ON THE CHARACTERIZATION OF NON-RADIATING SOURCES FOR THE ELASTIC WAVES IN ANISOTROPIC INHOMOGENEOUS MEDIA

ON THE CHARACTERIZATION OF NON-RADIATING SOURCES FOR THE ELASTIC WAVES IN ANISOTROPIC INHOMOGENEOUS MEDIA

PU-ZHAO KOW AND JENN-NAN WANG

Abstract. In this paper, we would like to characterize non-radiating volume and surface (faulting) sources for the elastic waves in anisotropic inhomogeneous media. Each type of the source can be decomposed into a radiating part and a non-radiating part. The radiating part can be unique determined by an explicit formula containing the near-field measurements. On the other hand, the non-radiating part does not induce scattered waves at a certain frequency.

In other words, such non-radiating source can not be detected by measuring field at one single frequency in a region outside of the domain where the source is located.

1. Motivation of study and mathematical setup

Seismic waves in earth are typically generated by two types of sources. One is the external sources including winds, volcanic eruptions, vented explosions, meteorite impacts, etc. The other is the internal sources such as earthquakes or underground explosions. Seismic waves generated by external sources are usually foreseeable, while those induced by internal sources are hard to predict and often cause massive destruction. In this paper, we are interested in characterizing internal sources in terms of the scattering theory.

Internal sources can be grouped into two categories: volume sources and surface sources. In the seismic terminology, surface sources are called faulting sources resulting from slips across fracture planes [AR02, Chapter 3]. It was generally recognized that earthquakes are due to waves radiated from spontaneous slippage on active geological faults. According to U. S.

Geological Survey¹, there are more than a hundred of significant earthquakes around the world each year. Seismic waves generated by non-radiating sources are only confined in a bounded region. In this sense, the existence of non-radiating sources poses little threats to the nature environment. The main theme of this paper is to characterize non-radiating internal sources, both volume and surface sources. As a byproduct, we also derive reconstruction formulae of determining radiating volume and surface sources by the near-field measurements at one single frequency. For our problem, we will consider the stationary elastic wave equation in anisotropic inhomogeneous media.

1991 Mathematics Subject Classification. 35J47; 35Q74; 35R30; 74B05.

Key words and phrases. Non-radiating sources, Volume sources, Surface sources, Elastic waves.

1https://earthquake.usgs.gov/earthquakes/browse/significant.php?year=2019 1

To set up our mathematical problem, let Ω ⊂ R³ be a domain with Lipschitz boundary

∂Ω and R³\ Ω is connected. Before introducing the elasticity tensor, we first define



















(A : B)ijk` :=

3

X

p,q=1

A_ijpqB_pqk` for two tensors A and B,

A : B :=

3

X

i,j=1

a_ijb_ij for two matrices A, B,

|A|² := A : A for arbitrary matrix A.

Assumption 1.1 (Assumptions on the elasticity tensor). Let C(x) = (Cijk`(x))1≤i,j,k,`≤3

be a real-valued elasticity tensor such that each entry C_ijk` ∈ C^∞(R³) satisfies symmetry properties

C_{ijk`</sub>(x) = C<sub>k`ij}(x), C_{ijk`</sub>(x) = C<sub>jik`}(x) in R³.

for all 1 ≤ i, j, k, ` ≤ 3. Moreover, assume that the strong ellipticity holds, that is, there exist constants 0 < κ₁ < κ₂ such that

(1.1) κ₁|A|² ≤ A : C(x) : A ≤ κ2|A|² for all x ∈ R³,

and for all (complex-valued) matrix A. In addition, we assume that C is isotropic and homogeneous outside Ω, with Lamé constants λ and µ, that is,

(1.2) C_{ijk`</sub>(x) = λδ<sub>ij</sub>δ<sub>k`}+ µ(δ_ikδ_{j`</sub>+ δ<sub>i`}δ_jk) in R³\ Ω.

Remark 1.2. In R³\ Ω, (1.1) holds whenever the Lamé constants in (1.2) satisfy µ > 0 and 3λ + 2µ > 0. Moreover, from (1.2), we have

∇ · (C(x) : ∇u) = L^λ,µu := µ∆u + (λ + µ)∇(∇ · u) in R³\ Ω.

For general anisotropic media, we denote

L^Cu = ∇ · (C(x) : ∇u).

In the following section, we will describe the non-radiating sources, both volume and surface ones, in detail.

2. Main results and consequences

The theme of this section is to give precise definitions of non-radiating volume and surface sources and statements of main theorems. In view of main theorems, we then derive some interesting consequences in the case of homogeneous isotropic media. The proofs of theorems will be deferred to the later sections.

2.1. Volume sources. Suppose f ∈ [L²(R³)]³ with supp (f ) ⊂ Ω (possibly complex-valued), denoted by [L²_Ω(R³)]³. Let ω > 0 be a frequency and consider the following time-harmonic elasticity equation:

(2.1)

(∇ · (C(x) : ∇u) + ω²u = −f in R³, u satisfies the Kupradze radiation condition at |x| → ∞, with u ∈ [H_loc¹ (R³)]³.

Before we explain the Kupradze radiation condition, we first recall the following well-known fact, which can be found in [KGBB79, Theorem III.2.2.5 (p.123)].

Lemma 2.1. Given any open set D ⊂ R³ with smooth boundary, if u is a (smooth) solution to

(2.2) (L^λ,µ+ ω²)u = 0 in D,

then u can be decomposed into compression and shear components, that is, it can be repre- sented as the sum of vectors

u(x) = u^(p)(x) + u^(s)(x), where u^(p), u^(s) satisfy Helmholtz equations

((∆ + k²_p)u^(p) = 0, curl u^(p) = 0

(∆ + k²_s)u^(s)= 0, div u^(s) = 0 in D, where k_p² = ω²/(λ + 2µ) and k_s² = ω²/µ.

The following Definition can be found in [KGBB79, Definition III.2.2.6 (p.124)].

Definition 2.2. Suppose that u is a solution to (2.2) with D = R³ \ B_r(0) for some r > 0.

Let u^(p) and u^(s) be given in Lemma 2.1. We say that u satisfies the Kupradze radiation condition at |x| → ∞, if

lim

|x|→∞u^(p)(x) = 0, lim

|x|→∞|x|



∂_|x|u^(p)(x) − ik_pu^(p)(x)



= 0, lim

|x|→∞u^(s)(x) = 0, lim

|x|→∞|x|



∂|x|u^(s)(x) − ik_su^(s)(x)



= 0,

where ∂|x| = ˆx · ∇ (see, for example, [KGBB79]). Here and after, we denote i =√

−1.

