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-eld 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 eld at one single frequency in a region outside of the domain where the source is located.

1. Introduction

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 Survey^{1}, there are more than a hundred of signicant earthquakes around the world
each year. Seismic waves generated by non-radiating sources are only conned 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-eld measurements at one
single frequency. For our problem, we will consider the stationary elastic wave equation in
anisotropic inhomogeneous media.

2020 Mathematics Subject Classication. 35J47; 35Q74; 35R30; 74B05.

Key words and phrases. volume sources, surface sources, seismic waves, elastic waves.

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

To set up our mathematical problem, let Ω ⊂ R^{3} be a domain with Lipschitz boundary

∂Ω and R^{3}\ Ω is connected. Before introducing the elasticity tensor, we rst dene

(A : B)ijk` :=

3

X

p,q=1

A_{ijpq}B_{pqk`} for two tensors A and B,

A : B :=

3

X

i,j=1

aijbij 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 Cijk` ∈ C^{∞}(R^{3}) satises symmetry
properties

C_{ijk`}(x) = C_{k`ij}(x), C_{ijk`}(x) = C_{jik`}(x) in R^{3}.

for all 1 ≤ i, j, k, ` ≤ 3. Moreover, assume that the strong ellipticity holds, that is, there
exist constants 0 < κ1 < κ_{2} such that

(1.1) κ_{1}|A|^{2} ≤ A : C(x) : A ≤ κ2|A|^{2} for all x ∈ R^{3}

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,

C_{ijk`}(x) = λδ_{ij}δ_{k`}+ µ(δ_{ik}δ_{j`}+ δ_{i`}δ_{jk}) in R^{3}\ Ω.

Remark 1.2. In R^{3}\ Ω, (1.1) holds whenever the Lamé constants in (4) satisfy µ > 0 and
3λ + 2µ > 0. Moreover, from (4), we have

∇ · (C(x) : ∇u) = L^{λ,µ}u := µ∆u + (λ + µ)∇(∇ · u) in R^{3}\ Ω.

For general anisotropic media, we denote

L^{C}u = ∇ · (C(x) : ∇u)

In the following, we will describe the non-radiating sources in detail.

1.1. Volume sources. Suppose f ∈ [L^{2}(R^{3})]^{3} with supp (f) ⊂ Ω (possibly complex-valued),
denoted by [L^{2}Ω(R^{3})]^{3}. Let ω > 0 be a frequency and consider the following time-harmonic
elasticity equation:

(1.2)

(∇ · (C(x) : ∇u) + ω^{2}u = −f in R^{3},
u satises the Kupradze radiation condition at |x| → ∞

with u ∈ [H_{loc}^{1} (R^{3})]^{3}. We shall explain the Kupradze radiation condition in Denition 2.2
later.

Denition 1.3. We say that f ∈ [L^{2}Ω(R^{3})]^{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 1.4. By Rellich's lemma, f is non-radiating if and only if the far-eld pattern of u in (1.2) 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 ∈ (H^{1}(Ω))^{3} ∇ · (C(x) : ∇v) + ω^{2}v = 0 in Ω

and E(Ω) be the completion of E(Ω) in [L^{2}(Ω)]^{3}. Now we state the rst main result of the
paper.

Theorem 1.5. f ∈ [L^{2}_{Ω}(R^{3})]^{3} is a non-radiating volume source if and only if f ∈ E(Ω)^{⊥},
that is,

(1.3) Z Z Z

Ω

f · v dx = 0 for all v ∈ E(Ω).

For a general volume source f ∈ [L^{2}Ω(R^{3})]^{3}, let f = fk+f⊥, where f⊥ is the (L^{2}(Ω))^{3}-orthogonal
projection of f onto E(Ω)^{⊥}. Then fk ∈ E(Ω) is uniquely determined by a single measurement
of Cauchy data (u|∂Ω, (C(x) : ∇u)ν|^{∂Ω}) with the explicit formula

(1.4) 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 1.6. For the unique determination of fk, one only needs to measure u|∂Ω since
u in R^{3} \ ¯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 1.7. We want to elaborate (1.3) in a simple case. Suppose that C is isotropic and
homogeneous with Lamé constants λ, µ throughout R^{3}. 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 Yn^{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 [L^{2}(S^{2})]^{3}. In particular, one can
write

(1.5) [L^{2}(S^{2})]^{3} = [L^{2}_{r}(S^{2})]^{3}⊕ [L^{2}_{t}(S^{2})]^{3},

where [L^{2}r(S^{2})]^{3} is the subspace spanned by {P^{m}n}n,m and [L^{2}t(S^{2})]^{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

