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 Survey1, there are more than a hundred of signi cant earthquakes around the world each year. Seismic waves generated by non-radiating sources are only con ned 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 Classi cation. 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 ⊂ R3 be a domain with Lipschitz boundary
∂Ω and R3\ Ω is connected. Before introducing the elasticity tensor, we rst de ne
(A : B)ijk` :=
3
X
p,q=1
AijpqBpqk` 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∞(R3) satis es symmetry properties
Cijk`(x) = Ck`ij(x), Cijk`(x) = Cjik`(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,
Cijk`(x) = λδijδk`+ µ(δikδj`+ δi`δjk) in R3\ Ω.
Remark 1.2. In R3\ 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) = Lλ,µu := µ∆u + (λ + µ)∇(∇ · u) in R3\ Ω.
For general anisotropic media, we denote
LCu = ∇ · (C(x) : ∇u)
In the following, we will describe the non-radiating sources in detail.
1.1. Volume sources. Suppose f ∈ [L2(R3)]3 with supp (f) ⊂ Ω (possibly complex-valued), denoted by [L2Ω(R3)]3. Let ω > 0 be a frequency and consider the following time-harmonic elasticity equation:
(1.2)
(∇ · (C(x) : ∇u) + ω2u = −f in R3, u satis es the Kupradze radiation condition at |x| → ∞
with u ∈ [Hloc1 (R3)]3. We shall explain the Kupradze radiation condition in De nition 2.2 later.
De nition 1.3. We say that f ∈ [L2Ω(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 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 ∈ (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 rst main result of the paper.
Theorem 1.5. f ∈ [L2Ω(R3)]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 ∈ [L2Ω(R3)]3, let f = fk+f⊥, where f⊥ is the (L2(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 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 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 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)
Pmn(ˆx) := ˆxYnm(ˆx), Cmn(ˆx) := 1 pn(n + 1)
θˆ sin θ
∂
∂ϕ − ˆϕ ∂
∂θ
!
Ynm(ˆx),
Bmn(ˆx) := 1 pn(n + 1)
θˆ ∂
∂θ + ϕˆ sin θ
∂
∂ϕ
Ynm(ˆx), where Ynm 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
(1.5) [L2(S2)]3 = [L2r(S2)]3⊕ [L2t(S2)]3,
where [L2r(S2)]3 is the subspace spanned by {Pmn}n,m and [L2t(S2)]3 is the subspace spanned by {Cmn}n,m∪ {Bmn}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
vnmL Lmn(x) + vnmMMmn(x) + vnmN Nmn(x)
,
where
Lmn(x) = i 4πin
Z Z
S2
eikpξ·xˆ Pmn( ˆξ) ds( ˆξ), kp = ω
√λ + 2µ,
Mmn(x) = pn(n + 1) 4πin
Z Z
S2
eiksξ·xˆ Cmn( ˆξ) ds( ˆξ), ks = ω
õ,
Nmn(x) = ipn(n + 1) 4πin
Z Z
S2
eiksξ·xˆ Bmn( 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) = vP(x) + vC(x) + vB(x), where
vP(x) =
∞
X
n=0 n
X
m=−n
vPnm Z Z
S2
eikpξ·xˆ Pmn( ˆξ) ds( ˆξ),
vC(x) =
∞
X
n=0 n
X
m=−n
vCnm Z Z
S2
eiksξ·xˆ Cmn( ˆξ) ds( ˆξ),
vB(x) =
∞
X
n=0 n
X
m=−n
vBnm Z Z
S2
eiksξ·xˆ Bmn( ˆξ) ds( ˆξ).
Here vnmP , vnmC , and vnmB are given by
vnmP = i 4πinvnmL , vnmC = pn(n + 1)
4πin vnmM, vnmB = ipn(n + 1)
4πin vnmN .