Definition 2.3. We say that f ∈ [L²_Ω(R³)]³ is a non-radiating volume source, if there exists a number R > 0 be such that u(x) = 0 for all |x| > R.

Remark 2.4. By Rellich’s lemma, f is non-radiating if and only if the far-field pattern of u in (2.1) vanishes identically. Indeed, non-radiating sources also can be written in the form of interior transmission problem, see Appendix A.

Let us denote

E(Ω) := v ∈ (H¹(Ω))³ ∇ · (C(x) : ∇v) + ω²v = 0 in Ω ,

and E(Ω) be the completion of E(Ω) in [L²(Ω)]³. Now we state the first main result of the paper.

Theorem 2.5. f ∈ [L²_Ω(R³)]³ is a non-radiating volume source if and only if f ∈ E(Ω)^⊥, that is,

(2.3)

Z Z Z

Ω

f · v dx = 0 for all v ∈ E(Ω).

For a general volume source f ∈ [L²_Ω(R³)]³, let f = f_k+f⊥, where f⊥ is the (L²(Ω))³-orthogonal projection of f onto E(Ω)^⊥. Then f_k ∈ E(Ω) is uniquely determined by a single measurement of Cauchy data (u|_∂Ω, (C(x) : ∇u)ν|∂Ω) with the explicit formula

(2.4)

Z Z Z

Ω

fk· v dx = − Z Z

∂Ω

(C(x) : ∇u)ν · v⁻ds(x) + Z Z

∂Ω

u · (C(x) : ∇v⁻)ν ds(x)

for all v ∈ E (Ω), where v−(x) = lim

y∈Ω,y→xv(y) for all x ∈ ∂Ω, and ν is the unit outward normal vector on ∂Ω.

Remark 2.6. For the unique determination of fk, one only needs to measure u|∂Ω since u in R³ \ ¯Ω is uniquely determined by u|_∂Ω and the radiation condition. In other words, (C(x) : ∇u)ν|∂Ω is determined by u|_∂Ω.

Remark 2.7. We want to elaborate (2.3) in a simple case. Suppose that C is isotropic and homogeneous with Lamé constants λ, µ throughout R³. Let

x = (|x| sin θ cos ϕ, |x| sin θ sin ϕ, |x| cos θ)

be the spherical coordinates and ˆx = (sin θ cos ϕ, sin θ sin ϕ, cos θ). We consider the vector spherical harmonics (VSH)

P^m_n(ˆx) := ˆxY_n^m(ˆx), C^m_n(ˆx) := 1 pn(n + 1)

θˆ sin θ

∂

∂ϕ − ˆϕ ∂

∂θ

!

Y_n^m(ˆx),

B^m_n(ˆx) := 1 pn(n + 1)

 θˆ ∂

∂θ + ϕˆ sin θ

∂

∂ϕ



Y_n^m(ˆx),

where  Y_n^m n = 0, 1, 2, · · · , |m| ≤ n are the standard spherical harmonics, see [DR95, (28)–(30)] or [BEG85] (or see [Han35] for the earliest result).

The set of VSH forms a complete orthogonal basis in [L²(S²)]³. In particular, one can write

(2.5) [L²(S²)]³ = [L²_r(S²)]³⊕ [L²_t(S²)]³,

where [L²_r(S²)]³ is the subspace spanned by {P^m_n}_n,m and [L²_t(S²)]³ is the subspace spanned by {C^m_n}_n,m∪ {B^m_n}_n,m (see [DR95, Lemma 1]). Using [DR95, (39)–(47)], we can show that if v ∈ E(Ω), then v can be expressed by

(2.6) v(x) =

∞

X

n=0 n

X

m=−n



v_nm^L L^m_n(x) + v_nm^MM^m_n(x) + v_nm^N N^m_n(x)

 ,

where

L^m_n(x) = i 4πiⁿ

Z Z

S²

e^ikpξ·xˆ P^m_n( ˆξ) ds( ˆξ), k_p = ω

√λ + 2µ,

M^m_n(x) = pn(n + 1) 4πiⁿ

Z Z

S²

e^iksξ·xˆ C^m_n( ˆξ) ds( ˆξ), k_s = ω

õ,

N^m_n(x) = ipn(n + 1) 4πiⁿ

Z Z

S²

e^iksξ·xˆ B^m_n( ˆξ) ds( ˆξ),

which are known as Navier eigenvectors, see [DR95, Lemma 4,5,6]. Here, we also refer to some classical monographs [MF53, Str41] for more details on the Navier eigenvectors. In other words, for v ∈ E(Ω), we can write

v(x) = v_P(x) + v_C(x) + v_B(x),

where

vP(x) =

∞

X

n=0 n

X

m=−n

v^P_nm Z Z

S²

e^ikpξ·xˆ P^m_n( ˆξ) ds( ˆξ),

v_C(x) =

∞

X

n=0 n

X

m=−n

v^C_nm Z Z

S²

e^iksξ·xˆ C^m_n( ˆξ) ds( ˆξ),

v_B(x) =

∞

X

n=0 n

X

m=−n

v^B_nm Z Z

S²

e^iksξ·xˆ B^m_n( ˆξ) ds( ˆξ).

Here v_nm^P , v_nm^C , and v_nm^B are given by



















v_nm^P = i 4πiⁿv^L_nm, v_nm^C = pn(n + 1)

4πiⁿ v^M_nm, v_nm^B = ipn(n + 1)

4πiⁿ v^N_nm.

Therefore, we only need to test (2.3) using the following choices:

v^nm_P (x) :=

Z Z

S²

e^ikpξ·xˆ P^m_n( ˆξ) ds( ˆξ), v^nm_C (x) :=

Z Z

S²

e^iksξ·xˆ C^m_n( ˆξ) ds( ˆξ), v^nm_B (x) :=

Z Z

S²

e^iksξ·xˆ B^m_n( ˆξ) ds( ˆξ).

Consequently, we can derive another characterization of non-radiating volume source, namely, f is a non-radiating volume source if and only if the following three conditions hold:

(2.7)

















 Z Z

S²

˜f (k_pξ) · Pˆ ^m_n( ˆξ) ds( ˆξ) = 0, Z Z

S²

˜f (k_sξ) · Cˆ ^m_n( ˆξ) ds( ˆξ) = 0, Z Z

S²

˜f (ksξ) · Bˆ ^m_n( ˆξ) ds( ˆξ) = 0, for all n = 0, 1, 2, · · · and |m| ≤ n, where

(2.8) ˜f (k) =

Z Z Z

Ω

f (x)e^−ik·xdx

denotes the Fourier transform of f . In view of (2.5), (2.7) implies that ˜f (k_pξ) ∈ [Lˆ ²_t(S²)]³ and

˜f (k_sξ) ∈ [Lˆ ²_r(S²)]³. In other words, the vector field ˜f (k) does not have the radial component at |k| = k_p and has no tangential component at |k| = k_s.