(1.6) v(x) =

∞

X

n=0 n

X

m=−n

v_{nm}^{L} L^{m}_{n}(x) + v_{nm}^{M}M^{m}_{n}(x) + v_{nm}^{N} N^{m}_{n}(x)

,

where

L^{m}_{n}(x) = i
4πi^{n}

Z Z

S^{2}

e^{ik}^{p}^{ξ·x}^{ˆ} P^{m}_{n}( ˆξ) ds( ˆξ), k_{p} = ω

√λ + 2µ,

M^{m}_{n}(x) = pn(n + 1)
4πi^{n}

Z Z

S^{2}

e^{ik}^{s}^{ξ·x}^{ˆ} C^{m}_{n}( ˆξ) ds( ˆξ), ks = ω

√µ,

N^{m}_{n}(x) = ipn(n + 1)
4πi^{n}

Z Z

S^{2}

e^{ik}^{s}^{ξ·x}^{ˆ} 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

v_{P}(x) =

∞

X

n=0 n

X

m=−n

v^{P}_{nm}
Z Z

S^{2}

e^{ik}^{p}^{ξ·x}^{ˆ} P^{m}_{n}( ˆξ) ds( ˆξ),

vC(x) =

∞

X

n=0 n

X

m=−n

v^{C}_{nm}
Z Z

S^{2}

e^{ik}^{s}^{ξ·x}^{ˆ} C^{m}_{n}( ˆξ) ds( ˆξ),

v_{B}(x) =

∞

X

n=0 n

X

m=−n

v^{B}_{nm}
Z Z

S^{2}

e^{ik}^{s}^{ξ·x}^{ˆ} B^{m}_{n}( ˆξ) ds( ˆξ).

Here vnm^{P} , vnm^{C} , and vnm^{B} are given by

v_{nm}^{P} = i
4πi^{n}v_{nm}^{L} ,
v_{nm}^{C} = pn(n + 1)

4πi^{n} v_{nm}^{M},
v_{nm}^{B} = ipn(n + 1)

4πi^{n} v_{nm}^{N} .

Therefore, we only need to test (1.3) using the following choices:

v^{nm}_{P} (x) :=

Z Z

S^{2}

e^{ik}^{p}^{ξ·x}^{ˆ} P^{m}_{n}( ˆξ) ds( ˆξ),
v^{nm}_{C} (x) :=

Z Z

S^{2}

e^{ik}^{s}^{ξ·x}^{ˆ} C^{m}_{n}( ˆξ) ds( ˆξ),
v^{nm}_{B} (x) :=

Z Z

S^{2}

e^{ik}^{s}^{ξ·x}^{ˆ} 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:

(1.7)

Z Z

S^{2}

˜f (k_{p}ξ) · Pˆ ^{m}_{n}( ˆξ) ds( ˆξ) = 0,
Z Z

S^{2}

˜f (k_{s}ξ) · Cˆ ^{m}_{n}( ˆξ) ds( ˆξ) = 0,
Z Z

S^{2}

˜f (k_{s}ξ) · Bˆ ^{m}_{n}( ˆξ) ds( ˆξ) = 0,
for all n = 0, 1, 2, · · · and |m| ≤ n, where

˜f (k) = Z Z Z

Ω

f (x)e^{−ik·x}dx

denotes the Fourier transform of f. In view of (1.5), (1.7) implies that ˜f(kpξ) ∈ [Lˆ ^{2}_{t}(S^{2})]^{3} and

˜f (ksξ) ∈ [Lˆ ^{2}_{r}(S^{2})]^{3}. In other words, the vector eld ˜f(k) does not have the radial component
at |k| = kp and has no tangential component at |k| = ks.

Remark 1.8. Now we want to discuss the reconstruction formula (1.4) in the homogeneous
isotropic media as in Remark1.7. Here we further assume Ω = B1(0) := x ∈ R^{3} |x| < 1
and thus ∂Ω = S^{2}. Note that fk ∈ E(Ω) and f^{k} is uniquely expressed by (1.6) with suitable,
coecients v^{L}_{nm}, v^{M}_{nm}, v^{N}_{nm}. Since ν = ˆx on S^{2}, the tractions on ∂Ω can be written by

(C : ∇v^{nm}P )ν = ik_{p}
Z Z

S^{2}

e^{ik}^{p}^{ξ·ˆ}^{ˆ}^{x}x · C : ( ˆˆ ξ ⊗ P^{m}_{n}( ˆξ)) ds( ˆξ),
(C : ∇v^{nm}C )ν = iks

Z Z

S^{2}

e^{ik}^{s}^{ξ·ˆ}^{ˆ}^{x}x · C : ( ˆˆ ξ ⊗ C^{m}_{n}( ˆξ)) ds( ˆξ),
(C : ∇v^{nm}B )ν = ik_{s}

Z Z

S^{2}

e^{ik}^{s}^{ξ·ˆ}^{ˆ}^{x}x · C : ( ˆˆ ξ ⊗ B^{m}_{n}( ˆξ)) ds( ˆξ).

