EIGENVALUE PROBLEM WITH APPLICATIONS HUAIAN DIAO HONGJIE LI HONGYU LIU AND JIEXIN TANG

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arXiv:2210.16617v2 [math.AP] 3 May 2023

SPECTRAL PROPERTIES OF AN ACOUSTIC-ELASTIC TRANSMISSION

EIGENVALUE PROBLEM WITH APPLICATIONS

HUAIAN DIAO, HONGJIE LI, HONGYU LIU, AND JIEXIN TANG

Abstract. We are concerned with a coupled-physics spectral problem arising in the cou-

pled propagation of acoustic and elastic waves, which is referred to as the acoustic-elastic

transmission eigenvalue problem. There are two major contributions in this work which are

new to the literature. First, under a mild condition on the medium parameters, we prove the

existence of an acoustic-elastic transmission eigenvalue. Second, we establish a geometric

rigidity result of the transmission eigenfunctions by showing that they tend to localize on

the boundary of the underlying domain. Moreover, we also consider interesting implications

of the obtained results to the eﬀective construction of metamaterials by using bubbly elastic

structures and to the inverse problem associated with the ﬂuid-structure interaction.

Keywords: acoustic-elastic; coupled-physics; transmission eigenvalues; transmission eigen-

functions; spectral geometry; boundary localization; bubbly elastic medium; ﬂuid-structure

interaction

2010 Mathematics Subject Classiﬁcation: 35C20, 35M10, 35M30, 35P25

1. Introduction

1.1. Mathematical setup and summary of major ﬁndings. Initially focusing on the

mathematics, but not the physics, we present the mathematical setup of our study. Let Ω

be a simply connected domain with ∂Ω∈C1,s for some 0 < s < 1 in RN, N = 2,3. The

complement RN\Ω is connected. Henceforth, we let ν∈SN−1signify the exterior unit normal

vector to ∂Ω. Let ρb, ρe∈R+and κ∈R+. Let ˜

λ, ˜µbe the Lam´e constants which satisfy the

following strong convexity conditions

˜µ > 0 and N˜

λ+ 2˜µ > 0.(1.1)

Let u(x), x∈Ω, be a CN-valued function.

Deﬁne the Lam´e operator L˜

λ,˜µand the traction operator T˜νrespectively as follows:

L˜

λ,˜µu(x) :=˜

λ∆u(x) + (˜

λ+ ˜µ)∇(∇ · u(x)),x∈Ω,

T˜νu(x) :=˜

λ(∇ · u(x))ν(x) + 2˜µ(∇su(x))ν(x),x∈∂Ω,(1.2)

where

∇su:= 1

2(∇u+∇u⊤).(1.3)

Let ω∈R+and v(x), x∈Ω, be a C-valued function. We are concerned with the following

spectral problem for (u, v)∈H1(Ω)N×H1(Ω):











L˜

λ,˜µu(x) + ω2ρeu(x) = 0 in Ω,

∇ · (1

ρb∇v(x)) + ω2

κv(x) = 0 in Ω,

u(x)·ν−1

ρbω2∇v(x)·ν= 0 on ∂Ω,

T˜νu+v(x)ν= 0 on ∂Ω.

(1.4)

In the physical setup, the ﬁrst equation in (1.4) is known as the Lam´e equation which describes

the propagation of elastic deformation, whereas the second one is the Helmholtz equation

SPECTRAL PROPERTIES OF AN ACOUSTIC-ELASTIC TRANSMISSION EIGENVALUE PROBLEM 2

which governs the acoustic wave propagation. Hence, (1.4) is a coupled-physics spectral

problem and shall be referred to as the acoustic-elastic transmission eigenvalue problem in

what follows. In (1.4), κand ρbdescribe the bulk modulus and density of the acoustic

medium. The physical parameters ˜

λand ˜µcharacterize the compressional modulus and the

shear modulus of the elastic material, respectively. Furthermore, ρespeciﬁes the density of

the elastic medium.

It is clear that (u, v)≡(0,0) is a pair of trivial solutions to (1.4). If there exists a nontrivial

pair of solutions to (1.4), ω∈R+is called an AE (acoustic-elastic) transmission eigenvalue

and (u, v) is the associated pair of transmission eigenfunctions. It is particularly noted that

if v≡0, one has from (1.4) that

(L˜

λ,˜µu(x) + ω2ρeu(x) = 0 in Ω,

u(x)·ν= 0, T˜νu= 0 on ∂Ω,(1.5)

which is the classical Jones eigenvalue problem. The Jones eigenvalue problem arises in

studying the ﬂuid-structure interaction [23] and has been extensively studied in the liter-

ature [20,21,32,34]. It is known that under certain conditions of the medium parameters