Remark 2.8. Now we want to discuss the reconstruction formula (2.4) in the homogeneous isotropic media as in Remark2.7. Here we further assume Ω = B₁(0) := x ∈ R³ |x| < 1 ,

and thus ∂Ω = S². Note that fk ∈ E(Ω) and f^k is uniquely expressed by (2.6) with suitable coefficients v^L_nm, v^M_nm, v^N_nm. Since ν = ˆx on S², the tractions on ∂Ω can be written by

(C : ∇v^nmP )ν = ik_p Z Z

S²

e^ikpξ·ˆˆxx · C : ( ˆˆ ξ ⊗ P^m_n( ˆξ)) ds( ˆξ), (C : ∇v^nmC )ν = ik_s

Z Z

S²

e^iksξ·ˆˆxx · C : ( ˆˆ ξ ⊗ C^m_n( ˆξ)) ds( ˆξ), (C : ∇v^nmB )ν = ik_s

Z Z

S²

e^iksξ·ˆˆxx · C : ( ˆˆ ξ ⊗ B^m_n( ˆξ)) ds( ˆξ),

where the operator ⊗ denotes the exterior product of vectors, defined by a ⊗ b := ab^T, which is a 3 × 3 matrix. Therefore, we have

(2.9)

− Z Z

∂Ω

(C : ∇u)ν · vP^nmds(x) + Z Z

∂Ω

u · (C : ∇v^nmP )ν ds(x)

= − Z Z

S²

Z Z

S²

e^{−ikpξ·ˆˆx}



∇u(ˆx) : C : (ˆx ⊗ P^m_n( ˆξ))



ds(ˆx) ds( ˆξ)

− ik_p Z Z

S²

Z Z

S²

e^{−ikpξ·ˆˆx}



(ˆx ⊗ u(ˆx)) : C : ( ˆξ ⊗ P^m_n( ˆξ))



ds(ˆx) ds( ˆξ).

Similar expressions also hold for v^nm_C and v^nm_B . Therefore, (2.4) is equivalent to the following three conditions:

Z Z

S²

f˜k(kpξ) · Pˆ ^m_n( ˆξ) ds( ˆξ)

= − Z Z

S²

Z Z

S²

e^{−ikpξ·ˆˆx}



∇u(ˆx) : C : (ˆx ⊗ P^m_n( ˆξ))



ds(ˆx) ds( ˆξ)

− ik_p Z Z

S²

Z Z

S²

e^{−ikpξ·ˆˆx}



(ˆx ⊗ u(ˆx)) : C : ( ˆξ ⊗ P^m_n( ˆξ))



ds(ˆx) ds( ˆξ), (2.10)

Z Z

S²

f˜k(k_sξ) · Cˆ ^m_n( ˆξ) ds( ˆξ)

= − Z Z

S²

Z Z

S²

e^{−iksξ·ˆˆx}



∇u(ˆx) : C : (ˆx ⊗ C^m_n( ˆξ))



ds(ˆx) ds( ˆξ)

− ik_s Z Z

S²

Z Z

S²

e^{−iksξ·ˆˆx}



(ˆx ⊗ u(ˆx)) : C : ( ˆξ ⊗ C^m_n( ˆξ))



ds(ˆx) ds( ˆξ), (2.11)

and

Z Z

S²

f˜k(k_sξ) · Bˆ ^m_n( ˆξ) ds( ˆξ)

= − Z Z

S²

Z Z

S²

e^{−iksξ·ˆˆx}



∇u(ˆx) : C : (ˆx ⊗ B^m_n( ˆξ))



ds(ˆx) ds( ˆξ)

− iks

Z Z

S²

Z Z

S²

e^{−iksξ·ˆˆx}



(ˆx ⊗ u(ˆx)) : C : ( ˆξ ⊗ B^m_n( ˆξ))



ds(ˆx) ds( ˆξ).

(2.12)

Formulae (2.10)-(2.12) uniquely determine v_nm^L , v_nm^M, v_nm^N in (2.6) and hence fk.

Besides the formula (2.3), we can give another explicit characterization of the non-radiating volume source for each frequency.

Theorem 2.9. f ∈ [L²_Ω(R³)]³ and f ∈ E(Ω)^⊥ if and only if (2.13) f := ∇ · (C(x) : ∇w) + ω²w ∈ [L²(Ω)]³ for some

(2.14) w ∈



w ∈ [H₀¹(Ω)]³ ∇ · (C : ∇w) ∈ [L²(Ω)]³ (C : ∇w)ν = 0 on ∂Ω

 . That is, f ∈ [L²_Ω(R³)]³ is non-radiating if and only if (2.13) holds.

Remark 2.10. As above, we will elaborate (2.13) for the homogeneous isotropic media. Due to (2.14), let w_ext be the zero extension of w, then w_ext ∈ [H¹(R³)]³ and

∇ · (C(x) : ∇wext) + ω²w_ext = f in R³.

In this case, we have ∇ · (C(x) : ∇w^ext) ≡ L^λ,µw_ext throughout R³. Let ˜w_ext(ξ) and ˜f (ξ) be the Fourier transforms of w_ext(x) and f (x), respectively. Then (2.13) is equivalent to

˜f (ξ) = −



µ|ξ|² + (λ + µ)ξ ⊗ ξ



˜

w_ext(ξ) + ω²w˜_ext(ξ)

=



(ω²− µ|ξ|²)I − (λ + µ)ξ ⊗ ξ



˜ w_ext(ξ).

Now we observe that

˜f (k_pξ) · ˆˆ ξ =



(ω²− k_p²µ)I − (λ + µ)k²pξ ⊗ ˆˆ ξ



ξ · ˜ˆ w_ext(ξ)

= (ω²− k_p²(λ + 2µ))( ˆξ · ˜w_ext(ξ)) = 0,

which implies ˜f (k_pξ) ∈ [Lˆ ²_t(S²)]³. Similarly, we can show that ˜f (k_sξ) ∈ [Lˆ ²_r(S²)]³. We hence obtain the same result as in Remark 2.7 for non-radiation volume sources.