Therefore, we have

(1.8)

− Z Z

∂Ω

(C : ∇u)ν · v^{nm}P ds(x) +
Z Z

∂Ω

u · (C : ∇v^{nm}P )ν ds(x)

= − Z Z

S^{2}

Z Z

S^{2}

e^{−ik}^{p}^{ξ·ˆ}^{ˆ}^{x}

∇u(ˆx) : C : (ˆx ⊗ P^{m}_{n}( ˆξ))

ds(ˆx) ds( ˆξ)

− ik_{p}
Z Z

S^{2}

Z Z

S^{2}

e^{−ik}^{p}^{ξ·ˆ}^{ˆ}^{x}

(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, (1.4) is equivalent to the following
three conditions:

Z Z

S^{2}

f˜k(kpξ) · Pˆ ^{m}_{n}( ˆξ) ds( ˆξ)

= − Z Z

S^{2}

Z Z

S^{2}

e^{−ik}^{p}^{ξ·ˆ}^{ˆ}^{x}

∇u(ˆx) : C : (ˆx ⊗ P^{m}_{n}( ˆξ))

ds(ˆx) ds( ˆξ)

− ik_{p}
Z Z

S^{2}

Z Z

S^{2}

e^{−ik}^{p}^{ξ·ˆ}^{ˆ}^{x}

(ˆx ⊗ u(ˆx)) : C : ( ˆξ ⊗ P^{m}_{n}( ˆξ))

ds(ˆx) ds( ˆξ), (1.9)

Z Z

S^{2}

f˜_{k}(k_{s}ξ) · Cˆ ^{m}_{n}( ˆξ) ds( ˆξ)

= − Z Z

S^{2}

Z Z

S^{2}

e^{−ik}^{s}^{ξ·ˆ}^{ˆ}^{x}

∇u(ˆx) : C : (ˆx ⊗ C^{m}_{n}( ˆξ))

ds(ˆx) ds( ˆξ)

− ik_{s}
Z Z

S^{2}

Z Z

S^{2}

e^{−ik}^{s}^{ξ·ˆ}^{ˆ}^{x}

(ˆx ⊗ u(ˆx)) : C : ( ˆξ ⊗ C^{m}_{n}( ˆξ))

ds(ˆx) ds( ˆξ), (1.10)

and

Z Z

S^{2}

f˜k(k_{s}ξ) · Bˆ ^{m}_{n}( ˆξ) ds( ˆξ)

= − Z Z

S^{2}

Z Z

S^{2}

e^{−ik}^{s}^{ξ·ˆ}^{ˆ}^{x}

∇u(ˆx) : C : (ˆx ⊗ B^{m}_{n}( ˆξ))

ds(ˆx) ds( ˆξ)

− ik_{s}
Z Z

S^{2}

Z Z

S^{2}

e^{−ik}^{s}^{ξ·ˆ}^{ˆ}^{x}

(ˆx ⊗ u(ˆx)) : C : ( ˆξ ⊗ B^{m}_{n}( ˆξ))

ds(ˆx) ds( ˆξ).

(1.11)

Formulae (1.9)-(1.11) uniquely determine vnm^{L} , v_{nm}^{M}, v_{nm}^{N} in (1.6) and hence fk.

Besides the formula (1.3), we can give another explicit characterization of the non-radiating volume source for each frequency.

Theorem 1.9. f ∈ [L^{2}_{Ω}(R^{3})]^{3} and f ∈ E(Ω)^{⊥} if and only if
(1.12) f := ∇ · (C(x) : ∇w) + ω^{2}w ∈ [L^{2}(Ω)]^{3}
for some

(1.13) w ∈

w ∈ [H_{0}^{1}(Ω)]^{3} ∇ · (C : ∇w) ∈ [L^{2}(Ω)]^{3}
(C : ∇w)ν = 0 on ∂Ω

.
That is, f ∈ [L^{2}Ω(R^{3})]^{3} is non-radiating if and only if (1.12) holds.

Remark 1.10. As above, we will elaborate (1.12) for the homogeneous isotropic media. Due
to (1.13), let wext be the zero extension of w, then wext ∈ [H^{1}(R^{3})]^{3} and

∇ · (C(x) : ∇wext) + ω^{2}w_{ext} = f in R^{3}.

