Radiating and non-radiating sources in elasticity

In this work, we study the inverse source problem of a fixed frequency for the Navier's equation. We investigate that nonradiating external forces. If the support of such a force has a convex or non-convex corner or edge on their boundary, the force must be vanishing there. The vanishing property at corners and edges holds also for sufficiently smooth transmission eigenfunctions in elasticity. The idea originates from the enclosure method: The energy identity and new type exponential solutions for the Navier's equation.


Motivation and background
The inverse source problem is an important research topic in scattering theory. Its goal is to determine the shape of unknown sources by measuring the radiated wave patterns in the far field. This problem is motivated by several scientific and industrial areas, such as medical imaging [21] and photo-acoustic tomography [2]. A classical example outside of elasticity is to utilize electric or magnetic fields on the periphery of the human body, such as in noninvasive brain image reconstruction. Mathematically, the inverse source problem for acoustic, electromagnetic and elastic waves has been studied widely by many researchers [1,3,5,17,18,28,33]. Furthermore, the inverse source problem can also be regarded as a basic mathematical method to study diffusion-based optical tomography, lidar imaging and fluorescence microscopy.
The inverse source problem of a fixed frequency is ill-posed and its solution requires a priori knowledge. These types of problems have attracted interest before too, but for different equations. The umbrella term Schiffer's problem [16] captures the essence of these problems: from the knowledge of one far-field pattern, or the Cauchy data of one solution, determine the shape of a scatterer. Friedman and Isakov [22] showed the unique determination of penetrable inclusions for the conductivity equation by the Cauchy boundary data of one solution, with reconstruction via the enclosure method appearing in [29]. In [28], the enclosure method is used to reconstruct the the convex hull of a polyhedral source or potential for the Helmholtz equation given a single measurement. In [30], the authors adapted the enclosure method for an inverse obstacle problem by applying a single set of the Cauchy data to the Navier equation. Furthermore, in a slightly different context, Schiffer's problem was studied in a non-local setup, see [14].
A new approach to this problem began from the realization that an acoustic potential that has a corner jump would scatter any incident wave at any wavenumber [11]. This applies to the potential scattering setting without a source. However, trying to apply it to source scattering produces an integration by parts formula used in the enclosure method [28]. The corner scattering method was then extended and used for the shape determination of polyhedral potential scatterers [27], and to sources in [7]. The latter used a new type of complex geometrical optics solution and the enclosure method. These techniques also led to the surprising discovery that interior transmission eigenfunctions [15] vanish at convex corners [8,9]. For more corner and edge scattering results, we also refer readers to [19,20]. In this paper we extend the single measurement enclosure method technique to the elastic setting.

The mathematical formulation of elastic waves
Let λ, μ be the Lamé constants satisfying the following strong convexity condition µ > 0 and nλ + 2µ > 0, for n = 2, 3. (1.1) Let f ∈ C n be an external force, which is assumed to be compactly supported. More specifically, the function f = χ Ω ϕ, where χ Ω is the characteristic function of a bounded Lipschitz domain Ω in R n and ϕ ∈ L ∞ (R n ; C n ). Given an angular frequency ω > 0, let u(x) = (u (x)) n =1 be the displacement vector field. Then the time-harmonic elastic system is Via the well-known Helmholtz decomposition in R n \Ω, one can see that the scattered field can be decomposed as where ω p and ω s are the compressional and shear wave numbers, respectively, which are given by Above rot = ∇ ⊥ represents π 2 clockwise rotation of the gradient when n = 2, and rot = ∇× stands for the curl operator when n = 3. The vector fields u p and u s are called the compressional and shear parts of the scattered vector field u, respectively. In addition, recall that f = 0 in R n \ Ω. Then u p and u s satisfy the Helmholtz equation Therefore, for the elastic scattering problem of equation (1.2), we need to pose the Kupradze radiation condition (1.4) uniformly in all directions x = x/|x|. Moreover, one can also expand the functions u s and u p as as |x| → ∞, (1.5) for n = 2, 3, where u ∞ s and u ∞ p denote the transversal and longitudinal elastic far fields radiated by the source f . Furthermore, u ∞ s and u ∞ p can be explicitly represented by for any unit vector e ∈ S n−1 , where Π e is the projection operator with respect to e. Notice that the vector fields u ∞ s and u ∞ p are the tangential and the normal components of the Fourier transform of f evaluated on S n−1 . Note that the elastic far fields (1.5) of the Navier equation are derived using the Helmholtz decomposition of equation (1.2) and the far-field patterns for the Helmholtz equations of (1.3), which is allowed by the radiation conditions of equation (1.4). For a more detailed discussion, we refer readers to [24][25][26].
It is known that for a given source function f ∈ L ∞ (Ω; C n ) ⊂ L 2 (Ω; C n ), the scattering problem of equations (1.2) and (1.4) has a unique solution (see [4] for instance): where G(x, y; ω) ∈ C n×n is the Green tensor for the Navier equation in equation (1.2). More precisely, the Green tensor can be expressed as where I n denotes an n × n identity matrix and is the fundamental solution for the Helmholtz equation. Here H (1) 0 is the Hankel function of the first kind with order zero.
We prove the following three theorems. The first one is for the inverse source problem. It states that if an external force is applied to a region having a corner or edge on an elastic body, it creates a propagating elastic wave at any wavenumber. 3} and bounded vector function ϕ ∈ L ∞ (R n ). Let ω, µ > 0, nλ + 2µ > 0 and u ∈ H 2 loc (R n ) satisfy equation (1.2) and the radiation condition of equation (1.4).
Assume that Ω has a corner (2D) or an edge (3D) that can be connected to infinity by a path in R n \ Ω, and that ϕ is Hölder continuous near it. If u has zero far-field pattern, then ϕ = 0 on the corner or edge, i.e. ϕ is the zero vector. In other words, f has no jumps at these locations.

