Localised Boundary-Domain Singular Integral Equations of Acoustic Scattering by Inhomogeneous Anisotropic Obstacle

# Localised Boundary-Domain Singular Integral Equations of Acoustic Scattering by Inhomogeneous Anisotropic Obstacle

[    [    [ \orgdivA.Razmadze Mathematical Institute, \orgnameI.Javakhishvili Tbilisi State University, \orgaddress\stateTbilisi, \countryGeorgia \orgdivDepartment of Mathematics, \orgnameBrunel University London, \orgaddress\stateLondon, \countryUK \orgdivDepartment of Mathematics, \orgnameGeorgian Technical University, \orgaddress\stateTbilisi, \countryGeorgia
xxxxxxxxx
xxxxxxxxx
xxxxxxxxx
###### Abstract

[Summary] We consider the time-harmonic acoustic wave scattering by a bounded anisotropic inhomogeneity embedded in an unbounded anisotropic homogeneous medium. The material parameters may have discontinuities across the interface between the inhomogeneous interior and homogeneous exterior regions. The corresponding mathematical problem is formulated as a transmission problems for a second order elliptic partial differential equation of Helmholtz type with discontinuous variable coefficients. Using a localised quasi-parametrix based on the harmonic fundamental solution, the transmission problem for arbitrary values of the frequency parameter is reduced equivalently to a system of localised boundary-domain singular integral equations. Fredholm properties of the corresponding localised boundary-domain integral operator is studied and its invertibility is established in appropriate Sobolev-Slobodetskii and Bessel potential spaces, which implies existence and uniqueness results for the localised boundary-domain integral equations system and the corresponding acoustic scattering transmission problem.

Acoustic scattering, partial differential equations, transmission problems, localized parametrix, localized boundary-domain integral equations, pseudodifferential equations.
\articletype

Article Type

1]Otar Chkadua 2]Sergey E. Mikhailov* 3]David Natroshvili

\authormark

\corres

*S.E. Mikhailov, Department of Mathematics, Brunel University London, Uxbridge, UB8 3PH, UK.

## 1 Introduction

We consider the time-harmonic acoustic wave scattering by a bounded anisotropic inhomogeneous obstacle embedded in an unbounded anisotropic homogeneous medium. We assume that the material parameters and speed of sound are functions of position within the inhomogeneous bounded obstacle. The physical model problem with a frequency parameter is formulated mathematically as a transmission problem for a second order elliptic partial differential equation with variable coefficients in the inhomogeneous anisotropic bounded region and for a Helmholtz type equation with constant coefficients in the homogeneous anisotropic unbounded region . The material parameters and are not assumed to be continuous across the interface between the inhomogeneous interior and homogeneous exterior regions. The transmission conditions are assumed on the interface, relating the interior and exterior traces of the wave amplitude and its co-normal derivative on .

The isotropic transmission problems, when and is the Helmholtz operator is well presented in the literature (see Colton et al6, Nédélec 20, and the references therein). The acoustic scattering problem in the whole space corresponding to a more general isotropic case, when , where is Kronecker’s delta, and , was analysed by the indirect boundary-domain integral equation method by P.Werner 27-28. Applying the potential method based on the Helmholtz fundamental solution, P.Werner reduced the problem to the Fredholm-Riesz type integral equations system and proved its unique solvability. The same problem by the direct method was considered by P.Martin 13, where the problem was reduced to a singular integro-differential equation in the inhomogeneous bounded region . Using the uniqueness and existence results obtained in the references 27-28, the equivalence of the integro-differential equation to the initial transmission problem and its unique solvability were shown for special type right-hand side functions associated with Green’s third formula.

Note that the wave scattering problems for the general inhomogeneous anisotropic case described above can be studied by the variational method incorporated with the non-local approach and also by the classical potential method when the corresponding fundamental solution is available in an explicit form. However, fundamental solutions for second order elliptic partial differential equations with variable coefficients are not available in explicit form, in general. Application of the potential method based on the corresponding Levi function, which always can be constructed explicitly, leads to Fredholm-Riesz type integral equations but invertibility of the corresponding integral operators can be proved only for particular cases (see Miranda18).