Therefore, we only need to test (1.3) using the following choices:
vnmP (x) :=
Z Z
S2
eikpξ·xˆ Pmn( ˆξ) ds( ˆξ), vnmC (x) :=
Z Z
S2
eiksξ·xˆ Cmn( ˆξ) ds( ˆξ), vnmB (x) :=
Z Z
S2
eiksξ·xˆ Bmn( 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
S2
˜f (kpξ) · Pˆ mn( ˆξ) ds( ˆξ) = 0, Z Z
S2
˜f (ksξ) · Cˆ mn( ˆξ) ds( ˆξ) = 0, Z Z
S2
˜f (ksξ) · Bˆ mn( ˆξ) ds( ˆξ) = 0, for all n = 0, 1, 2, · · · and |m| ≤ n, where
˜f (k) = Z Z Z
Ω
f (x)e−ik·xdx
denotes the Fourier transform of f. In view of (1.5), (1.7) implies that ˜f(kpξ) ∈ [Lˆ 2t(S2)]3 and
˜f (ksξ) ∈ [Lˆ 2r(S2)]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 ∈ R3 |x| < 1 and thus ∂Ω = S2. Note that fk ∈ E(Ω) and fk is uniquely expressed by (1.6) with suitable, coe cients vLnm, vMnm, vNnm. Since ν = ˆx on S2, the tractions on ∂Ω can be written by
(C : ∇vnmP )ν = ikp Z Z
S2
eikpξ·ˆˆxx · C : ( ˆˆ ξ ⊗ Pmn( ˆξ)) ds( ˆξ), (C : ∇vnmC )ν = iks
Z Z
S2
eiksξ·ˆˆxx · C : ( ˆˆ ξ ⊗ Cmn( ˆξ)) ds( ˆξ), (C : ∇vnmB )ν = iks
Z Z
S2
eiksξ·ˆˆxx · C : ( ˆˆ ξ ⊗ Bmn( ˆξ)) ds( ˆξ).
Therefore, we have
(1.8)
− Z Z
∂Ω
(C : ∇u)ν · vnmP ds(x) + Z Z
∂Ω
u · (C : ∇vnmP )ν ds(x)
= − Z Z
S2
Z Z
S2
e−ikpξ·ˆˆx
∇u(ˆx) : C : (ˆx ⊗ Pmn( ˆξ))
ds(ˆx) ds( ˆξ)
− ikp Z Z
S2
Z Z
S2
e−ikpξ·ˆˆx
(ˆx ⊗ u(ˆx)) : C : ( ˆξ ⊗ Pmn( ˆξ))
ds(ˆx) ds( ˆξ).
Similar expressions also hold for vnmC and vnmB . Therefore, (1.4) is equivalent to the following three conditions:
Z Z
S2
f˜k(kpξ) · Pˆ mn( ˆξ) ds( ˆξ)
= − Z Z
S2
Z Z
S2
e−ikpξ·ˆˆx
∇u(ˆx) : C : (ˆx ⊗ Pmn( ˆξ))
ds(ˆx) ds( ˆξ)
− ikp Z Z
S2
Z Z
S2
e−ikpξ·ˆˆx
(ˆx ⊗ u(ˆx)) : C : ( ˆξ ⊗ Pmn( ˆξ))
ds(ˆx) ds( ˆξ), (1.9)
Z Z
S2
f˜k(ksξ) · Cˆ mn( ˆξ) ds( ˆξ)
= − Z Z
S2
Z Z
S2
e−iksξ·ˆˆx
∇u(ˆx) : C : (ˆx ⊗ Cmn( ˆξ))
ds(ˆx) ds( ˆξ)
− iks Z Z
S2
Z Z
S2
e−iksξ·ˆˆx
(ˆx ⊗ u(ˆx)) : C : ( ˆξ ⊗ Cmn( ˆξ))
ds(ˆx) ds( ˆξ), (1.10)
and
Z Z
S2
f˜k(ksξ) · Bˆ mn( ˆξ) ds( ˆξ)
= − Z Z
S2
Z Z
S2
e−iksξ·ˆˆx
∇u(ˆx) : C : (ˆx ⊗ Bmn( ˆξ))
ds(ˆx) ds( ˆξ)
− iks Z Z
S2
Z Z
S2
e−iksξ·ˆˆx
(ˆx ⊗ u(ˆx)) : C : ( ˆξ ⊗ Bmn( ˆξ))
ds(ˆx) ds( ˆξ).
(1.11)
Formulae (1.9)-(1.11) uniquely determine vnmL , vnmM, vnmN 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 ∈ [L2Ω(R3)]3 and f ∈ E(Ω)⊥ if and only if (1.12) f := ∇ · (C(x) : ∇w) + ω2w ∈ [L2(Ω)]3 for some
(1.13) w ∈
w ∈ [H01(Ω)]3 ∇ · (C : ∇w) ∈ [L2(Ω)]3 (C : ∇w)ν = 0 on ∂Ω
. That is, f ∈ [L2Ω(R3)]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 ∈ [H1(R3)]3 and
∇ · (C(x) : ∇wext) + ω2wext = f in R3.