2.2. Surface sources. Let Σ be a Lipschitz closed surface in R³, modeling a buried fault across which discontinuities may arise. That is, displacements or traction may differ from inside and outside of Σ. Let Ω₀ be an open set such that ∂Ω₀ = Σ. Let α represents the jump of displacement, while β describes the jump of traction, across the interface Σ. We remark that, for spontaneous rupture, the traction must be continuous, that is, β ≡ 0.

Theorem 2.11. Let ω > 0. Given any (α, β) ∈ [H^1/2(Σ)]³ × [H^−1/2(Σ)]³, there exists a unique u ∈ [H¹(Ω₀)]³ ∩ [H_loc¹ (R³ \ Ω₀)]³ satisfies the following time-harmonic equations of elasticity:

(2.15)











∇ · (C(x) : ∇u) + ω²u = 0 in R³\ Σ, u satisfies the Kupradze radiation condition at |x| → ∞,

[u]_Σ = α

[(C(x) : ∇u)ν]Σ = β on Σ.

Here, ν denotes the unit outer normal on Σ and ( [u]_Σ= u−− u₊

[(C(x) : ∇u)ν]Σ = (C(x) : ∇u⁻)ν − (C(x) : ∇u+)ν on Σ,

where







u±(x) = lim

h→0+

u(x ± hν(x)) (C(x) : ∇u^±(x))ν(x) = lim

h→0+



(C(x ± hν(x)) : ∇u(x ± hν(x)))

 ν(x)

on Σ.

Definition 2.12. The pair (α, β) is called a non-radiating surface source, if there exists a number R > 0 such that u(x) = 0 for all |x| > R.

As in the case of volume sources, we can give a variational characterization of a non- radiating surface source. Let Ω = B_R(0) and Ω₀ ⊂ Ω.

Theorem 2.13. (α, β) ∈ [H^1/2(Σ)]³× [H^−1/2(Σ)]³ is a non-radiating source if and only if (2.16)

Z Z

Σ

α · (C(x) : ∇v)ν ds(x) = Z Z

Σ

β · v ds(x) for all v ∈ E (Ω).

Indeed, for the general surface source (α, β) ∈ [H^1/2(Σ)]³×[H^−1/2(Σ)]³, the following relation holds:

Z Z

Σ



α · (C(x) : ∇v)ν − β · v

 ds(x)

= Z Z

∂Ω



u · (C(x) : ∇v)ν − (C(x) : ∇u)ν · v

 ds(x) (2.17)

for all v ∈ E (Ω).

Interestingly, under the assumption that the elasticity system poses the unique continuation property (UCP), a non-radiating surface source can be characterized by a more explicit formula.

Theorem 2.14. If (α, β) ∈ [H^1/2(Σ)]³× [H^−1/2(Σ)]³ is a Cauchy data of ∇ · (C(x) : ∇u) + ω²u = 0 in Ω₀, that is, there exists a u ∈ [H¹(Ω₀)]³ be such that

(2.18)







∇ · (C(x) : ∇u) + ω²u = 0 in Ω₀, u = α

(C(x) : ∇u)ν = β on Σ,

then (α, β) must be a non-radiating. Conversely, if we additionally assume that (2.19) the UCP is satisfied for ∇ · (C(x) : ∇w) + ω²w = 0 in R³\ Ω₀,

then any non-radiating source (α, β) ∈ [H^1/2(Σ)]³×[H^−1/2(Σ)]³is a Cauchy data of ∇·(C(x) :

∇u) + ω²u = 0 in Ω₀.

Remark 2.15. We say that w that solves the equation in (2.19) satisfies the UCP if there exists a nonempty open set U ⊂ R³\ Ω₀ such that w = 0 in U , then w ≡ 0 in the whole R³ \ Ω0. Under some regularity assumptions, the UCP for the isotropic elasticity system (Lamé system) is known, see [DLW20, LNUW11, LW15]. In addition, the UCP for Lamé eigen-functions also holds, see [DiLW20]. However, the UCP for the general elasticity system remains an open problem. It is worth-mentioning that the UCP may not hold for general elliptic systems, see [KW16] for counterexamples.

Remark 2.16 (Explicit characterization of radiating surface sources). Under the assumption of the UCP (2.19), we can characterize a radiating surface source using Calderón’s projectors.

Indeed, any (α, β) ∈ [H^1/2(Σ)]³× [H^−1/2(Σ)]³ can be uniquely decomposed into (α, β) = (α⊥, β⊥) + (αk, βk),

with

(α⊥, β⊥) := C_int(α, β) ∈ [H^1/2(Σ)]³× [H^−1/2(Σ)]³, (αk, βk) := C_ext(α, β) ∈ [H^1/2(Σ)]³ × [H^−1/2(Σ)]³,

where C_int and C_ext are Calderón’s projectors introduced in (3.3). From Theorem 2.14, it is clear that (α⊥, β⊥) is non-radiating. The pair (αk, βk) is the radiating part satisfying that there exists u_ext ∈ [H_loc¹ (R³\ Ω₀)]³ such that

(2.20)











∇ · (C(x) : ∇uext) + ω²u_ext = 0 in R³ \ Ω₀, u_ext satisfies Kupradze radiation condition at |x| → ∞,

u_ext = −αk

(C(x) : ∇uext)ν = −βk

on Σ.

Therefore, (αk, βk) can be uniquely determined by the measurements (u|_∂Ω, (C(x) : ∇u)ν|∂Ω) = (uext|∂Ω, (C(x) : ∇u^ext)ν|∂Ω), via formula (2.17), that is,

Z Z

Σ



αk(x) · (C(x) : ∇v(x))ν − β^k(x) · v(x)

 ds(x)

= Z Z

∂Ω



u(x) · (C(x) : ∇v(x))ν − ((C(x) : ∇u(x))ν) · v(x)

 ds(x)

for all v ∈ E (Ω). Here we again remark that (C(x) : ∇u)ν|∂Ω is uniquely determined by u|_∂Ω.

2.3. Some related results. The investigation of radiating and non-radiating sources for the acoustic and electromagnetic waves has a long history. We refer the reader to Devaney and Wolf’s work [DW73] for the early development. Later generalizations including the inverse source problem can be found in [AM06, BC77, Dev04]. Due to the non-uniqueness of the inverse source problem using only one frequency, there are growing interests in the study of the inverse source problem by the measurements at multi-frequency.

