A Note on Resonant Frequencies for a System of Elastic Wave Equations

We present a rather simple proof of the existence of resonant frequencies for the direct scattering problem associated to a system of elastic wave equations with Dirichlet boundary condition. Our approach follows techniques similar to those in Cortés-Vega (2003). The proposed technique relies on a stationary approach of resonant frequencies, that is, the poles of the analytic continuation of the solution. 1. Introduction. The existence of resonant frequencies (or poles) of the analytic continuation of the so-called S-matrix (which connects the asymptotic behaviors of the incident and scattered waves), associated with symmetric hyperbolic systems of first order in exterior domains and coercive boundary condition and enjoying the unique continuation property is a problem of significant interest in the time-dependent scattering theory of Lax and Phillips; these complex numbers present resonant properties for the wave motion. A good discussion of this problem and important results on the subject may be found in [21, Chapter IV] and the survey articles (see [22, Theorems 5.5 and 5.6]). Apart from their intrinsic interest, the resonant frequencies are also relevant as a very rich source of interesting problems. For instance, in the so-called inverse problems the existence and location in the complex plane should give some information about the fashion and size of the obstacle. This kind of results for perturbations of the scalar wave equation appears in [19, 20, 25, 27] and the references therein. Subsequently, extensive attention in this and other aspects for the direct scattering problems associated to the system of elastic waves and the scalar wave equation ap-for recent results. In this context, our goal in this work is to develop a rather simple proof of the existence of resonant frequencies associated with a phenomenon described by a system of elastic waves with prescribed Dirichlet operator on the boundary ∂Ω ∈ C 2 of an

1. Introduction. The existence of resonant frequencies (or poles) of the analytic continuation of the so-called S-matrix (which connects the asymptotic behaviors of the incident and scattered waves), associated with symmetric hyperbolic systems of first order in exterior domains and coercive boundary condition and enjoying the unique continuation property is a problem of significant interest in the time-dependent scattering theory of Lax and Phillips; these complex numbers present resonant properties for the wave motion. A good discussion of this problem and important results on the subject may be found in [21, Chapter IV] and the survey articles (see [22,Theorems 5.5 and 5.6]).
Apart from their intrinsic interest, the resonant frequencies are also relevant as a very rich source of interesting problems. For instance, in the so-called inverse problems the existence and location in the complex plane should give some information about the fashion and size of the obstacle. This kind of results for perturbations of the scalar wave equation appears in [19,20,25,27] and the references therein.
In this context, our goal in this work is to develop a rather simple proof of the existence of resonant frequencies associated with a phenomenon described by a system of elastic waves with prescribed Dirichlet operator on the boundary ∂Ω ∈ C 2 of an arbitrary domain Ω = R 3 /D: (1.1) and the Kupradze-Sommerfeld radiation conditions [18] Kup L In this context, a resonant frequency is a complex number σ for which system (1.1) and (1.2) with h ≡ 0 has a nontrivial solution v. Our proof follows similar lines to the arguments in [6,7], the analysis is based on a stationary approach of resonant frequencies, that is, the poles of the analytic continuation of the solution operator. In my view, the technique combines the attributes of both simplicity and flexibility. Indeed, as pointed out in [6], this method can be used in situations not included in the time-dependent theory of Lax and Phillips [21], for instance, the impedance problem with absorbing boundary conditions [20] or acoustic resonators [11].
The linearized system equations of the time-dependent problem from which one obtains (1.1) and (1.2) are the following (a mathematical formulation of this problem in terms of semigroups of linear operators is studied in [2]): is the displacement at the time t and location x ∈ R 3 scattered by the obstacle D, f = (f 0 , f 1 ) is the initial value for this Cauchy problem, h = (h 1 ,h 2 ,h 3 ) is a given function, and σ ∈ C.
In general context, the resonant frequencies associated to the model (1.3) are complex numbers, which are, in some sense, eigenfrequencies of the generator and characterize the asymptotic behavior of the solutions as time approaches infinity.
To state our main result, we introduce some notations which will be used throughout the note: let Ω = R 3 \D be the exterior of D with boundary ∂Ω ∈ C 2 . Also, we denote by ∇ the gradient, by ∇ x × v the rotational vector of v, ∇ x · v is the usual divergence of v (see, above), and where is the usual Laplacian operator. 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 be 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 denote [E] 3 = ⊕ 3 i=1 E and the norm of a vector v which belongs to [E] 3 will be denoted by · [E] 3 . C ∞ 0 (R 3 ) 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 : If R > 0, then B(R) is the ball centered at zero and of radius R. Also, we denote by For any two vectors A and B of R 3 , we denote by A · B the usual inner product between A and B. If v : R 3 → R has partial derivatives and x = 0, then ∂v/∂|x| denotes the radial derivative of v, that is, Now, if w : R 3 → R 3 is such that each component has partial derivatives, then (1.8) Outline of the work. In Section 2, we present the formulation of the main result. Section 3 contains the proof of the main theorem. Finally, in Section 4, we present the meromorphic extension of the solution for every σ ∈ C with (σ ) ≤ 0. With the notations above, we establish our main theorem.