Remark.
Recall that by the Helmholtz decomposition, one can reduce the Navier equation of (1.2) into two Helmholtz equations, as in (1.3), outside of Ω. One might wonder if it were possible to prove theorem 1.1 using the same decomposition for the source term and using a version of theorem 1.1 proven in [7] that holds for the two Helmholtz equations of (1.3) separately. After all, by the Helmholtz decomposition, one could split f = f p + f s and rewrite equation (1.2) as However there is a problem of smoothness. Theorem 1.1 assumes only that the function f is Hölder continuous near a corner. This does not guarantee that both of f p and f s are Hölder continuous near the same corner. This prevents the use of [7, theorem 1.1] to derive the same conclusion for the Navier equation.
In order to prove theorem 1.1, we need to construct a new type of exponential solution for the Navier equation in the plane. The classical exponential solutions for the Navier equation are called the complex geometrical optic (CGO) solutions. As a matter of fact, the CGO solutions [31] were utilized to show global uniqueness for the inverse problem of the isotropic elasticity system with infinite measurements. In [31], the authors introduced CGO solutions for the isotropic elasticity system by decoupling the system into weakly coupled systems. Their principal part is an exponential of quadratic complex-valued function.
The second theorem shows source shape and boundary value determination from a single far-field pattern. 3} and Ω, Ω ⊂ R n be bounded convex polyhedral domains. Let ϕ, ϕ ∈ C α (R n ), for some α ∈ (0, 1) and have non-zero value on ∂Ω, ∂Ω .

In other words they satisfy equation (1.2) with the radiation condition of equation (1.4).
If u and u have the same far-field pattern then Ω = Ω and ϕ = ϕ at each of their vertices and in three dimensions, edges.
Strictly speaking, we do not need to have ϕ, ϕ Hölder continuous everywhere or nonvanishing on the whole boundary for the unique determination of the shape. Near the corners is enough. In three dimensions, if they are in addition non-vanishing and Hölder continuous near the edges, then we can also deduce that ϕ = ϕ on them.

Definition 1.3 (Interior transmission eigenfunctions). A pair
Above T ν is the boundary traction operator.
Definition 1.4. The boundary traction operator T ν is defined as follows. In the two-dimensional case it is is a unit outer normal on ∂Ω and ν ⊥ := (−ν 2 , ν 1 ). In the three dimensional case, The interior transmission problem is a well studied problem in inverse scattering theory. In particular the sampling method for solving the inverse scattering problem fails at wavenumbers that are transmission eigenvalues [16]. Notable results [12,15,36,38] and a recent survey [13] have so far focused mostly on the spectral properties of the transmission eigenvalues. Only recently, e.g. [10,32,37], have results about the eigenfunctions themselves started surfacing, in the form of completeness of the eigenfunctions in some sense. These past results are mostly in the context of acoustic scattering for the Helmholtz equation. Bellis et al [6] considers the fundamental properties in the context of elasticity, and shows the existence and discreteness of the transmission eigenvalues.
Recently in [7][8][9] it was shown that under given smoothness and geometric assumptions the transmission eigenfunctions for the Helmholtz equation vanish at convex corners of the domain Ω. We show the same conclusion for the interior transmission problem for an elastic material with varying density. 3} and Ω ⊂ R n be a bounded domain. Let V ∈ L ∞ (Ω) be the material density, and µ > 0, nλ + 2µ > 0 be constant Lamé parameters. Assume that ω > 0 is an interior transmission eigenvalue and v, w ∈ L 2 (Ω) are the corresponding transmission eigenfunctions defined by equation (1.6). Let x c be any vertex or edge of ∂Ω around which V and either one of v, w are C α smooth, for some α ∈ (0, 1). Then so is the other, and v( The paper is structured as follows. In section 2, we construct a new type of exponential solution for the two-dimensional isotropic elastic case and develop the integration by parts formula for the domain with corners. In section 3, we discuss the corner scattering in a plane, and we use the dimensional reduction technique to solve the three-dimensional case. In addition, we prove our theorems in section 4. Finally, in appendix, we show that there is a nonradiating source f that generates elastic far fields that vanish in all directions.