Our work is closely related to the results in [AM06] (electromagnetics) and in [Dev04]

(acoustics). In this section, we would like to compare our results with those in [AM06]

and [Dev04]. Some characterizations of non-radiating volume and surface currents were investigated in [AM06]. In the case of volume currents, a variational characterization of a non-radiating current and the decomposition of a general volume current into non-radiating and radiating parts, similar to Theorem 2.5 here, was proved in [AM06, Theorem 2.2]. We provide another characterization of non-radiating volume sources (necessary and sufficient condition) in terms of some special functions, see Theorem2.9. A similar characterization of non-radiating volume currents, but only sufficiency, was established in [AM06, Lemma 2.1].

The non-radiating surface current was also studied in [AM06] where the magnetic field admits current jump, while the electric field is continuous. Precisely, the authors considered

the following Maxwell system:















−iωE + σE − ∇ × H = 0

−iωµH + ∇ × E = 0 in R³\ Σ,

(E, H) satisfies the Silver-Müller radiation condition at |x| → ∞, [H × ν] = −J

[E × ν] = 0 on Σ,

where, as above, Σ = ∂Ω0. In [AM06, Theorem 3.1], they ruled out nontrivial non-radiating surface current J whenever ω is not an eigenvalue of the Maxwell operator in Ω0. The proof of [AM06, Theorem 3.1] relies on the denseness of the tangential trace of the Herglotz wave functions [Mon03]. The nonexistence of nontrivial J can also be seen from the UCP for the Maxwell equations (see, for example, [NW12]). In view of Theorem 2.14 above, nontrivial electromagnetic surface sources can exist only if both [H × ν] and [E × ν] are not zero on Σ. On the other hand, similar to [AM06, Theorem 3.1], it is easy to see that nontrivial surface source (α, 0) or (0, β) does not exist if ω² is not an Neumann eigenvalue or Dirichlet eigenvalue for the elastic operator in Ω₀, respectively.

We now discuss the result of [Dev04] where surface sources for the acoustic equation (Helmholtz equation) was studied. Even though it was not written explicitly, the set up of the transmission problem in [Dev04] was similar to (2.15) above. In [Dev04], a full charac- terization of non-radiating surface sources analogous to Theorem 2.14 was proved. Devaney [Dev04] named the Cauchy data of solutions of the Helmholtz equation secondary sources.

Example A of [Dev04, page 2219-2220] demonstrates that any non-radiating secondary source on a sphere must be trivial. This example is nothing but the obvious fact that the trivial solution of the Helmholtz equation must have zero Cauchy data. To our best knowledge, the detailed characterization of radiating and non-radiating elastic volume and surface sources has not been studied before. Our results, Theorem 2.13 and Theorem 2.14, give a complete characterization of non-radiating elastic surface sources without or with the UCP.

It is known that the inverse source problem is ill-posed. However, in recent studies, we observe that the stability improves as we increase the frequency. Results for such increasing stability phenomena in the inverse source problems for the acoustic, electromagnetic, and elastic waves can be found in [ABF02,BLT10, BHKY18,BLZ20,CIL16,EI18, EI20,IW20].

2.4. Organization of this paper. In addition to Section1and2, the paper is organized as follows. Before proving several characterization results about the volume and surface sources, we first prove Theorem 2.11 in Section 3. Theorem 2.11 will be useful in our proofs. Then, characterizations of volume and surface sources are proved in detail in Section4and Section5, respectively. We present some interesting observations in Appendix A and Appendix B.

3. The well-posedness of the transmission problems for the system of inhomogeneous anisotropic elasticity

Now, we want to prove Theorem 2.11 by modifying the ideas in [CS90]. Let G(x, y), for x 6= y, be the Green’s 3 × 3 dyadic [DHM18], which satisfies

∇x· (C(x) : ∇^xG(x, y)) + ω²G(x, y) = −δ(x − y) in R³, where δ is the Dirac function.

Uniqueness. The uniqueness result can be proved by using the Somigliana representation formula (see [CS90, (2.7)]) and follows the ideas in [CS90, Lemma 2.2]. For brevity, we omit the detail here.

Existence. We will focus on the proof of the existence. We define the single layer and double layer potentials [McL00, (6.16),(6.17)] by







(SLf )(x) :=

Z Z

Σ

G(x, y)f (y) ds(y) (DLf )(x) :=

Z Z

Σ

(B_ν,yG(x, y)^∗)^∗f (y) ds(y)

for x ∈ R³\ Σ,

where the superscript ∗ denotes the conjugate transpose and the traction operator B_ν,x is given by

B_ν,xu(x) := (C(x) : ∇u)ν for x ∈ Σ.

The following lemma is a R³ special case of [McL00, Theorem 6.11].

Lemma 3.1. Let γ⁺ and γ⁻ be exterior and interior trace operator, respectively. The fol- lowing linear operators are bounded and satisfy the following jump relations:











γSL : [H^−1/2(Σ)]³ → [H^1/2(Σ)]³, [SLψ]_Σ= 0 for ψ ∈ [H^−1/2(Σ)]³, γ^±DL : [H^1/2(Σ)]³ → [H^1/2(Σ)]³, [DLφ]_Σ = −φ for φ ∈ [H^1/2(Σ)]³, B^±_νSL : [H^−1/2(Σ)]³ → [H^−1/2(Σ)]³, [B_νSLψ]_Σ = ψ for ψ ∈ [H^−1/2(Σ)]³, B_νDL : [H^1/2(Σ)]³ → [H^−1/2(Σ)]³, [B_νDLφ]_Σ = 0 for φ ∈ [H^1/2(Σ)]³. As in [McL00, p.218-219], we define the following bounded operators:

(3.1)











S := γSL (single layer potential),

T := 2γ⁺DL − Id = 2γ⁻DL + Id (double layer potential),

T^∗ := 2B_ν⁺SL + Id = 2B_ν⁻SL − Id (adjoint double layer potential), R := −B_νDL (hypersingular layer potential), which satisfy the relations

(3.2) S = S^∗, R = R^∗, SR = 1

4(Id−T²), ST^∗ = TS, RT = T^∗R, RS = 1

4(Id−(T^∗)²).

