTIONS FOR A SYSTEM OF ELASTIC WAVE EQUATIONS

A simple and short proof of the existence of solutions for the direct scattering problem associated with the system of elastic wave equations is shown. Subjclass : Primary 35B40, 76Q05, Secondary 73D35, 73C02


Introduction
The propagation of time-harmonic elastic waves by impenetrable bounded obstacle D ⊂ R 3 with Ω = R 3 \D its complement in R 3 and polarization p 0 ∈ R 3 leads to exterior boundary value problems for the system here σ ∈ C is the frequency of the incident wave v in and v = v in +v sc , where v =(v 1 , v 2 ,v 3 ), v sc = (v sc 1 , v sc 2 , v sc 3 ), denote the displacement of the refracted and scattered wave, respectively.
The total wave v it is required to fulfill the Kupradze-Sommerfeld radiation conditions uniformly for all directions x = ( 1 |x| ) x, where v = v L + v T it is a sum of an irrotational (lamellar) vector v T and a solenoidal vector v L .Here b 2 = µ and a 2 = λ + 2 µ, where λ, µ are the Lamé coeficientes of the Elastycity Theory, σ L ∈ C is the longitudinal (dilational) wave number, σ L = σ b and transverse (shear) wave number σ T = σ a ∈ C, with a 2 > ( 43 ) b 2 > 0. For a soft obstacle the unknown scattered wave v sc = v − v in has to satisfy the Dirichlet boundary condition v sc = − v in on ∂Ω whereas for hard obstacle v sc has to satisfy a Neumann boundary condition T n v sc = −T n v in on ∂Ω, where n = (n 1 , n 2 , n 3 ) denotes the unit outward normal to the boundary ∂Ω and T n is the stress vector calculated on the surface element is the derivative with respect to the outer normal n on ∂Ω.
Problems involving the propagation of time-harmonic elastic waves as above arise naturally in many situations, particularly those involving fluid-structure interaction (see for instance [7,14] and [16]) and in the existence (localization) of the scattering frequencies (see for instance [3,4] and [17]) which are problems of significant interest.
The existence of solutions for the exterior boundary value problems above based on boundary integral equations, appear, for example in [1,2,11,16] and by other methods in [7,8], and references therein.Here, the basic results needed to devolp efficient tools for inverse scattering problems are provided.This is of great practical interest (see for instance [9] and [10]).
In this work we consider the scattering of time-harmonic plane elastic waves in a homogeneus isotropic medium at an obstacle D. We hence present a simple and short proof of the existence of solutions for the exterior boundary value problem associated with the reduced system of elastic wave equations together with the Dirichlet boundary condition v(x) = 0, x ∈ ∂Ω (1.4) and the Kupradze-Sommerfeld radiation condition (1.1) and (1.2), respectively.In (1.3) h = (h 1 , h 2, h 3 ) is a given function and σ ∈ C. To this end, we use a tecnique similar to the one discussed in ( [18], p.p, 35-36) and [5,6], in this sense our approach is new.
Outline of the work: In section §2 we present the formulation of the main result.In section §3 we give the proof the main theorem.Finally, in the section §4 we present the meromorphic extension of the solution for every σ ∈ C with (σ) ≤ 0.
We shall use the standart notation: Here and throughout this work we assume that Ω = R 3 \D is the exterior of D with smooth boundary ∂Ω.Also, we denote by grad the gradient, by rot the rotational vector, ∆v = ( v 1 , v 2 , v 3 ), where is the usual Laplacian operator and div the divergence.For any positive integer p and 1 ≤ s ≤ ∞ we consider the Sobolev space W p,s (Ω) of (classes of) functions in L s (Ω) which together with their derivatives up to order p belong to L s (Ω).The norm of W p,s (Ω) will denoted by • p,s in the case s = 2 we write H p (Ω) instead of W p,2 (Ω).If E is a vector space then we denoted [E] 3 = ⊕ 3 i=1 E and the norm of a vector v wich belong to [E] 3 will be denoted by ) denotes the space of all C ∞ functions defined on R 3 with compact support.If E is a Banach space, we consider the space B(E, E) of linear bounded operators in E. If h :R 3 → R 3 , h = (h 1, h 2 , h 3 ) then we denoted by supp h = ∩ 3 i=1 supp h i the support of h and If R > 0 then B(R) is the ball centered at zero and radius R. Also, we denoted by With all these notations we stablish now our main theorem