Our goal here is to show that the acoustic transmission problems for anisotropic layered structures can be equivalently reformulated as systems of localized boundary-domain singular integral equations (LBDIEs) with the help of a localized harmonic paramerix based on the harmonic fundamental solution, which is a quasi-parametrix for the considered PDEs of acoustics, and to prove that the corresponding localized singular boundary-domain integral operators (LBDIO) are invertible for an arbitrary value of the frequency parameter. Beside a pure mathematical interest, these results seem to be important from the point of view of applications, since LBDIE system can be applied in constructing convenient numerical algorithms (cf. the references 15-30). Novelty of the paper is in development of the localized boundary-domain singular integral equations method in the acoustic scattering theory for anisotropic inhomogeneous layered structures.

The paper is organized as follows. First, after mathematical formulation of the problem, we introduce layer and volume potentials based on a localized harmonic parametrix and derive basic integral relations in bounded inhomogeneous and unbounded homogeneous anisotropic regions. Then we reduce the transmission problem under consideration to the localized boundary-domain singular integral equations system and prove the equivalence theorem for arbitrary values of the frequency parameter, which plays a crucial role in our analysis. Afterwards, applying the Vishik-Eskin approach, we investigate Fredholm properties of the corresponding matrix LBDIO, containing singular integral operators over the interface surface and the bounded region occupied by the inhomogeneous obstacle, and prove invertibility of the LBDIO in appropriate Sobolev-Slobodetskii and Bessel potential spaces. This invertibility property implies then, in particular, existence and uniqueness results for the LBDIE system and the corresponding original transmission problem.

Next, we analyze also an alternative non-local approach based on coupling of variational and boundary integral equation methods, which reduces the transmission problem for unbounded composite structure to the variational equation containing a coercive sesquilinear form which lives on the bounded inhomogeneous region and the interface manifold. Both approaches presented in the paper can be applied in the study of similar wave scattering problems for multi-layer piecewise inhomogeneous anisotropic structures.

Finally, for the readers convenience, we collected necessary auxiliary material related to classes of localizing functions, properties of localized potentials and anisotropic radiating potentials in three brief appendices.

## 2 Formulation of the transmission problem

Let be a bounded domain in with a simply connected boundary , and . For simplicity, we assume that if not otherwise stated. Throughout the paper denotes the unit normal vector to directed outward the domain .

We assume that the propagation region of a time harmonic acoustic wave is the whole space which consists of an inhomogeneous part and a homogeneous part . Acoustic wave propagation is governed by the uniformly elliptic second order scalar partial differential equation

 Autot(x)≡∂k(akj(x)∂jutot(x))+ω2κ(x)utot(x)=f(x),x∈Ω2∪Ω1, (1)

where , , and are real-valued functions, is a frequency parameter, while is the volume force amplitude. Here and in what follows, the Einstein summation by repeated indices from to is assumed.

Note that in the mathematical model of an inhomogeneous absorbing medium the function is complex-valued, with nonzero real and imaginary parts, in general (see, e.g., Colton et al6, Ch. 8). Here we treat only the case when the is a real-valued function but it should be mentioned that the complex-valued case can be also considered by the approach developed here.

In our further analysis, it is assumed that the real-valued variable coefficients and are constant in the homogeneous unbounded region and the following relations hold:

 akj(x)=ajk(x)=⎧⎨⎩a(1)kjforx∈Ω1,a(2)kj(x)forx∈Ω2,κ(x)={κ1>0forx∈Ω1,κ2(x)>0forx∈Ω2, (2)

where and are constants, while and are smooth function in ,

 a(2)kj,κ2∈C2(¯¯¯¯Ω2),j,k=1,2,3. (3)

Moreover, the matrices are uniformly positive definite, i.e., there are positive constants and such that

 (4)

We do not assume that the coefficients and are continuous across in general, i.e., the case and for is covered by our analysis. Further, let us denote

 A1v(x):=a(1)kj∂xk∂xjv(x)+ω2κ1v(x)forx∈Ω1, (5) A2v(x):=∂xk(a(2)kj(x)∂xjv(x))+ω2κ2(x)v(x)forx∈Ω2.