Therefore, similar to [McL00, p.243], the interior and exterior Calderón projectors are defined by

(3.3) C_int :=

 ₁

2(Id − T) S R ¹₂(Id + T^∗)



and C_ext :=

 ₁

2(Id + T) −S

−R ¹₂(Id − T^∗)

 . Clearly, C_int + C_ext = Id. Relations (3.2) imply that C_intC_ext = 0, C_extC_int = 0, C_int² = C_int, and C_ext² = C_ext, see also [Ces96, Def. 4.3.4, Def. 4.3.5] for analogue ideas for the Maxwell equations. From Lemma 3.1, it is clear that

C_int :[H^1/2(Σ)]³ × [H^−1/2(Σ)]³ → [H^1/2(Σ)]³× [H^−1/2(Σ)]³, C_ext :[H^1/2(Σ)]³ × [H^−1/2(Σ)]³ → [H^1/2(Σ)]³× [H^−1/2(Σ)]³,

and both operators are bounded. Indeed, combining (3.1) and (3.3), we can easily compute C_int ψ1

ψ₂



=

 γ⁻(−DLψ₁+ SLψ₂) B_ν⁻(−DLψ₁+ SLψ₂)



and C_ext ψ1

ψ₂



=

 γ⁺(DLψ₁ − SLψ₂) B⁺_ν(DLψ₁− SLψ₂)

 .

For u ∈ [H¹(Ω₀)]³∩ [H¹(R³\ Ω₀)]³ satisfies (2.15), we write u =

(u_ext in R³\ Ω₀, u_int in Ω₀.

Recall the Somigliana representation formula, see e.g. [CS90, (2.7)]:

u_int = −DLu−+ SL((C : ∇u⁻)ν), (3.4)

u_ext = DLu₊− SL((C : ∇u+)ν).

(3.5)

Therefore, u ∈ [H¹(Ω0)]³∩ [H_loc¹ (R³\ Ω0)]³ satisfies (2.15) if and only if

(3.6)























u−

(C : ∇u⁻)ν

!

= C_int u−

(C : ∇u⁻)ν

! , u₊

(C : ∇u+)ν

!

= C_ext u₊ (C : ∇u+)ν

! , u−− u₊= α,

(C : ∇u⁻)ν − (C : ∇u+)ν = β.

Now we want to eliminate the unknowns u₊ and (C : ∇u+)ν. Let A := C_int− C_ext = −T 2S

2R T^∗

 , then

 α β



= C_int

 u−

(C : ∇u⁻)ν



− C_ext

 u+

(C : ∇u⁺)ν



= C_int

 u₋

(C : ∇u⁻)ν



+ C_ext α β



− C_ext

 u₋

(C : ∇u⁻)ν



= C_ext α β

 + A

 u−

(C : ∇u⁻)ν

 , that is,

(3.7) A

 u−

(C : ∇u⁻)ν



= (Id − Cext) α β

 ,

which is equivalent to both (2.15) and (3.6), see [CS90, (2.14)]. We define the pairing h•, •i by

  ψ φ



, ψ₀ φ₀

  :=

Z Z

Σ



φ₀· ψ + φ · ψ₀

 ds

for all  ψ φ



, ψ₀ φ₀



∈ [H^1/2(Σ)]³× [H^−1/2(Σ)]³. In view of (3.1), we can write

A = A−+ A₊ with A± = −γ^±DL S R B^±_νSL

 ,

which is exactly the first line of the proof of [CS90, Theorem 2.6]. Therefore, following the arguments in [CS90, Theorem 2.6], we can show that

(3.8) <

(A + T ) ψ φ

 , ψ

φ

 

≥ κ(kψk²_H1/2(Σ)+ kφk²_H−1/2(Σ))

for some positive constant κ and some compact operator T : [H^1/2(Σ)]³ × [H^−1/2(Σ)]³ → [H^1/2(Σ)]³ × [H^−1/2(Σ)]³. Therefore, A is of Fredholm of index zero. Since the solution of (2.15) (equivalenly, (3.6) or (3.7)) is unique (see [CS90, Lemma 2.2]), by the Fredholm theory, we conclude the existence of the solution, which completes the proof of Theorem 2.11.

4. Characterization of non-radiating volume sources

It is often convenient to reformulate the scattering problem into an equivalent boundary value problem in a bounded domain. It is done by the exterior Dirichlet-to-Neumann map.

The following well-posdedness of the scattering problem is well-known, e.g., [BP08, Theorem 1].

Lemma 4.1. Given any λ ∈ [H¹²(∂Ω)]³, there exists a unique v ∈ [H_loc¹ (R³\ Ω)]³ such that







∇ · (C(x) : ∇v) + ω²v = L^λ,µv + ω²v = 0 in R³\ Ω,

v+= λ on ∂Ω,

v satisfies the Kupradze radiation condition at |x| → ∞.

Therefore, we can define the exterior Dirichlet-to-Neumann map Λ^ext_DN : [H¹²(∂Ω)]³ → [H⁻¹²(∂Ω)]³ by

Λ^ext_DN(λ) := (C(x) : ∇v+)ν, where ν is the outer unit normal on ∂Ω.

Lemma 4.2. Λ^ext_DN : [H¹²(∂Ω)]³ → [H⁻¹²(∂Ω)]³ is self-adjoint.

Proof. Given any η ∈ [H¹²(∂Ω)]³ and let w ∈ [H_loc¹ (R³\ Ω)]³ solve







∇ · (C(x) : ∇w) + ω²w = L^λ,µw + ω²w = 0 in R³\ Ω,

w₊= η on ∂Ω,

w satisfies the Kupradze radiation condition at |x| → ∞.

The existence and uniqueness of w follows from Lemma 4.1. Let λ, η ∈ [H¹²(∂Ω)]³ and v be given in Lemma 4.1, then we have

Z Z

∂Ω

λ · (Λ^ext_DN)^∗η ds(x) = Z Z

∂Ω

Λ^ext_DN(λ) · η ds(x) = Z Z

∂Ω

(C : ∇v+)ν · w₊ds(x)

= − Z Z

∂(R³\Ω)

(C : ∇v)ν · w ds(x)

= − Z Z Z

R³\Ω

∇ · (C : ∇v) · w dx − Z Z Z

R³\Ω

∇v : C : ∇w dx

= ω² Z Z Z

R³\Ω

v · w dx − Z Z

∂(R³\Ω)v · (C : ∇w)ν ds(x) + Z Z Z

R³\Ωv : ∇ · (C : ∇w) dx

= Z Z

∂Ω

λ · (C : ∇w+)ν ds(x),

By the arbitrariness of λ, η ∈ [H¹²(∂Ω)]³, we obtain our desired lemma.  Testing (2.1) by a function v ∈ [H¹(Ω)]³ satisfying ∇ · (C(x) : ∇v) ∈ [L²(Ω)]³ gives

Z Z Z

Ω

f · v dx

= − Z Z Z

Ω

∇ · (C(x) : ∇u) · v dx − Z Z Z

Ω

ω²u · v dx

= − Z Z

∂Ω

(C(x) : ∇u)ν · v⁻ds(x) + Z Z Z

Ω

∇u : C(x) : ∇v dx − Z Z Z

Ω

ω²u · v dx (4.1)

= − Z Z

∂Ω

(C(x) : ∇u)ν · v⁻ds(x) + Z Z

∂Ω

u · (C(x) : ∇v⁻)ν ds(x)

− Z Z Z

Ω

u · (∇ · (C(x) : ∇v) + ω²v) dx.