Exponential solutions of the Navier equation in the plane
We write L := λ∆ + (λ + µ)∇(∇·) for the second order elliptic operator. If u solves the elasticity system of equation (1.2) then Lu = f in R n for n = 2, 3, where f is the given compactly supported bounded source function. Next, let v be a solution of Lv = 0. Via the standard regularity theory for the second order elliptic system (see [35, theorem 4.16] for instance), one can obtain that u, v are H 2 loc (R n ) functions. Then integrating by parts twice yields that where T ν stands for the boundary traction operator (definition 1.4) of equation (1.2).
In two dimensions, note that the system of equation (1.2) can be expressed componentwise as From now on we identify R 2 with the complex plane C, and we have the following lemma.
where z = x 1 + ix 2 and s ∈ R +. The complex square root is defined as where −π < θ π is the argument of z. Then v satisfies Lv = 0 in Ω.
Proof. The complex square root is complex analytic in C except at the branch cut composed of the origin and the negative real axis. Thus the components of v = (v 1 , v 2 ) T are complex analytic in Ω. Let us rewrite the differential operator in equation (2.2) using the complex derivatives These imply that ∆ = 4∂∂ and that Hence the above-mentioned operator becomes and thus Lv = 0 if and only if Recall that our choice of v implies that ∂v 1 = ∂v 2 = 0 in Ω. Hence the above reduces to The following lemma provides the integration by parts formula around the corner, where v is not smooth.

Corner scattering Proposition 3.1. Let v : R 2 → C be the function given in lemma 2.1 and define the open sector
In addition for α, h > 0 and j ∈ {1, 2} we have the upper bounds Proof. This result is in [7, section 2], however note that the parameter s is outside of the square root in this article. □
Proof. Write the equation as Lu + ω 2 u = f where L is as in equation (2.2). We may assume that f is real valued. If not, then split it and u into their real and imaginary parts f R , f I , u R , u I which would then satisfy Let v be as in lemma 2.1 and so Lv = 0 in Ω ∩ B. Let f = f − ω 2 u and keep in mind that (3.2) and moreover the left-hand side of equation (3.2) can be split as As in the proof of lemma 2.2, we have where h is the radius of B and θ the argument of x 1 + ix 2 . Let δ K := min θm<θ<θM cos(θ/2) and it is positive. Hence for some positive constant C K,B,µ (depending on K, B and μ), when x ∈ K ∩ ∂B. By a similar consideration on the same part of the boundary. Thus, equations (3.4) and (3.5), the Cauchy-Schwarz and trace inequalities give that which decays exponentially as s → ∞. Let us estimate the three terms in equation (3.3) next. However, before that, we recall f = f − ω 2 u, and that u ∈ H 2 , which embeds continuously into C α in two dimensions. Hence f ∈ C α (K ∩ B). Then proposition 3.1 gives where C K = 0 and the other constants C K , C K,α are finite and positive. Multiplying equations (3.2) and (3.3) by s 4 and letting s → ∞ implies that and since f is real-valued and C K = 0, that f (0) = 0. □
In order to derive the equation for R ξ u, note that in the three-dimensional case, the isotropic elastic operator L can be rewritten as The Lebesuge dominated convergence theorem and an integration by parts formula yield that