Formulation of result
In this section we shall establish the existence of solutions for the systems of elastic waves This will be done based on [5,6].Our starting point is the following lemma whose proof appears in [6,11].
Existence of solutions for a system of elastic wave equations 309 satisfying the Kupradze-Sommerfeld radiation condition for a 2 > ( 43 ) b 2 .Then we have is a linear continuous operator.In particular, if v 1 and v 2 solves (2.1) and satisfies the Kupradze-Sommerfeld radiation condition, then See [5] for the proof. 3and take f 0 given by where ψ is the function
Next we summarize the well known result given, for example in [15].
Then w(x) = 0 for every x ∈ B(R).See [15] or [6] for the proof.At this point, we derive from the above lemmas the proof the main theorem. 3.Furthermore, v can be extended in a meromorphic way to σ ∈C with (σ) ≤ 0 except, for some countable number of poles in Ξ = {σ ∈ C : (σ) ≤ 0}.

Proof of theorem 1
The proof of Theorem 1 is divided into two parts.
Proof.Uniqueness: Let v be the difference of two solutions v 1 and v 2 of (2.4), then v satisfies (2.4) with h = 0. Now, let R > 0 be such that ∂B(R) is contained in Ω and denoted by Ω R = {x ∈ Ω : |x| ≤ R}, the Bettis-Green formula (see for instance [11] or [13]) yields to Now, using (3.1) toghether with Lemma 1, the homogeneus boundary condition and passing to the limit as R → ∞ we get and Hence, v = 0 on Ω.Therefore, (a) and (b) implies v 1 = v 2 .And the uniqueness is proved for all σ ∈ C with (σ) > 0.
Existence: Now we study the existence of solutions for the system (2.4), to this end, we assume that ∂Ω is sufficiently regular for the use of the Betti-Green formula.Let R > 0 and R 0 > 0 be such that B(R 0 ) ⊂ D, ∂Ω ⊂ B(R).We start with an arbitrary function In order to analyze our existence problem we introduce here the following function where u is the Calderón extension to R 3 of a (see for instance [12] Here, v 0 satisfies (see Lemma 2) the differential equations and the Kupradze-Sommerfeld radiation condition.From (3.5) and (w2) we obtain Furthermore, it is easy to see from (ζ1) and (3.5) that v(x) = v 0 (x), for every x ∈ R 3 /B(R).Now, the function v 0 satisfies the Kupradze-Sommerfeld radiation condition (1.1) and (1.2).In view of this, the function v has this property.Thus, for any h ∈[L 2 (Ω)] 3 with supp h ⊂ Ω R and σ ∈C with (σ) > 0 the function v(x) = v 0 (x) + ζ(x) u(x), x ∈ R 3 , will be a solution of the system (2.4) if only if, for every x ∈ Ω, we obtain It is simple to see from (ζ1), (ζ3) and ( 2. well be solution of the system (3.6) if only if, for every x ∈ Ω R , we have Applying to div w the operator grad, on Ω R we find rot (rot w(x)) = −∆ w(x) + grad (div w(x)).
Therefore, the ansatz (3.7) takes the form given by the formula On the other hand, the solution operator P (σ) associated with the system (w1-w3) above, that is, P (σ) g = w, where and depends analytically from σ ∈ C, because v 0 has this property.In a similar fashion, the trace is a continuous linear operator.Thus, with this operators and taking into account the fact that v 0 = v 0|Ω R on Ω R , (3.8) can be written in the form is a restriction, continuous linear operator.Also, is a continuous linear operator given by the compotition A(σ)r = A(σ)M ψ r, where A(σ) it is the solution operator of the system b 2 ∆v 0 (x) + (a 2 − b 2 ) grad(div v 0 (x)) + σ 2 v 0 (x) = f 0 (x), x ∈ R 3 , (see Lemma 2) and M ψ it is the multiplication operator Note that Thus, (3.13) can be written as From these considerations we see that the theorem will be proved if: Proof.

Meromorphic Extension
In the previous sections the existence and uniqueness of solutions for the system x ∈ ∂Ω, Radiation condition, (4.1) with σ ∈ C such that (σ) > 0 is proved.Now, in this section we present the extension of the solution for all σ ∈ C such that (σ) ≤ 0 except, for some countable number of complex singularities, called "resonant frequencies".Our approach follows the main ideas of the previous sections and the subjet iniciated in [5] and [6], but it is related to some other works mainly [3], [4], [17] among other.The basic tools for the proof is the Steinberg theorem [20] about families of compact operators depending on a complex parameter (see, also [19]).With the same notations of the section §2 and §3, we stablish the following