In this case, we have ∇ · (C(x) : ∇wext) ≡ L^{λ,µ}w_{ext} throughout R^{3}. Let ˜w_{ext}(ξ) and ˜f(ξ) be
the Fourier transforms of wext(x) and f(x), respectively. Then (1.12) is equivalent to

˜f (ξ) = −

µ|ξ|^{2} + (λ + µ)ξ ⊗ ξ

w˜ext(ξ) + ω^{2}w˜ext(ξ)

=

(ω^{2}− µ|ξ|^{2})I − (λ + µ)ξ ⊗ ξ

˜
w_{ext}(ξ).

Now we observe that

˜f (k_{p}ξ) · ˆˆ ξ =

(ω^{2}− k^{2}_{p}µ)I − (λ + µ)kp^{2}ξ ⊗ ˆˆ ξ

ξ · ˜ˆ w_{ext}(ξ)

= (ω^{2}− k^{2}_{p}(λ + 2µ))( ˆξ · ˜w_{ext}(ξ)) = 0,

which implies ˜f(kpξ) ∈ [Lˆ ^{2}_{t}(S^{2})]^{3}. Similarly, we can show that ˜f(ksξ) ∈ [Lˆ ^{2}_{r}(S^{2})]^{3}. We hence
obtain the same result as in Remark 1.7 for non-radiation volume sources.

1.2. Surface sources. Let Σ be a Lipschitz closed surface in R^{3}, modeling a buried fault
across which discontinuities may arise. That is, displacements or traction may dier 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 1.11. Let ω > 0. Given any (α, β) ∈ [H^{1/2}(Σ)]^{3} × [H^{−1/2}(Σ)]^{3}, there exists a
unique u ∈ [H^{1}(Ω_{0})]^{3} ∩ [H_{loc}^{1} (R^{3} \ Ω_{0})]^{3} satises the following time-harmonic equations of
elasticity:

(1.14)

∇ · (C(x) : ∇u) + ω^{2}u = 0 in R^{3}\ Σ,
u satises 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.

Denition 1.12. 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 = BR(0) and Ω0 ⊂ Ω.

Theorem 1.13. (α, β) ∈ [H^{1/2}(Σ)]^{3}× [H^{−1/2}(Σ)]^{3} is a non-radiating source if and only if
(1.15) 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) (1.16)

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

(1.17)

∇ · (C(x) : ∇u) + ω^{2}u = 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
(1.18) the UCP is satised for ∇ · (C(x) : ∇w) + ω^{2}w = 0 in R^{3}\ Ω_{0},

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

∇u) + ω^{2}u = 0in Ω0.

Remark 1.15 (Explicit characterization of radiating surface sources). Under the assumption of the UCP (1.18), 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) := Cext(α, β) ∈ [H^{1/2}(Σ)]^{3} × [H^{−1/2}(Σ)]^{3},

where Cint and Cext are Calderón's projectors introduced in (4.3). From Theorem 1.14, it is
clear that (α⊥, β⊥) is non-radiating. The pair (αk, βk) is the radiating part satisfying that
there exists uext ∈ [H_{loc}^{1} (R^{3}\ Ω_{0})]^{3} such that

(1.19)

∇ · (C(x) : ∇uext) + ω^{2}u_{ext} = 0 in R^{3} \ Ω_{0},
u_{ext} satises Kupradze radiation condition at |x| → ∞,

u_{ext} = −αk

(C(x) : ∇uext)ν = −β_{k} on Σ.

Therefore, (αk, βk)can be uniquely determined by the measurements (u|∂Ω, (C(x) : ∇u)ν|∂Ω) =
(u_{ext}|_{∂Ω}, (C(x) : ∇uext)ν|_{∂Ω}), via formula (1.16), that is,

Z Z

Σ

α_{k}(x) · (C(x) : ∇v(x))ν − βk(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}.

Remark 1.16. Under some regularity assumptions, the UCP for solutions of isotropic elasticity system (Lamé system) is known, see [DLW20a, LNUW11, LW15]. The UCP for Lamé eigen-functions also hold, see [DLW20b]. However, the UCP for general elasticity system remains open. It is worth-mentioning that the UCP does not holds for general elliptic systems, see [KW16] for counterexamples.

1.3. 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. 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, EI18a, IW20]. Nonetheless, to our best knowledge, the detailed characterization of radiating and non-radiating elastic volume and surface sources has not been studied before.

1.4. Organization of this paper. In Section 2, we give a precise denition of Kupradze radiation condition. In Section 3, we introduce the exterior Dirichlet-to-Neumann map related to the scattering problem. Before proving several characterization results about the volume and surface sources, we rst prove Theorem 1.11 in Section 4. Theorem 1.11 will be useful in our proofs. Then, characterizations of volume and surface sources are proved in detail in Section5and Section 6, respectively. Finally, we present some interesting observations in Appendix A and Appendix B.