We will often write instead of and instead of , when this does not lead to a confusion.
For a function sufficiently smooth in and , the classical co-normal derivative operators, are well defined as

 T±cqv(x):=a(q)kjnk(x)γ±(∂xjv(x)),x∈S,q=1,2; (6)

here the symbols and denote one-sided boundary trace operators on from the interior and exterior domains respectively. Their continuous right inverse operators, which are non-uniquely defined, are denoted by symbols .

By , , and , , we denote the -based Bessel potential spaces on an open domain and on a closed manifold without boundary, while stands for the space of infinitely differentiable test functions with support in . Recall that is a space of square integrable functions in . Let the symbol denote the restriction operator onto .

Since the boundary traces of gradients, are generally not well defined on functions from , the classical co-normal derivatives (6) are not well defined on such functions either, cf. Mikhailov17, Appendix A, where an example of such function, for which the classical co-normal derivative exists at no boundary point. Let us introduce the following subspaces of and to which the classical co-normal derivatives can be continuously extended, cf., e.g., Grisvard9, Costabel7, Mikhailov16:

 H1,0(Ω2;A2):={v∈H1(Ω2):A2v∈H0(Ω2)},H1,0loc(Ω1;A1):={v∈H1loc(Ω1):A1v∈H0loc(Ω1)}.

We will also use the corresponding spaces with the Laplace operator instead of .

Motivated by the first Green identity well known for smooth functions, the classical co-normal derivative operators (6) can be extended by continuity to functions from the spaces and giving the canonical co-normal derivative operators, and , defined in the weak form as

 ⟨T+2u,g⟩S:= ∫Ω2[a(2)kj(x)∂ju(x)∂k(γ+)−1g(x)−ω2κ2(x)u(x)(γ+)−1g(x)]dx +∫Ω2A2u(x)(γ+)−1g(x)dx,u∈H1,0(Ω2;A2),∀ g∈H12(S), (7) ⟨T−1u,g⟩S:= −∫Ω1[a(1)kj∂ju(x)∂k(γ−)−1g(x)−ω2κ1u(x)(γ−)−1g(x)]dx −∫Ω1A1u(x)(γ−)−1g(x)dx,u∈H1,0loc(Ω1;A1),∀ g∈H12(S), (8)

where and are the right inverse operators to the trace operators , and the angular brackets should be understood as duality pairing of with which extends the usual bilinear inner product.

The canonical co-normal derivatives and can be defined analogously for functions from the spaces and , respectively, provided that the variable coefficients and are continuously extended from to the whole space preserving the smoothness. It is evident that for functions from the space and the classical and canonical co-normal derivative operators coincide. Concerning the canonical and generalized co-normal derivatives in wider functional spaces see Mikhailov16.

For two times continuously differentiable function in a neighbourhood of , we employ also the notation , , to denote the restriction of to , which coincides with both the classical and the canonical co-normal derivatives.

Recall that, the definitions of the co-normal derivatives do not depend on the choice of the right inverse operators and the following Green’s first and second identities hold (cf. Theorem 3.9 in Mikhailov16),

 ⟨T+qu,γ+v⟩S=∫Ω2[a(q)kj∂ju∂kv−ω2κquv]dx+∫Ω2vAqudx, u∈H1,0(Ω2;Aq), v∈H1(Ω2),q=1,2, (9) ⟨T−1u,γ−v⟩S=−∫Ω1[a(1)kj∂ju∂kv−ω2κ1uv]dx−∫Ω1vA1udx, u∈H1,0loc(Ω1;A1), v∈H1comp(Ω1). (10)

By we denote a sub-class of complex-valued functions from satisfying the Sommerfeld radiation conditions at infinity (see Vekua26, Colton et al6 for the Helmholtz operator and Vainberg25, Jentsch et al11 for the “anisotropic" operator defined by (5)). Denote by the characteristic surface (ellipsoid) associated with the operator ,

 a(1)kjξkξj−ω2κ1=0,ξ∈R3.