λ, ˜µ, ρeas well as the domain Ω, there exist Jones eigenvalues [20]. Clearly, Jones eigenvalues

to (1.4) are a special subset of the AE transmission eigenvalues to (1.5). In this paper, we

show that in addition to the Jones eigenvalues, there exist AE transmission eigenvalues to

(1.4) under two generic scenarios. In the ﬁrst scenario with no geometric restriction on Ω

but a minor condition on the medium parameters, we show the existence of AE transmission

eigenvalues. We employ the layer potential theory and Gohberg-Sigal theory to establish the

aforementioned result. In the second scenario, if Ω is of a radial shape and no restriction is

imposed on the medium parameters, we show the existence of inﬁnitely many AE transmis-

sion eigenvalues. The derivation of the second result employs Fourier series expansions and

involves highly technical and subtle calculations. The results are contained in Sections 2and

3in what follows.

In addition to the spectral properties of the transmission eigenvalues, we further show that

the transmission eigenfunctions tend to localize on ∂Ω in the sense that their L2-energies tend

to concentrate on ∂Ω. In fact, we rigorously show the existence of a sequence of transmission

eigenfunctions (um, vm)m∈Nassociated with ωm→ ∞ such that the aforementioned boundary

localization pattern occurs. Furthermore, it is intriguing to note that depending on the

conﬁguration of the medium parameters, it may happen that both umand vmare boundary-

localized, which are referred to as a pair of bi-localized eigen-modes; and it may also happen

that only vmis boundary-localized whereas umis not, which are referred to as a pair of

mono-localized eigen-modes. The results of this part are rigorously proved in the radial case

and numerically veriﬁed in the non-radial case, which are contained in Section 3.

1.2. Physical relevance and implications. We discuss the physical background and mo-

tivation of our study and the interesting implications that it may have. We ﬁrst consider

the ﬂuid-structure interaction problem which is described by the following PDE system

(cf. [20,23,32]):











L˜

λ,˜µu(x) + ω2ρeu(x) = 0 in Ω,

∇ · (1

ρb∇vt(x)) + ω2

κvt(x) = 0 in RN\Ω,

u(x)·ν=1

ρbω2∇vt(x)·ν, T˜νu=−vt(x)νon ∂Ω,

vt−visatisﬁes the Sommerfeld radiation condition,

(1.6)

SPECTRAL PROPERTIES OF AN ACOUSTIC-ELASTIC TRANSMISSION EIGENVALUE PROBLEM 3

where viis an entire solution to (∆ + m2)vi= 0 in RNwith m:= ω/cband cb:= pκ/ρb.

The last condition in (1.6) signiﬁes that the scattered ﬁeld vs:= vt−visatisﬁes

lim

r→∞ r(N−1)/2(∂rvs−imvs) = 0, r := |x|.(1.7)

The well-posedness of the forward problem (1.6) is known. In the physical setup, (Ω; ˜

λ, ˜µ, ρe)

signiﬁes an elastic body which is embedded in a ﬂuid whose acoustic property is characterized

by κand ρb.videnotes an incident acoustic wave with ω∈R+being its angular frequency and

its impingement on the elastic body generates the scattering phenomenon where the elastic

response inside the elastic body is governed by the ﬁrst equation in (1.6) and the acoustic wave

propagation outside Ω is governed by the second equation in (1.6). The elastic and acoustic

ﬁelds are coupled together via the transmission conditions across ∂Ω. An inverse problem of

practical importance in sonar technology is to identify the solid structure Ω by knowledge of

the acoustic ﬁeld vsaway from Ω. It is clear that if vs≡0, i.e. vt=viin RN\Ω, the solid

body is invisible with respect to the acoustic scanning by using vi. In such a case, one can

directly verify that v=vi|Ωand ufulﬁls (1.4), namely they form a pair of AE transmission

eigenfunctions. That is, if invisibility occurs, the scattering pattern, namely the perturbative

wave pattern, is trapped inside the solid body. Hence, in order to understand the invisibility

phenomenon, one needs to study the AE transmission eigenvalue problem (1.4). In fact, by

following a similar argument in [4] for the acoustic transmission eigenvalue problem, one can

show that if (v, u) is a pair of AE transmission eigenfunctions, then vcan be extended by the

Herglotz approximation to form an incident ﬁeld whose impingement on Ω generates a nearly-

vanishing scattered ﬁeld. Hence, our results on the existence of AE transmission eigenvalues

indicates that (near) invisibility is not a sporadic phenomenon. Moreover, our results on

the boundary localization properties of the AE transmission eigenfunctions characterize the

quantitative behaviours of both the elastic and acoustic ﬁelds when (near) invisibility occurs.