2. Kupradze radiation condition

Before we explain the Kupradze radiation condition, we rst 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 ⊂ R^{3} with smooth boundary, if u is a (smooth) solution
to

(2.1) (L^{λ,µ}+ ω^{2})u = 0 in D,

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

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

((∆ + k^{2}_{p})u^{(p)} = 0, curl u^{(p)} = 0

(∆ + k^{2}_{s})u^{(s)}= 0, div u^{(s)} = 0 in D,
where kp^{2} = ω^{2}/(λ + 2µ) and ks^{2} = ω^{2}/µ.

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

Denition 2.2. Suppose that u is a solution to (2.1) with D = R^{3} \ B_{r}(0) for some r > 0.

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

lim

|x|→∞u^{(p)}(x) = 0, lim

|x|→∞|x|

∂_{|x|}u^{(p)}(x) − ik_{p}u^{(p)}(x)

= 0, lim

|x|→∞u^{(s)}(x) = 0, lim

|x|→∞|x|

∂|x|u^{(s)}(x) − ik_{s}u^{(s)}(x)

= 0,

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

−1. 3. Exterior Dirichlet-to-Neumann map

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 3.1. Given any λ ∈ [H^{1}^{2}(∂Ω)]^{3}, there exists a unique v ∈ [Hloc^{1} (R^{3}\ Ω)]^{3} such that

∇ · (C(x) : ∇v) + ω^{2}v = L^{λ,µ}v + ω^{2}v = 0 in R^{3}\ Ω,

v+= λ on ∂Ω,

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

Therefore, we can dene the exterior Dirichlet-to-Neumann map Λ^{ext}_{DN} : [H^{1}^{2}(∂Ω)]^{3} →
[H^{−}^{1}^{2}(∂Ω)]^{3} by

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

Lemma 3.2. Λ^{ext}DN : [H^{1}^{2}(∂Ω)]^{3} → [H^{−}^{1}^{2}(∂Ω)]^{3} is self-adjoint.

Proof. Given any η ∈ [H^{1}^{2}(∂Ω)]^{3} and let w ∈ [H_{loc}^{1} (R^{3}\ Ω)]^{3} solve

∇ · (C(x) : ∇w) + ω^{2}w = L^{λ,µ}w + ω^{2}w = 0 in R^{3}\ Ω,

w_{+}= η on ∂Ω,

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

The existence and uniqueness of w follows from Lemma 3.1. Let λ, η ∈ [H^{1}^{2}(∂Ω)]^{3} and v be
given in Lemma 3.1, 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

∂(R^{3}\Ω)(C : ∇v)ν · w ds(x)

= − Z Z Z

R^{3}\Ω

∇ · (C : ∇v) · w dx − Z Z Z

R^{3}\Ω

∇v : C : ∇w dx

= ω^{2}
Z Z Z

R^{3}\Ω

v · w dx − Z Z

∂(R^{3}\Ω)

v · (C : ∇w)ν ds(x) + Z Z Z

R^{3}\Ω

v : ∇ · (C : ∇w) dx

= Z Z

∂Ω

λ · (C : ∇w+)ν ds(x),

By the arbitrariness of λ, η ∈ [H^{1}^{2}(∂Ω)]^{3}, we obtain our desired lemma.
4. The well-posedness of the transmission problems for the system of

inhomogeneous anisotropic elasticity

Now, we want to prove Theorem 1.11 by modifying the ideas in [CS90]. Let G(x, y), for x 6= y, be the Green's 3 × 3 dyadic [DHM18], which satises

∇_{x}· (C(x) : ∇xG(x, y)) + ω^{2}G(x, y) = δ(x − y) in R^{3},

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 dene 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,y}G(x, y)^{∗})^{∗}f (y) ds(y)

for x ∈ R^{3}\ Σ,

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

B_{ν,x}u(x) := (C(x) : ∇u)ν for x ∈ Σ.

The following lemma is a R^{3} special case of [McL00, Theorem 6.11].

Lemma 4.1. Let γ^{+} and γ^{−} be exterior and interior trace operator, respectively. The
following 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^{±}_{ν}SL : [H^{−1/2}(Σ)]^{3} → [H^{−1/2}(Σ)]^{3}, [B_{ν}SLψ]_{Σ} = ψ for ψ ∈ [H^{−1/2}(Σ)]^{3},
B_{ν}DL : [H^{1/2}(Σ)]^{3} → [H^{−1/2}(Σ)]^{3}, [B_{ν}DLφ]_{Σ} = 0 for φ ∈ [H^{1/2}(Σ)]^{3}.
As in [McL00, p.218-219], we dene the following bounded operators:

(4.1)

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_{ν}DL (hypersingular layer potential),
which satisfy the relations

(4.2) S = S^{∗}, R = R^{∗}, SR = 1

4(Id−T^{2}), ST^{∗} = TS, RT = T^{∗}R, RS = 1

4(Id−(T^{∗})^{2}).