In this case, we have ∇ · (C(x) : ∇wext) ≡ Lλ,µwext throughout R3. Let ˜wext(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) + ω2w˜ext(ξ)
=
(ω2− µ|ξ|2)I − (λ + µ)ξ ⊗ ξ
˜ wext(ξ).
Now we observe that
˜f (kpξ) · ˆˆ ξ =
(ω2− k2pµ)I − (λ + µ)kp2ξ ⊗ ˆˆ ξ
ξ · ˜ˆ wext(ξ)
= (ω2− k2p(λ + 2µ))( ˆξ · ˜wext(ξ)) = 0,
which implies ˜f(kpξ) ∈ [Lˆ 2t(S2)]3. Similarly, we can show that ˜f(ksξ) ∈ [Lˆ 2r(S2)]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 R3, 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) ∈ [H1/2(Σ)]3 × [H−1/2(Σ)]3, there exists a unique u ∈ [H1(Ω0)]3 ∩ [Hloc1 (R3 \ Ω0)]3 satis es the following time-harmonic equations of elasticity:
(1.14)
∇ · (C(x) : ∇u) + ω2u = 0 in R3\ Σ, u satis es 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.
De nition 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) ∈ [H1/2(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) ∈ [H1/2(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) ∈ [H1/2(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
(1.17)
∇ · (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 (1.18) the UCP is satis ed for ∇ · (C(x) : ∇w) + ω2w = 0 in R3\ Ω0,
then any non-radiating source (α, β) ∈ [H1/2(Σ)]3×[H−1/2(Σ)]3is a Cauchy data of ∇·(C(x) :
∇u) + ω2u = 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) ∈ [H1/2(Σ)]3× [H−1/2(Σ)]3 can be uniquely decomposed into (α, β) = (α⊥, β⊥) + (αk, βk),
with
(α⊥, β⊥) := Cint(α, β) ∈ [H1/2(Σ)]3× [H−1/2(Σ)]3, (αk, βk) := Cext(α, β) ∈ [H1/2(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 ∈ [Hloc1 (R3\ Ω0)]3 such that
(1.19)
∇ · (C(x) : ∇uext) + ω2uext = 0 in R3 \ Ω0, uext satis es Kupradze radiation condition at |x| → ∞,
uext = −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) : ∇uext)ν|∂Ω), via formula (1.16), that is,
Z Z
Σ
αk(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 de nition 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 ⊂ R3 with smooth boundary, if u is a (smooth) solution to
(2.1) (Lλ,µ+ ω2)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
((∆ + k2p)u(p) = 0, curl u(p) = 0
(∆ + k2s)u(s)= 0, div u(s) = 0 in D, where kp2 = ω2/(λ + 2µ) and ks2 = ω2/µ.
The following De nition can be found in [KGBB79, De nition III.2.2.6 (p.124)].
De nition 2.2. Suppose that u is a solution to (2.1) with D = R3 \ Br(0) for some r > 0.
Let u(p) and u(s) be given in Lemma 2.1. We say that u satis es the Kupradze radiation condition at |x| → ∞, if
lim
|x|→∞u(p)(x) = 0, lim
|x|→∞|x|
∂|x|u(p)(x) − ikpu(p)(x)
= 0, lim
|x|→∞u(s)(x) = 0, lim
|x|→∞|x|
∂|x|u(s)(x) − iksu(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 ∈ [H12(∂Ω)]3, there exists a unique v ∈ [Hloc1 (R3\ Ω)]3 such that
∇ · (C(x) : ∇v) + ω2v = Lλ,µv + ω2v = 0 in R3\ Ω,
v+= λ on ∂Ω,
v satis es the Kupradze radiation condition at |x| → ∞.
Therefore, we can de ne the exterior Dirichlet-to-Neumann map ΛextDN : [H12(∂Ω)]3 → [H−12(∂Ω)]3 by
ΛextDN(λ) := (C(x) : ∇v+)ν, where ν is the outer unit normal on ∂Ω.
Lemma 3.2. ΛextDN : [H12(∂Ω)]3 → [H−12(∂Ω)]3 is self-adjoint.