It is also interesting to note the mono-localized eigen-modes (i.e. vis boundary-localized

but uis not) can be used for the design of one-way information transmission in that the

acoustic observable is void outside the solid, but the elastic observable inside the solid is non-

void. Finally, we would like to mention that the spectral pattern of boundary localization

was recently investigated in [8,12] for the acoustic transmission eigenfunctions and it has

been used to produce a super-resolution scheme for acoustic imaging. By following a similar

spirit, one can also use the spectral pattern discovered in the current article to develop novel

imaging scheme for the ﬂuid-structure interaction problem. We shall consider these and other

developments in a forthcoming paper.

Next, we consider the elastodynamics in bubbly elastic media. For simplicity, we consider

the case with a single air bubble (Ω; κ, ρe) embedded in an elastic medium (RN\Ω; ˜

λ, ˜µ, ρb).

The linear elastic deformation is governed by the following PDE system (cf. [30]):











L˜

λ,˜µut(x) + ω2ρeut(x) = 0 in RN\Ω,

∇ · (1

ρb∇v(x)) + ω2

κv(x) = 0 in Ω,

ut(x)·ν=1

ρbω2∇v(x)·ν, T˜νut=−v(x)νon ∂Ω,

ut−uisatisﬁes the Kupradze radiation condition,

(1.8)

where uiis an entire solution to L˜

λ,˜µui+ω2ρeui= 0 in RN. The Kupradze radiation condition

is given by decomposing the elastic ﬁeld us:= ut−uiinto its shear and compressional parts

and requiring that each part fulﬁls the Sommerfeld radiation condition (1.7); see e.g. [28,29]

for a more detailed description. Clearly, one can consider the invisibility issue for (1.8) and

the identically vanishing of usleads again to the transmission eigenvalue problem (1.4) with

u=ui|Ω. However, we are more interested in understanding under what conditions such

SPECTRAL PROPERTIES OF AN ACOUSTIC-ELASTIC TRANSMISSION EIGENVALUE PROBLEM 4

that when ui≡0, there exists a nontrivial solution to (1.8). In fact, a systematic study was

provided in [30] and it is shown that if ρe/ρb≫1 and ω≪1, the aforementioned resonance

phenomenon indeed may happen (at least asymptotically). This low-frequency resonance

phenomenon is referred to as the Minnaert resonance and forms the fundamental basis for

the eﬀective realisation of elastic metamaterials by bubble-elastic structures (cf. [24,30,38]).

Intriguingly, the Minnaert resonant mode possesses the same boundary-localization pattern

as the AE transmission eigenfunctions (cf. [19]), which deﬁnes the polarizability of the nano-

bubbles. Hence, we expect that the discovery in the current article may produce interesting

applications in eﬀective construction of elastic metamaterials, which is deﬁnitely worth our

further study.

1.3. Comments on our study. The AE transmission eigenvalue problem (1.4) was ﬁrst

considered in [25], and it is proved that if the AE transmission eigenvalue exist, they form a

discrete set and can accumulate only at ∞. To our best knowledge, we proved the ﬁrst gen-

eral result in the literature that under generic scenarios, there indeed exist AE transmission

eigenvalues with a mil dcondition on the medium parameters. The boundary-localization

of transmission eigenfunctions was ﬁrst discovered in [8] for the acoustic transmission eigen-

value problem. It is further extended to the Maxwell system for electromagnetic transmission

eigenfunctions in [15], and to the Lam´e system for elastic transmission eigenfunctions [22]. It

seems that the boundary-localization phenomenon seems to be universal for diﬀerent waves

when invisibility/non-scattering occurs. However, we would like to emphasize that the dif-

ferent physics underlying this intriguing phenomenon leads to diﬀerent technical challenges

and mathematical treatments, as well as diﬀerent applications. Indeed, we note that the AE

transmission eigen-system (1.4) consists of two PDEs with one vector-valued and the other

one scalar-valued, and moreover the eigenvalue ωis also coupled in the boundary transmission

conditions, which give rise to signiﬁcant technical diﬃculties compared to the studies in [8,15].

The boundary-localization properties in [8] lead to a super-resolution imaging scheme and

generating the so-called pseudo plasmon modes, and in [15] lead to an artiﬁcial mirage scheme.