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

(4.3) C_{int} :=

_{1}

2(Id − T) S
R ^{1}_{2}(Id + T^{∗})

and Cext :=

_{1}

2(Id + T) −S

−R ^{1}_{2}(Id − T^{∗})

.
Clearly, Cint + C_{ext} = Id. Relations (4.2) imply that CintC_{ext} = 0, CextC_{int} = 0, C_{int}^{2} = C_{int},
and Cext^{2} = C_{ext}, see also [Ces96, Def. 4.3.4, Def. 4.3.5] for analogue ideas for the Maxwell
equations. From Lemma 4.1, 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},
Cext :[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 (4.1) and (4.3), we can easily compute
C_{int} ψ1

ψ_{2}

=

γ^{−}(−DLψ_{1}+ SLψ_{2})
B_{ν}^{−}(−DLψ_{1}+ SLψ_{2})

and Cext

ψ1

ψ_{2}

=

γ^{+}(DLψ_{1} − SLψ_{2})
B^{+}_{ν}(DLψ_{1}− SLψ_{2})

.

For u ∈ [H^{1}(Ω_{0})]^{3}∩ [H^{1}(R^{3}\ Ω_{0})]^{3} satises (1.14), we write

u =

(u_{ext} in R^{3}\ Ω_{0},
u_{int} in Ω0.

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

u_{int} = −DLu−+ SL((C : ∇u^{−})ν)
(4.4)

u_{ext} = DLu_{+}− SL((C : ∇u+)ν).

(4.5)

Therefore, u ∈ [H^{1}(Ω0)]^{3}∩ [H_{loc}^{1} (R^{3}\ Ω0)]^{3} satises (1.14) if and only if

(4.6)

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,

(4.7) A

u−

(C : ∇u^{−})ν

= (Id − Cext) α β

,

which is equivalent to both (1.14) and (4.6), see [CS90, (2.14)]. We dene the pairing h•, •i by

ψ φ

, ψ_{0}
φ_{0}

:=

Z Z

Σ

φ_{0}· ψ + φ · ψ_{0}

ds for all ψ

φ

, ψ_{0}
φ_{0}

∈ [H^{1/2}(Σ)]^{3}× [H^{−1/2}(Σ)]^{3}.In view of (4.1), we can write

A = A−+ A_{+} with A^{±} = −γ^{±}DL S
R B^{±}_{ν}SL

,

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

(4.8) <

(A + T ) ψ φ

, ψ

φ

≥ κ(kψk^{2}_{H}1/2(Σ)+ kφk^{2}_{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
(1.14) (equivalenly, (4.6) or (4.7)) is unique (see [CS90, Lemma 2.2]), by the Fredholm theory,
we conclude the existence of the solution, which completes the proof of Theorem 1.11.

5. Characterization of non-radiating volume sources

Testing (1.2) by a function v ∈ [H^{1}(Ω)]^{3} satisfying ∇ · (C(x) : ∇v) ∈ [L^{2}(Ω)]^{3} gives
Z Z Z

Ω

f · v dx

= − Z Z Z

Ω

∇ · (C(x) : ∇u) · v dx − Z Z Z

Ω

ω^{2}u · 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

ω^{2}u · v dx
(5.1)

= − 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) + ω^{2}v) dx.

Substituting v ∈ E(Ω) into (5.1), we immediately obtain (1.4), i.e., Lemma 5.1. For any v ∈ E(Ω), the following identity holds (5.2) 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 1.5.

Proof of Theorem 1.5. Suppose that f is a non-radiating source. Since R^{3} \ Ω is connected,
by unique continuation property for isotropic elasticity system (see Remark 1.16), we have
u = 0 in R^{3}\ Ω, then

(C(x) : ∇u)ν = u = 0 on ∂Ω.

From (5.2), 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 (5.2) that
Z Z Z

Ω

f · v dx = 0 for all v ∈ E(Ω),

and hence (5.3) 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 (1.2) in the following form:

(5.4)

∇ · (C(x) : ∇u) + ω^{2}u = −f in Ω,
u = u_{+}

(C(x) : ∇u)ν = Λ^{ext}DN(u|_{∂Ω})(= Λ^{ext}_{DN}(u_{+}|_{∂Ω})) on ∂Ω.