For an arbitrary vector with there exists only one point such that the outward unit normal vector to at the point has the same direction as , i.e., . Note that and It can easily be verified that

 ξ(η)=ωκ1/21(a−11η⋅η)−1/2a−11η, (11)

where is the matrix inverse to .

{definition}

A complex-valued function belongs to the class if there exists a ball of radius centered at the origin such that , and satisfies the Sommerfeld radiation conditions associated with the operator for sufficiently large ,

 v(x)=O(|x|−1),∂kv(x)−iξk(η)v(x)=O(|x|−2),k=1,2,3, (12)

where corresponds to the vector (i.e., is given by (11) with ). Notice that due to the ellipticity of the operator , any solution to the constant coefficient homogeneous equation in an open region is a real analytic function of in .

Conditions (12) are equivalent to the classical Sommerfeld radiation conditions for the Helmholtz equation if , i.e., if and , where is the Kronecker delta. There holds the following analogue of the classical Rellich-Vekua lemma (for details see Jentsch et al11, Natroshvili et al19). {lemma} Let be a solution of the equation in and let

 limR→+∞\rm Im{∫ΣR¯¯¯¯¯¯¯¯¯¯v(x)T1(x,∂x)v(x)dΣR}=0, (13)

where is the sphere with radius centered at the origin. Then in . {remark} For and we have and in view of (6) and (12) for a function we get

 T1(x,∂x)v(x)=a(1)kjnk(x)[iξj(η)v(x)]+O(|x|−2)=ia(1)kjηkξj(η)v(x)+O(|x|−2).

Therefore, by (11) and the symmetry condition , we arrive at the relation

 ¯¯¯¯¯¯¯¯¯¯v(x)T1(x,∂)v(x)=iωκ1/21|v(x)|2(a−11η⋅η)−1/2a1η⋅a−1η+O(|x|−3)=iωκ1/21(a−11η⋅η)−1/2|v(x)|2+O(|x|−3),

On the other hand, matrix is positive definite, cf. (4), which implies positive definiteness of the inverse matrix . Hence there are positive constants and such that the inequality holds for all . Consequently, (13) for is equivalent to the condition in the well known Rellich-Vekua lemma in the theory of the Helmholtz equation, Vekua26, Rellich21, Colton et al6,

 limR→+∞∫ΣR|v(x)|2dΣR=0.

In the unbounded region , we have a total wave field , where is a wave motion initiating known incident field and is a radiating unknown scattered field. It is often assumed that the incident field is defined in the whole of , being for example a corresponding plane wave which solves the homogeneous equation in but does not satisfy the Sommerfeld radiation conditions at infinity. Motivated by relations (2), let us set for and for .

Now we formulate the transmission problem associated with the time-harmonic acoustic wave scattering by a bounded anisotropic inhomogeneity embedded in an unbounded anisotropic homogeneous medium:

Find complex-valued functions and satisfying the differential equations

 A1u1(x)=f1(x)forx∈Ω1, (14) A2u2(x)=f2(x)forx∈Ω2, (15)

and the transmission conditions on the interface ,

 γ+u2−γ−u1=φ0onS, (16) T+2u2−T−1u1=ψ0onS, (17)

where

 f2:=rΩ2f∈H0(Ω2),f1:=rΩ1f∈H0comp(Ω1),f∈H0comp(R3),φ0∈H12(S),ψ0∈H−12(S). (18)

In the above setting, equations (14) and (15) are understood in the distributional sense, the Dirichlet type transmission condition (16) is understood in the usual trace sense, while the Neumann type transmission condition (16) is understood in the canonical co-normal derivative sense defined by the relations (2)-(2).
If the interface continuity of and its co-normal derivatives is assumed, then , .