Proof. Given any η ∈ [H12(∂Ω)]3 and let w ∈ [Hloc1 (R3\ Ω)]3 solve
∇ · (C(x) : ∇w) + ω2w = Lλ,µw + ω2w = 0 in R3\ Ω,
w+= η on ∂Ω,
w satis es the Kupradze radiation condition at |x| → ∞.
The existence and uniqueness of w follows from Lemma 3.1. Let λ, η ∈ [H12(∂Ω)]3 and v be given in Lemma 3.1, then we have
Z Z
∂Ω
λ · (ΛextDN)∗η ds(x) = Z Z
∂Ω
ΛextDN(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\Ω
v : ∇ · (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. 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 satis es
∇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 de ne 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
(Bν,yG(x, y)∗)∗f (y) ds(y)
for x ∈ R3\ Σ,
where the superscript ∗ denotes the conjugate transpose and the traction operator Bν,x is given by
Bν,xu(x) := (C(x) : ∇u)ν for x ∈ Σ.
The following lemma is a R3 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 → [H1/2(Σ)]3, [SLψ]Σ= 0 for ψ ∈ [H−1/2(Σ)]3, γ±DL : [H1/2(Σ)]3 → [H1/2(Σ)]3, [DLφ]Σ = −φ for φ ∈ [H1/2(Σ)]3, B±νSL : [H−1/2(Σ)]3 → [H−1/2(Σ)]3, [BνSLψ]Σ = ψ for ψ ∈ [H−1/2(Σ)]3, BνDL : [H1/2(Σ)]3 → [H−1/2(Σ)]3, [BνDLφ]Σ = 0 for φ ∈ [H1/2(Σ)]3. As in [McL00, p.218-219], we de ne the following bounded operators:
(4.1)
S := γSL (single layer potential),
T := 2γ+DL − Id = 2γ−DL + Id (double layer potential),
T∗ := 2Bν+SL + Id = 2Bν−SL − Id (adjoint double layer potential), R := −BνDL (hypersingular layer potential), which satisfy the relations
(4.2) 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 de ned by
(4.3) Cint :=
1
2(Id − T) S R 12(Id + T∗)
and Cext :=
1
2(Id + T) −S
−R 12(Id − T∗)
. Clearly, Cint + Cext = Id. Relations (4.2) imply that CintCext = 0, CextCint = 0, Cint2 = Cint, and Cext2 = Cext, 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
Cint :[H1/2(Σ)]3 × [H−1/2(Σ)]3 → [H1/2(Σ)]3× [H−1/2(Σ)]3, Cext :[H1/2(Σ)]3 × [H−1/2(Σ)]3 → [H1/2(Σ)]3× [H−1/2(Σ)]3,
and both operators are bounded. Indeed, combining (4.1) and (4.3), we can easily compute Cint ψ1
ψ2
=
γ−(−DLψ1+ SLψ2) Bν−(−DLψ1+ SLψ2)
and Cext
ψ1
ψ2
=
γ+(DLψ1 − SLψ2) B+ν(DLψ1− SLψ2)
.
For u ∈ [H1(Ω0)]3∩ [H1(R3\ Ω0)]3 satis es (1.14), we write
u =
(uext in R3\ Ω0, uint in Ω0.
Recall the Somigliana representation formula, see e.g. [CS90, (2.7)]:
uint = −DLu−+ SL((C : ∇u−)ν) (4.4)
uext = DLu+− SL((C : ∇u+)ν).
(4.5)
Therefore, u ∈ [H1(Ω0)]3∩ [Hloc1 (R3\ Ω0)]3 satis es (1.14) if and only if
(4.6)
u−
(C : ∇u−)ν
!
= Cint u−
(C : ∇u−)ν
! , u+
(C : ∇u+)ν
!
= Cext 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 := Cint− Cext = −T 2S
2R T∗
, then
α β
= Cint
u−
(C : ∇u−)ν
− Cext
u+
(C : ∇u+)ν
= Cint
u−
(C : ∇u−)ν
+ Cext α β
− Cext
u−
(C : ∇u−)ν
= Cext 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 de ne the pairing h•, •i by
ψ φ
, ψ0 φ0
:=
Z Z
Σ
φ0· ψ + φ · ψ0
ds for all ψ
φ
, ψ0 φ0
∈ [H1/2(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±νSL
,
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(kψk2H1/2(Σ)+ kφk2H−1/2(Σ))
for some positive constant κ and some compact operator T : [H1/2(Σ)]3 × [H−1/2(Σ)]3 → [H1/2(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 ∈ [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 (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) + ω2v) 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 R3 \ Ω is connected, by unique continuation property for isotropic elasticity system (see Remark 1.16), we have u = 0 in R3\ 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) + ω2u = −f in Ω, u = u+
(C(x) : ∇u)ν = ΛextDN(u|∂Ω)(= ΛextDN(u+|∂Ω)) on ∂Ω.