By Theorem 1.11, we can choose v ∈ [H^{1}(Ω)]^{3}∩ [H_{loc}^{1} (R^{3}\ Ω)]^{3} be such that

∇ · (C(x) : ∇v) + ω^{2}v = 0 in R^{3}\ ∂Ω,
v satises 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 3.2 and (5.3), 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,

kuk^{2}_{L}2(∂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,µ}u + ω^{2}u = 0 in R^{3}\ Ω,

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

by the uniqueness result in Lemma3.1, we conclude that u = 0 in R^{3}\ Ω, which implies that

f is a non-radiating volume source.

Proof of Theorem 1.9. Clearly, if f is given in (1.12), then by Theorem 1.5, such f is non- radiating.

Conversely, given any f ∈ [L^{2}(Ω)]^{3}, using the Fredholm alternative, there exists a countable
set SpecDir,Ω(L^{C})(set of Dirichlet spectra in Ω), where L^{C}u = ∇ · (C(x) : ∇u), such that the
following holds:

ω^{2} 6∈ Spec_{Dir,Ω}(L^{C})
if and only if there exists a unique w ∈ [H0^{1}(Ω)]^{3} such that
(5.5) ∇ · (C(x) : ∇w) + ω^{2}w = f in Ω.

Since f ∈ [L^{2}(Ω)]^{3}, we know that ∇ · (C : ∇w) ∈ [L^{2}(Ω)]^{3}. If f is non-radiating, plugging f
into (1.3), we can see that w belongs to the space given in (1.13). Now we consider the case
when ω ∈ SpecDir,Ω(L^{C}). Since f is non-radiating, (1.3) implies that f is orthogonal to the

eigenfunction corresponding to the eigenvalue ω^{2}. Therefore, there exists w ∈ [H0^{1}(Ω)]^{3} (but
not unique) such that (5.5) holds. The proof of Theorem 1.9 is completed.

6. Characterization of non-radiating surface sources Similarly, we can reformulate (1.14) in the following form:

(6.1)

∇ · (C(x) : ∇u) + ω^{2}u = 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 (6.1) by a function v ∈ [H^{1}(Ω)]^{3} satisfying ∇ · (C(x) : ∇v) ∈ [L^{2}(Ω)]^{3}, we have
0 = −

Z Z Z

Ω

∇ · (C(x) : ∇u) · v dx − Z Z Z

Ω

ω^{2}u · 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

ω^{2}u · 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

ω^{2}u · 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) + ω^{2}v) 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) + ω^{2}v) 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 6.1. For v ∈ E(Ω), (1.16) 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).

(6.2)

Now we are ready to prove Theorem 1.13.

Proof of Theorem 1.13. Let (α, β) be a non-radiating source. Since R^{3}\ Ω is connected, by
the UCP for isotropic elasticity system (see Remark 1.16), we have that

u = (C(x) : ∇u)ν = 0 on ∂Ω.

Therefore, from (6.2), it yields (1.15).

Conversely, assume that (1.15) holds. Formula (6.2) 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 (5.3). Therefore, following exactly the same argument after (5.3), we obtain

Theorem 1.13.

Now we can prove Theorem 1.14.

Proof of Theorem 1.14. Assume that (α, β) ∈ [H^{1/2}(Σ)]^{3}× [H^{−1/2}(Σ)]^{3} is a Cauchy data of

∇ · (C(x) : ∇u) + ω^{2}u = 0 in Ω0, that is, there exists a u ∈ [H^{1}(Ω_{0})]^{3} be such that (1.17)
holds. Given any v ∈ E(Ω), we can compute

− ω^{2}
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 (1.17), we see that 0 = −

Z Z Z

Ω0

(∇ · (C(x) : ∇u(x)) + ω^{2}u(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 (1.15). It follows from Theorem 1.13 that (α, β) is a non-radiating surface source.

Now we prove the converse, with additional UCP assumption (1.18). 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 ∈ [H^{1}(Ω_{0})]^{3}∩ [H_{loc}^{1} (R^{3}\ Ω_{0})]^{3} such that

(6.3)

∇ · (C(x) : ∇˜u(x)) + ω^{2}u(x) = 0˜ in R^{3}\ Σ,

˜

u satises 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 (1.18), then ˜u(x) = 0 for all x ∈ R^{3}\ Ω_{0}, and thus

˜

u_{+} = (C(x) : ∇˜u_{+})ν = 0 on Σ.

Therefore, ˜u ∈ [H^{1}(Ω_{0})]^{3} of (6.3) satises

∇ · (C(x) : ∇˜u(x)) + ω^{2}u(x) = 0˜ in Ω0,

˜
u_{−} = α

(C(x) : ∇˜u_{−})ν = β on Σ,

which is exactly (1.17). 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 ∈ [L^{2}_{Ω}(R^{3})]^{3} is a non-radiating source if and
only if there exists a pair (u1, u_{2}) ∈ [H^{1}(Ω)]^{3}× [H^{1}(Ω)]^{3} such that

(A.1)

∇ · (C(x) : ∇u1) + ω^{2}u_{1} = f

∇ · (C(x) : ∇u2) + ω^{2}u_{2} = 0 in Ω,
u1 = u2

(C(x) : ∇u1)ν = (C(x) : ∇u2)ν on ∂Ω.