As discussed earlier, the boundary-localization properties in the current article may have the

potential to be applied to the ﬂuid-structure inverse problems and the eﬀective construction

of elastic metamaterials. Finally, we would like to mention that in all of the aforementioned

works [8,15,22], the boundary-localization properties were all rigorously veriﬁed for the ra-

dial geometry and numerically veriﬁed for the general geometry. The only exception is in [9]

where the general geometry is considered and the boudnary-localization is rigorously justiﬁed

for the acoustic transmission eigenfunctions via the theory of pseudo-diﬀerential operators

and generalised Wely’s law. However, due to the technical requirement, [9] actually studies

the so-called generalised transmission eigenfunctions and moreover the boundary-localization

properties are not so sharp and thorough compared to the relevant results for the radial

geometry. Hence, we shall follow a similar spirt in the current study by focusing on treat-

ing the boundary-localization for the radial geometry rigorously and the general geometry

numerically. We shall consider justifying the general geometry in a forthcoming paper.

2. Existence of infinitely many transmission eigenvalues

In this section, we show that in a certain generic scenario there exist transmission eigen-

values for the system (1.4).

2.1. Parameter conﬁguration and nondimensionalization. To facilitate analyzing the

coupled-physics system (1.4), we introduce the following nondimensional parameters. Deﬁne

δ=ρb/ρe, τ =cb

˜cp

=pκ/ρb

q(˜

λ+ 2˜µ)/ρe

, cb=pκ/ρb,˜cp=q(˜

λ+ 2˜µ)/ρe.(2.1)

SPECTRAL PROPERTIES OF AN ACOUSTIC-ELASTIC TRANSMISSION EIGENVALUE PROBLEM 5

Let lΩbe the average length of the domain Ω and we denote the following parameters by

x′=x

lΩ

, k =ω

lΩ,u′=u

lΩ

µ=˜µ

λ+ 2˜µ, λ =˜

λ+ 2˜µ, v′=v

ρbc2

(2.2)

We would like to mention that the parameters deﬁned in (2.1) and (2.2) are all nondimen-

sional. Through substituting these parameters into (1.4) and dropping their primes, we

obtain the following nondimensional coupled PDE system (cf. [30]):











Lλ,µu(x) + k2τ2u(x) = 0 in Ω,

∆v(x) + k2v(x) = 0 in Ω,

u(x)·ν−1

k2∇v(x)·ν= 0 on ∂Ω,

Tνu+δτ2v(x)ν= 0 on ∂Ω.

(2.3)

It is remarked that the system (2.3) is equivalent to the original system (1.4). Indeed, one can

obtain the system (1.4) from (2.3) by substituting the parameters in (2.1) and (2.2) into the

system (2.3). Consequently, in what follows we focus on studying the system (2.3) instead of

the system (1.4).

2.2. Layer potentials and integral reformulation. We shall rely on the layer potential

theory to reformulate the transmission eigenvalue problem (2.3) into an eigenvalue problem

associated with a system of integral equations. To that end, we ﬁrst introduce the layer

potential operators for our subsequent use.

Let Gk(x) be the fundamental solution of the operator △+k2(cf. [27]), namely

Gk(x) = 









−i

4H(1)

0(k|x|), N = 2,

−eik|x|

4π|x|, N = 3,

(2.4)

where H(1)

0is the zeroth-order Hankel function of the ﬁrst kind. The single layer potential

associated with the Helmholtz system is deﬁned by

∂Ω[ϕ](x) = Z∂Ω

Gk(x−y)ϕ(y)ds(y)x∈RN,(2.5)

with ϕ(x)∈H−1/2(∂Ω). Then the conormal derivative of the single layer potential enjoys

the jump formula

∇Sk

∂Ω[ϕ]·ν|±(x) = ±1

2I+Kk,∗

∂Ω[ϕ](x)x∈∂Ω,(2.6)

where Iis an identity operator and

Kk,∗

∂Ω[ϕ](x) = Z∂Ω∇xGk(x−y)·νxϕ(y)ds(y)x∈∂Ω,

which is also known as the Neumann-Poincar´e (N-P) operator associated with Helmholtz

system. Here and also in what follows, the subscript ±indicates the limits from outside and

inside Ω, respectively.

For the Lam´e system, the fundamental solution Γk= (Γk

i,j )N

i,j=1 of the operator Lλ,µ +ρek2

can be expressed as (cf. [30]):

Γk=Γk

s+Γk

p,(2.7)

where

Γk

p=−1

ρek2∇∇Gkpand Γk

s=1

ρek2(k2

sI+∇∇)Gks,

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摘要：

arXiv:2210.16617v2[math.AP]3May2023SPECTRALPROPERTIESOFANACOUSTIC-ELASTICTRANSMISSIONEIGENVALUEPROBLEMWITHAPPLICATIONSHUAIANDIAO,HONGJIELI,HONGYULIU,ANDJIEXINTANGAbstract.Weareconcernedwithacoupled-physicsspectralproblemarisinginthecou-pledpropagationofacousticandelasticwaves,whichisreferredtoasthea...

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