Substituting v ∈ E (Ω) into (4.1), we immediately obtain (2.4), i.e., Lemma 4.3. For any v ∈ E (Ω), the following identity holds (4.2)

Z Z Z

Ω

f · v dx = − Z Z

∂Ω

(C(x) : ∇u)ν · v⁻ds(x) + Z Z

∂Ω

u · (C(x) : ∇v⁻)ν ds(x).

Now we are ready to prove Theorem 2.5.

Proof of Theorem 2.5. Suppose that f is a non-radiating source. Since R³ \ Ω is connected, by unique continuation property for isotropic elasticity system (see Remark 2.15), we have u = 0 in R³\ Ω, then

(C(x) : ∇u)ν = u = 0 on ∂Ω.

From (4.2), we have

Z Z Z

Ω

f · v dx = 0 for all v ∈ E (Ω), which implies

Z Z Z

Ω

f · v dx = 0 for all v ∈ E(Ω).

that is, f ∈ E(Ω)^⊥.

Conversely, suppose that f ∈ E(Ω)^⊥. It follows from (4.2) that Z Z Z

Ω

f · v dx = 0 for all v ∈ E (Ω), and hence

(4.3)

Z Z

∂Ω

(C(x) : ∇u)ν · v⁻ds(x) = Z Z

∂Ω

u · (C(x) : ∇v⁻)ν ds(x) for all v ∈ E (Ω).

Since the inhomogeneity of C and supp(f) are in Ω, we can reformulate (2.1) in the following form:

(4.4)







∇ · (C(x) : ∇u) + ω²u = −f in Ω, u = u+

(C(x) : ∇u)ν = Λ^extDN(u|_∂Ω)(= Λ^ext_DN(u₊|_∂Ω)) on ∂Ω.

By Theorem 2.11, we can choose v ∈ [H¹(Ω)]³∩ [H_loc¹ (R³\ Ω)]³ be such that











∇ · (C(x) : ∇v) + ω²v = 0 in R³\ ∂Ω, v satisfies the Kupradze radiation condition at |x| → ∞,

[v]_∂Ω= 0

[(C(x) : ∇v)ν]∂Ω = u = u₊ on ∂Ω.

Choose g = v±|_∂Ω∈ [H^1/2(∂Ω)]. Using Lemma 4.2 and (4.3), we have Z Z

∂Ω

u · Λ^ext_DN(g) ds(x) = Z Z

∂Ω

Λ^ext_DN(u|_∂Ω) · g ds(x)

= Z Z

∂Ω

(C(x) : ∇u)ν · g ds(x)

= Z Z

∂Ω

u · (C(x) : ∇v⁻)ν ds(x).

Hence,

kuk²_L2(∂Ω) = Z Z

∂Ω

u · u ds(x) = Z Z

∂Ω

u · [(C(x) : ∇v)ν]∂Ωds(x) = 0.

Therefore, we conclude that u ≡ 0 on ∂Ω. Since

(L^λ,µu + ω²u = 0 in R³\ Ω,

u satisfies Kupradze radiation condition at |x| → ∞,

by the uniqueness result in Lemma4.1, we conclude that u = 0 in R³\ Ω, which implies that

f is a non-radiating volume source. 

Proof of Theorem 2.9. Clearly, if f is given in (2.13), then by Theorem 2.5, such f is non- radiating.

Conversely, given any f ∈ [L²(Ω)]³, using the Fredholm alternative, there exists a countable set Spec_Dir,Ω(L^C) (set of Dirichlet spectra in Ω), where L^Cu = ∇ · (C(x) : ∇u), such that the following holds:

ω² 6∈ Spec_Dir,Ω(L^C) if and only if there exists a unique w ∈ [H₀¹(Ω)]³ such that

(4.5) ∇ · (C(x) : ∇w) + ω²w = f in Ω.

Since f ∈ [L²(Ω)]³, we know that ∇ · (C : ∇w) ∈ [L²(Ω)]³. If f is non-radiating, plugging f into (2.3), we can see that w belongs to the space given in (2.14). Now we consider the case when ω² ∈ Spec_Dir,Ω(L^C). Since f is non-radiating, (2.3) implies that f is orthogonal to the eigenfunction corresponding to the eigenvalue ω². Therefore, there exists w ∈ [H₀¹(Ω)]³ (but not unique) such that (4.5) holds. The proof of Theorem 2.9 is completed. 

5. Characterization of non-radiating surface sources Similarly, we can reformulate (2.15) in the following form:

(5.1)















∇ · (C(x) : ∇u) + ω²u = 0 in Ω \ Σ, Λ^ext_DN(u₊|_∂Ω) = (C(x) : ∇u)ν

u+= u on ∂Ω,

[u]Σ = α

[(C(x) : ∇u)ν]Σ = β on Σ.

Testing (5.1) by a function v ∈ [H¹(Ω)]³ satisfying ∇ · (C(x) : ∇v) ∈ [L²(Ω)]³, we have 0 = −

Z Z Z

Ω

∇ · (C(x) : ∇u) · v dx − Z Z Z

Ω

ω²u · v dx

= − Z Z Z

Ω\Ω0

∇ · (C(x) : ∇u) · v dx − Z Z Z

Ω0

∇ · (C(x) : ∇u) · v dx − Z Z Z

Ω

ω²u · v dx

= − Z Z

∂(Ω\Ω0)

(C(x) : ∇u)ν · v ds(x) − Z Z

∂Ω0

(C(x) : ∇u)ν · v ds(x) +

Z Z Z

Ω\Ω0

∇u : C : ∇v dx + Z Z Z

Ω0

∇u : C : ∇v dx − Z Z Z

Ω

ω²u · v dx

= − Z Z

∂Ω

(C(x) : ∇u)ν · v ds(x) − Z Z

Σ

[(C(x) : ∇u)ν]Σ· v ds(x) +

Z Z

∂(Ω\Ω0)

u · (C : ∇v)ν ds(x) + Z Z

∂Ω0

u · (C : ∇v)ν ds(x)

− Z Z Z

Ω

u · (∇ · (C(x) : ∇v) + ω²v) dx

= − Z Z

∂Ω

(C(x) : ∇u)ν · v ds(x) + Z Z

∂Ω

u · (C(x) : ∇v)ν ds(x)

− Z Z

Σ

=β

z }| {

[(C(x) : ∇u)ν]Σ·v ds(x) + Z Z

Σ

=α

z}|{[u]_Σ·(C(x) : ∇v)ν ds(x)

− Z Z Z

Ω

u · (∇ · (C(x) : ∇v) + ω²v) dx.