The system (A.1) is an ITP.

By denition 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 ∈ [H^{1}(Ω)]^{3} such that

(A.2)

∇ · (C(x) : ∇u) + ω^{2}u = f in Ω,
u = 0

(C(x) : ∇u)ν = 0 on ∂Ω.

Choosing u1 = u and u2 = 0, it is obvious that (u1, u_{2}) solves (A.1). Conversely, suppose
that there exists a pair (u1, u_{2}) ∈ [H^{1}(Ω)]^{3} × [H^{1}(Ω)]^{3} such that (A.1) holds. Note that
u = u_{1}− u_{2} ∈ [H^{1}(Ω)]^{3} satises (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) + ω^{2}u = 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, u_{2}) ∈ [H^{1}(Ω \ Σ)]^{3}× [H^{1}(Ω)]^{3} such that

(A.3)

∇ · (C(x) : ∇u1) + ω^{2}u_{1} = 0 in Ω \ Σ,

∇ · (C(x) : ∇u2) + ω^{2}u_{2} = 0 in Ω,
[u_{1}]_{Σ} = α

[(C(x) : ∇u1)ν]_{Σ} = β on Σ.

u1 = u2

(C(x) : ∇u1)ν = (C(x) : ∇u2)ν on ∂Ω.

To prove this characterization, as in the previous case, we know that (α, β) is non-radiating
if and only if there exists u ∈ [H^{1}(Ω \ Σ)]^{3} such that

(A.4)

∇ · (C(x) : ∇u) + ω^{2}u = 0 in Ω \ Σ,
[u]_{Σ}= α

[(C(x) : ∇u)ν]Σ = β on Σ.

u = 0

(C(x) : ∇u)ν = 0 on ∂Ω.

This is exactly (A.3) with u1 = u and u2 ≡ 0. Conversely, suppose that there exists a pair
(u_{1}, u_{2}) ∈ [H^{1}(Ω \ Σ)]^{3} × [H^{1}(Ω)]^{3} such that (A.3) holds. The function u = u1 − u_{2} ∈
[H^{1}(Ω \ Σ)]^{3} will satisfy (A.4), and hence (α, β) is non-radiating.

Appendix B. Relation between surface sources and volume sources
It is known to the seismologists that a surface source can be reformulated into a singular
volume source. In this appendix, we would like to prove this statement rigorously. For
simplicity, we consider (0, β) with β ∈ [H^{−1/2}(Σ)]^{3} as the surface source. More precisely, let
u ∈ [H^{1}(Ω_{0})]^{3}∩ [H^{1}(R^{3} \ Ω_{0})]^{3} satisfy

∇ · (C(x) : ∇u) + ω^{2}u = 0 in R^{3}\ Σ,
u satises Kupradze radiation condition at |x| → ∞,

[u]_{Σ} = 0

[(C(x) : ∇u)ν]Σ= β on Σ,

equivalently, u ∈ [H^{1}(R^{3})]^{3} and satises

(B.1)

∇ · (C(x) : ∇u) + ω^{2}u = 0 in Ω \ Σ,
u satises Kupradze radiation condition at |x| → ∞,

[(C(x) : ∇u)ν]Σ = β on Σ,

Lemma B.1. u ∈ [H^{1}(R^{3})]^{3} satises (B.1) if and only if
(B.2) Z Z Z

R^{3}

∇u : C(x) : ∇w dx − ω^{2}
Z Z Z

R^{3}

u · w dx = Z Z

Σ

β · w ds(x)
for all w ∈ [H^{1}(R^{3})]^{3}.

Proof. Testing w on (B.1), we have

−ω^{2}
Z Z Z

R^{3}

u · w dx = Z Z Z

R^{3}

∇ · (C(x) : ∇u) · w dx

= Z Z Z

R^{3}\Ω0

∇ · (C(x) : ∇u) · w dx + Z Z Z

Ω0

∇ · (C(x) : ∇u) · w dx

= − Z Z

Σ

(C(x) : ∇u+)ν · w ds(x) + Z Z

Σ

(C(x) : ∇u−)ν · w ds(x)

− Z Z Z

R^{3}\Ω_{0}∇u : C(x) : ∇w dx −
Z Z Z

Ω0

∇u : C(x) : ∇w dx

= Z Z

Σ

β · w ds(x) − Z Z Z

R^{3}

∇u : C(x) : ∇w dx,