By Theorem 1.11, we can choose v ∈ [H1(Ω)]3∩ [Hloc1 (R3\ Ω)]3 be such that
∇ · (C(x) : ∇v) + ω2v = 0 in R3\ ∂Ω, v satis es the Kupradze radiation condition at |x| → ∞,
[v]∂Ω= 0
[(C(x) : ∇v)ν]∂Ω = u = u+ on ∂Ω.
Choose g = v±|∂Ω∈ [H1/2(∂Ω)]. Using Lemma 3.2 and (5.3), we have Z Z
∂Ω
u · ΛextDN(g) ds(x) = Z Z
∂Ω
ΛextDN(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,
kuk2L2(∂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
(Lλ,µu + ω2u = 0 in R3\ Ω,
u satis es Kupradze radiation condition at |x| → ∞,
by the uniqueness result in Lemma3.1, we conclude that u = 0 in R3\ 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 ∈ [L2(Ω)]3, using the Fredholm alternative, there exists a countable set SpecDir,Ω(LC)(set of Dirichlet spectra in Ω), where LCu = ∇ · (C(x) : ∇u), such that the following holds:
ω2 6∈ SpecDir,Ω(LC) if and only if there exists a unique w ∈ [H01(Ω)]3 such that (5.5) ∇ · (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 (1.3), we can see that w belongs to the space given in (1.13). Now we consider the case when ω ∈ SpecDir,Ω(LC). Since f is non-radiating, (1.3) 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 (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) + ω2u = 0 in Ω \ Σ, ΛextDN(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 ∈ [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 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 R3\ 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) ∈ [H1/2(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 (1.17) 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 (1.17), 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 (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) ∈ [H1/2(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∩ [Hloc1 (R3\ Ω0)]3 such that
(6.3)
∇ · (C(x) : ∇˜u(x)) + ω2u(x) = 0˜ in R3\ Σ,
˜
u satis es 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 ∈ R3\ Ω0, and thus
˜
u+ = (C(x) : ∇˜u+)ν = 0 on Σ.
Therefore, ˜u ∈ [H1(Ω0)]3 of (6.3) satis es
∇ · (C(x) : ∇˜u(x)) + ω2u(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 ∈ [L2Ω(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 de nition 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 satis es (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) ∈ [H1/2(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.
To prove this characterization, as in the previous case, we know that (α, β) is non-radiating if and only if there exists u ∈ [H1(Ω \ Σ)]3 such that
(A.4)
∇ · (C(x) : ∇u) + ω2u = 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 (u1, u2) ∈ [H1(Ω \ Σ)]3 × [H1(Ω)]3 such that (A.3) holds. The function u = u1 − u2 ∈ [H1(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 ∈ [H1(Ω0)]3∩ [H1(R3 \ Ω0)]3 satisfy
∇ · (C(x) : ∇u) + ω2u = 0 in R3\ Σ, u satis es Kupradze radiation condition at |x| → ∞,
[u]Σ = 0
[(C(x) : ∇u)ν]Σ= β on Σ,
equivalently, u ∈ [H1(R3)]3 and satis es
(B.1)
∇ · (C(x) : ∇u) + ω2u = 0 in Ω \ Σ, u satis es Kupradze radiation condition at |x| → ∞,
[(C(x) : ∇u)ν]Σ = β on Σ,
Lemma B.1. u ∈ [H1(R3)]3 satis es (B.1) if and only if (B.2) Z Z Z
R3
∇u : C(x) : ∇w dx − ω2 Z Z Z
R3
u · w dx = Z Z
Σ
β · w ds(x) for all w ∈ [H1(R3)]3.
Proof. Testing w on (B.1), we have
−ω2 Z Z Z
R3
u · w dx = Z Z Z
R3
∇ · (C(x) : ∇u) · w dx
= Z Z Z
R3\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
R3\Ω0∇u : C(x) : ∇w dx − Z Z Z
Ω0
∇u : C(x) : ∇w dx
= Z Z
Σ
β · w ds(x) − Z Z Z
R3
∇u : C(x) : ∇w dx,