It is easy to see that
Next, we want to show that II ξ (x ) = 0 for x ∈ Γ. By denoting Γ := S 1 ∪ S 2 , where S 1 , S 2 are segments and S 1 ∩ S 2 = {x 0 } is the corner point, we only need to demonstrate that II ξ (x ) = 0 on S 1 . By choosing suitable boundary normal coordinates, without loss of generality, we may assume that S 1 × (−L, L) ⊂ span{e 1 , e 2 } ⊂ R 3 with its normal direction ν = e 3 . Here {e 1 , e 2 , e 3 } forms the standard orthonormal basis in R 3 . Recall that u ∈ H 2 loc (R 3 ), then one has ∂uj ∂x k ∈ H 1 loc (R 3 ) for j, k ∈ {1, 2, 3}. Therefore, ∂uj ∂x k | Γ×(−L,L) is a well-defined L 2 (Γ × (−L, L)) function in the trace sense. Since u = 0 on S 1 × (−L, L), we have ∂uj ∂x k = 0 for j = 1, 2, 3 and k = 1, 2. Therefore, by using the boundary traction T ν u = 0 on S 1 × (−L, L), and that µ > 0, λ + 2µ > 0, which follow from the assumptions, one can easily see that ∂uj ∂x k = 0 on Γ × (−L, L) for j, k = 1, 2, 3. Similar arguments hold when x ∈ S 2 , which proves that II ξ (x ) = 0 on Γ. This demonstrates equation (3.9). □ Remark 3.4. The boundary normal coordinates are useful in studying the inverse boundary value problem. The idea is based on the invariance of the elasticity system via change of variables. For example, in [17,34], the authors utilized this technique to study the boundary determination for the isotropic elasticity system from the boundary measurements. Given f ∈ C α ((B × (−L, L)) ∩ Ω; C 3 ) for some α ∈ (0, 1), let u ∈ H 2 ((B× (−L, L)) ∩ Ω; C 3 ) be a solution of Lu = f in B ∩ Ω. Then Proof. The proof is similar to the proof of [7, proposition 3.5]. For the sake of completeness, we offer a detailed proof here. By Sobolev embedding, we know that H 2 embeds to C 1/2 and we may assume α 1/2 without loss of generality. By lemma 3.3, given any ξ ∈ R, there is a F ξ ∈ C α (B ∩ K), such that one can find a solution U ∈ H 2 (B ∩ K) ∩ C α (B ∩ K) fulfilling LU = F ξ in B ∩ K, where L is defined by equation (3.8). In addition, U = T ν U = 0 on B ∩ ∂K, via proposition 3.2, then we obtain F ξ (0) = 0. Finally, recall that for any smooth cut-off functions φ(x 3 ) ∈ C ∞ c ((−L, L)) and for any ξ ∈ R. The Fourier inversion formula implies that f (0) = 0.

Proof of theorems
At the end of this paper, we prove our theorems stated in section 1.
Proof of theorem 1.1. Rellich's lemma for the Helmholtz equation (see e.g. lemma 2.11 in [16]) and the unique continuation principle imply that u p = u s = 0 in the connected component of R n \ Ω that reaches infinity. Hence u = 0 and T ν u = 0 on the boundary of the corner or edge. The claim follows from propositions 3.2 or 3.5. □ Proof of theorem 1.2. By Rellich's lemma for the Helmholtz equation again and the unique continuation principle u p = u p , u s = u s in R n \ Ω ∪ Ω . Assume Ω ⊂ Ω . Then by convexity there is a corner (2D) or edge (3D) point x c ∈ ∂Ω \ Ω . Since u = u outside Ω ∪ Ω we have u = u and T ν u = T ν u on ∂Ω near x c . Set w = u − u . We have λ∆w + (λ + µ)∇∇ · w + ω 2 w = f in Ω near x c with w ∈ H 2 . Propositions 3.2 and 3.5 imply that ϕ(x c ) = 0. But this is a contradiction since ϕ = 0 on ∂Ω. Hence Ω ⊂ Ω . The same proof with Ω, Ω switched shows that Ω ⊂ Ω. Hence Ω = Ω . Next, let x c be a vertex (2D) or an edge point (3D) of ∂Ω = ∂Ω . If w = u − u then this time λ∆w + (λ + µ)∇∇ · w + ω 2 w = f − f in Ω with w ∈ H 2 . Rellich's lemma for the Helmholtz equation and the unique continuation principle for the Navier equations imply that w = 0 and T ν w = 0 on ∂Ω near x c in this case too. Propositions 3.2 and 3.5 imply f = f at x c . □ Finally, we can prove the third main theorem in this paper.
Proof of theorem 1.5. Set f = −ω 2 Vv and u = v − w. In two and three dimensions H 2 embeds into C α for 0 < α < 1/2 the latter of which we may assume. So v − w is Hölder continuous, and thus both v and w are too if either one is in C α . These functions satisfy λ∆u + (λ + µ)∇∇ · u + ω 2 u = f with u ∈ H 2 (Ω), u = T ν u = 0 on ∂Ω, and f ∈ C α (Ω). Propositions 3.2 and 3.5 imply that f (x c ) = 0, so if V(x c ) = 0 then v(x c ) = 0 and since v = w on ∂Ω, so is w(x c ) = 0. □ Therefore, from equations (A.1) and (A.2), having the elastic far fields u ∞ s = u ∞ p = 0 can be reduced to seeking the Lamé parameters (λ, µ) such that where ω s = ω/ √ µ and ω p = ω/ √ λ + 2µ. Let A, B > 0 be any solutions of J 3 2 (t) = 0 with B greater than A large enough. Combining with equation (A.3), without loss of generality, then we can choose This is equivalent to taking the Lamé parameters as Note that by choosing B large enough, the Lamé parameters λ and μ satisfy the strong convexity condition equation (1.1). Hence, for any frequency ω > 0, there are examples (λ, µ) and source f that is non-zero, but its far fields are zero.