Formulation of result.
In this section, we will establish the existence and uniqueness of the solution to a system of elastic waves that is presented in (1.1) and by the radiation conditions (Kup L ) and (Kup T ) in (1.2).
This will be done based on [6,7]. Our starting point is the following lemma whose proof appears in [6,18].
satisfying the Kupradze-Sommerfeld radiation condition for

2)
where "·" is the inner product in R 3 and T x is the stress vector calculated on the surface element is a linear continuous operator. In particular, if v 1 and v 2 solve (2.4) and satisfy the Kupradze-Sommerfeld radiation condition, 3 and take f 0 given by where ψ is the function and Ω R = {x ∈ Ω : |x| < R}. 3 and suppose ∂Ω ∈ C 2 . Then the system of elastic waves 3 . See [6] for the proof. For future reference the well-known result given, for example, in [24] is also needed.
At this point, we derive from the above lemmas the proof of the main theorem.
3. Proof of Theorem 2.5. The proof of Theorem 2.5 is divided into two steps.
Step 3.1 (uniqueness). The uniqueness of the only solution to (2.10) can be established by a standard argument. For the precise details we refer to the appendix.
Step 3.2 (existence). Here, we study the existence of solutions for the system (2.10); to this end, we assume that ∂Ω ∈ C 2 for the use of the Betti-Green formula. Let R > 0 In order to analyze our existence problem, we introduce here the following function:  3 and v 0 satisfies (see Lemma 2.2) the system will be a solution of the system (2.10) if and only if, for every x ∈ Ω, we obtain It is simple to see from (ζ1), (ζ2), (2.6), and (2.7) that (3.5) is valid on the set will be solution of the system (2.10) if and only if, for every x ∈ Ω R , we have Applying to ∇ x · w the operator ∇ on Ω R we find Therefore, the ansatz (3.5) takes the form where G ζ (σ ) is a continuous linear operator given by the formula On the other hand, the solution operator P (σ ) associated with the system (w1), (w2), and (w3), that is, 3 , is well defined, of course; P (σ ) is a continuous linear operator In a similar fashion, the trace where Λ n v 0 = g is a continuous linear operator. Thus, with this operator and taking into account the fact that v 0 = v 0|Ω R on Ω R , (3.10) can be written in the form is a restrictive, continuous linear operator. Also, is a continuous linear operator given by the composition where A(σ ) is the solution operator of the system (see Lemma 2.2), and M ψ is the multiplication operator Note that Therefore, M ψ is a continuous linear operator 3 for every r ∈ [L 2 (Ω R )] 3 . Let B ζ (σ ) be the operator defined by Thus, (3.17) can be written as From these considerations we see that the theorem will be proved if (I) the set of operators {B ζ (σ )}, σ ∈ C, with (σ ) > 0, given in (3.23) is a family of compact operators of [L 2 (Ω R )] 3 onto itself, and the homogeneous equation has only the trivial solution.

Proof (I).
We denote by are continuous applications. Therefore, item (I) is a simple consequence of (3.23) and of the compactness of i : 3 . See the operators in the following diagram: (3.28) We are now ready to prove (II).

Meromorphic extension.
In the previous sections and the appendix, the existence and uniqueness of the solution for the system that is presented in (1.1) and by the radiation conditions (Kup L ) and (Kup T ) in (1.2) with σ ∈ C with (σ ) > 0 is proved. Now, the goal of this section is to present the extension of the solution for all σ ∈ C such except for some countable number of complex singularities, called "resonant frequencies." Our approach follows the main ideas of the previous sections and the subject initiated in [6,7], but it is related to some other works, mainly [1,3,2,4,8,9,11,25]. The basic tool for the proof is the Steinberg theorem [31] about families of compact operators depending on a complex parameter (see also [28]). With the same notations of Sections 2 and 3, we establish the following. (4.1) solves the system (1.1) and (1.2) if only if f ∈ [L 2 (Ω R )] 3 solves Here, B ζ (σ ) is given by (3.23) where the operators are given in (3.13), (3.14), (3.15), (3.16), and (3.17), respectively.
Proof. The proof is implicit in Theorem 2.5. Proof. Since the solution v 0 from system  Proof. From Lemma 4.2, we have that the set {B ζ (σ )} with σ ∈ C and (σ ) > 0 is an analytic family of compact operators of [L 2 (Ω R )] 3 onto itself. By the Steinberg theorem [31], either (a) the operators [I +B ζ (σ )] −1 are never invertible for σ ∈ C, or (b) there is σ 0 ∈ C such that the operator is invertible. From Theorem 2.5 we have the existence and uniqueness of the solution for the system (1.1) and (1.2) for all σ ∈ C with (σ ) > 0; by the equivalence established in Lemma 4.2 we are in case (b). In this case, Steinberg's theorem also states that is defined analytically on C except for a countable set of poles. Now, Lemma 4.2 yields the equivalence statement.
Final remark. It can be thought that there is a reasonable parallelism between my former paper [7] and this note, but it is necessary to remark that a complete parallelism does not hold if we consider the imposed boundary conditions in both problems. Indeed, in [7] we studied the system of elastic waves with the Neumann boundary condition, whereas the condition that we impose here is of Dirichlet type. The results obtained in [7] and those of this work are analogous (which is a virtue of the technique). However, the models are different, because it is a known fact that in the Neumann case an interesting phenomenon related to the existence of surface waves exists (Rayleigh surface waves, who mathematically predicted the existence of this kind of waves in 1885) ; such waves remain near the border of the obstacle; this fact too stimulated the interest of many researchers in the influence of surface waves near scattering objects; see [32,33]. A first implication of this fact is that the uniform decay (in the local energy) of the solution is not preserved. As was already proved in [16,17], for the case of the isotropic elastic wave with Neumann boundary condition, the solution does not have uniform local energy decay. Important contributions in this direction appear in [29,30] and the references therein. In contrast to the Neumann case, it is a well-known fact that for the system studied here the phenomenon of Rayleigh is absent, and as a consequence, the uniform decay of the solutions may be guaranteed; see, for instance, [14,15]. Thus, independent of the differences in both models, the method presented here is still successful.
Appendix. In this appendix, we prove the uniqueness of the solution to the system that is presented in (1.1) and by the radiation conditions (Kup L ) and (Kup T ) in (1.2).