Consequently, we obtain the following lemma, which gives a link between the surface source (α, β) and the Cauchy data (u|∂Ω, (C(x) : ∇u)ν|^∂Ω).

Lemma 5.1. For v ∈ E (Ω), (2.17) holds, i.e., Z Z

Σ



α · (C(x) : ∇v)ν − β · v

 ds(x)

= Z Z

∂Ω



u · (C(x) : ∇v)ν − (C(x) : ∇u)ν · v



ds(x).

(5.2)

Now we are ready to prove Theorem 2.13.

Proof of Theorem 2.13. Let (α, β) be a non-radiating source. Since R³\ Ω is connected, by the UCP for isotropic elasticity system (see Remark 2.15), we have that

u = (C(x) : ∇u)ν = 0 on ∂Ω.

Therefore, from (5.2), it yields (2.16).

Conversely, assume that (2.16) holds. Formula (5.2) implies Z Z

∂Ω

(C : ∇u)ν · v−ds(x) = Z Z

∂Ω

u · (C(x) : ∇v−)ν ds(x) for all v ∈ E (Ω),

which is exactly (4.3). Therefore, following exactly the same argument after (4.3), we obtain

Theorem 2.13. 

Now we can prove Theorem 2.14.

Proof of Theorem 2.14. Assume that (α, β) ∈ [H^1/2(Σ)]³× [H^−1/2(Σ)]³ is a Cauchy data of

∇ · (C(x) : ∇u) + ω²u = 0 in Ω₀, that is, there exists a u ∈ [H¹(Ω₀)]³ be such that (2.18) holds. Given any v ∈ E (Ω), we can compute

− ω² Z Z Z

Ω0

u(x) · v(x) dx

= Z Z Z

Ω0

u(x) · (∇ · (C(x) : ∇v(x))) dx

= Z Z

Σ

u(x) · ((C(x) : ∇v(x))ν) ds(x) − Z Z Z

Ω0

∇u(x) : C(x) : ∇v(x) dx

= Z Z

Σ

u(x) · ((C(x) : ∇v(x))ν) ds(x) − Z Z

Σ

((C(x) : ∇u(x))ν) · v(x) ds(x) +

Z Z Z

Ω0

(∇ · (C(x) : ∇u(x))) · v(x) dx.

By (2.18), we see that 0 = −

 Z Z Z

Ω0

(∇ · (C(x) : ∇u(x)) + ω²u(x)) · v(x) dx



= Z Z

Σ

α(x) · ((C(x) : ∇v(x))ν) ds(x) − Z Z

Σ

β(x) · v(x) ds(x),

which is nothing but (2.16). It follows from Theorem 2.13 that (α, β) is a non-radiating surface source.

Now we prove the converse, with additional UCP assumption (2.19). Assume that (α, β) ∈ [H^1/2(Σ)]³×[H^−1/2(Σ)]³ is a non-radiating surface source. By the well-posedness assumption, there exists a unique ˜u ∈ [H¹(Ω₀)]³∩ [H_loc¹ (R³\ Ω₀)]³ such that

(5.3)











∇ · (C(x) : ∇˜u(x)) + ω²u(x) = 0˜ in R³\ Σ,

˜

u satisfies the Kupradze radiation condition at |x| → ∞,

˜

u−− ˜u₊= α

(C(x) : ∇˜u−)ν − (C(x) : ∇˜u₊)ν = β on Σ.

Since (α, β) is non-radiating, by (2.19), then ˜u(x) = 0 for all x ∈ R³\ Ω₀, and thus

˜

u₊ = (C(x) : ∇˜u₊)ν = 0 on Σ.

Therefore, ˜u ∈ [H¹(Ω₀)]³ of (5.3) satisfies







∇ · (C(x) : ∇˜u(x)) + ω²u(x) = 0˜ in Ω₀,

˜ u₋ = α

(C(x) : ∇˜u₋)ν = β on Σ,

which is exactly (2.18). The proof is completed. 

Appendix A. Characterization in terms of the interior transmission problem

In this appendix, we want to present another characterization of non-radiating sources.

The characterization is related to the interior transmission problem (ITP).

A.1. Volume sources. We can prove that f ∈ [L²_Ω(R³)]³ is a non-radiating source if and only if there exists a pair (u₁, u₂) ∈ [H¹(Ω)]³× [H¹(Ω)]³ such that

(A.1)











∇ · (C(x) : ∇u1) + ω²u₁ = f

∇ · (C(x) : ∇u2) + ω²u₂ = 0 in Ω, u1 = u2

(C(x) : ∇u1)ν = (C(x) : ∇u2)ν on ∂Ω.

The system (A.1) is an ITP.

By definition of a non-radiating source, together with the unique continuation property, we obtain that f is non-radiating if and only if there exists u ∈ [H¹(Ω)]³ such that

(A.2)







∇ · (C(x) : ∇u) + ω²u = f in Ω, u = 0

(C(x) : ∇u)ν = 0 on ∂Ω.

Choosing u₁ = u and u₂ = 0, it is obvious that (u₁, u₂) solves (A.1). Conversely, suppose that there exists a pair (u₁, u₂) ∈ [H¹(Ω)]³ × [H¹(Ω)]³ such that (A.1) holds. Note that u = u₁− u₂ ∈ [H¹(Ω)]³ satisfies (A.2), and hence f is non-radiating.

A.2. Surface sources. Similarly, we can also characterize non-radiating surface sources by an ITP. As above, assume that the UCP holds for solutions u of ∇ · (C(x)∇u) + ω²u = 0 in Ω. Then (α, β) ∈ [H^1/2(Σ)]³× [H^−1/2(Σ)]³ is a non-radiating surface source if and only if there exists a pair (u₁, u₂) ∈ [H¹(Ω \ Σ)]³× [H¹(Ω)]³ such that

(A.3)























∇ · (C(x) : ∇u1) + ω²u₁ = 0 in Ω \ Σ,

∇ · (C(x) : ∇u2) + ω²u₂ = 0 in Ω, [u₁]_Σ = α

[(C(x) : ∇u1)ν]_Σ = β on Σ.

u1 = u2

(C(x) : ∇u1)ν = (C(x) : ∇u2)ν on ∂Ω.