{remark}

If the variable coefficients and the function in (1) and (2) belong to and , then conditions (16) and (17) can be reduced to the homogeneous ones by introducing a new unknown function in , since on . For the function , the above formulated transmission problem is reduced then to the following one:
Find a solution to the differential equation

 A˜u(x)≡∂xk(akj(x)∂xj˜u(x))+ω2κ(x)˜u(x)=˜f(x),x∈R3, (19)

where due to the inclusions and in .

If in with as in (2), then equation (19) can be equivalently reduced to the Lippmann-Schwinger type integral equation (see, e.g. Colton et al6, Ch.8).

In our analysis, even for -smooth coefficients we always will keep the transmission conditions (16)–(17) which allow us to reduce the problem under consideration to the system of localized boundary-domain integral equations which live on the bounded domain and its boundary (cf. Nédélec20, Ch. 2).

Let us prove the uniqueness theorem for the transmission problem. {theorem} The homogeneous transmission problem (14)–(17) (with ) possesses only the trivial solution.

###### Proof.

Denote by a ball centred at the origin and radius , . We assume that is a sufficiently large positive number such that . Let a pair be a solution to the homogeneous transmission problem (14)–(17). Note that due to ellipticity of the constant coefficient operator . We can write the first Green identities for the domains and (see (9) and (10)),

 ∫Ω2[a(2)kj(x)∂ju2(x)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯∂ku2(x)−ω2κ2(x)|u2(x)|2]dx=⟨T+2u2,¯¯¯¯¯¯¯¯¯¯¯¯γ+u2⟩S, (20) ∫Ω1(R)[a(1)kj∂ju1(x)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯∂ku1(x)−ω2κ1|u1(x)|2]dx=−⟨T−1u1,¯¯¯¯¯¯¯¯¯¯¯¯γ−u1⟩S+⟨T+1u1,¯¯¯¯¯¯¯¯¯¯¯¯γ−u1⟩Σ(R). (21)

Since the matrices are symmetric and positive definite, in view of the homogeneous transmission conditions (16) and (17), after adding (20) and (21) and taking the imaginary part, we get

 \rm Im{∫ΣR¯¯¯¯¯¯¯¯¯¯¯¯¯u1(x)T1(x,∂x)u1(x)dΣR}=0.

Whence by Lemma 2 we deduce that in . In view of (16)–(17) then we see that the function solves the homogeneous Cauchy problem in for the elliptic partial differential equation with variable coefficients and being -smooth functions, see (3). By the interior and boundary regularity properties of solutions to elliptic problems we have and therefore in due to the well known uniqueness theorem for the Cauchy problem (see, e.g., Theorem 3 in Landis12, Theorem 6 in Calderon3). ∎

{remark}

Due to the recent results concerning the Cauchy problem for scalar elliptic operators one can reduce the smoothness of coefficients and to the Lipschitz continuity and require that is a Dini domain, see, e.g., Theorem 2.9 in Tao et al24.

## 3 Reduction to LBDIE system and equivalence theorem

### 3.1 Integral relations in the nonhomogeneous bounded domain

As it has already been mentioned, our goal is to reduce the above stated transmission problem to the corresponding system of localized boundary-domain integral equations. To this end let us define a localized parametrix associated with the fundamental solution of the Laplace operator,

 Pχ(x):=−χ(x)4π|x|,

where is a cut off function , see Appendix A. Throughout the paper we assume that this condition is satisfied and has a compact support if not otherwise stated.

Let us consider Green’s second identity for functions ,

where with being a ball centred at the point with radius . Substituting for the parametrix , by standard limiting arguments as one can derive Green’s third identity for (cf. Chkadua et al5),

 βu2+Nχu2−VχT+2u2+Wχγ+u2=PχA2u2inΩ2, (22)

where

 β(y)=13[a(2)11(y)+a(2)22(y)+a(2)33(y)], (23)

is a localized singular integral operator which is understood in the Cauchy principal value sense,

 Nχu2(y):= v.p.∫Ω2[A2(x,∂x)Pχ(x−y)]u2(x)dx=limε→0∫Ω2(y,ε)[A2(x,∂x)Pχ(x−y)]u2(x)dx,y∈R3, (24)

, , and are the localized single layer, double layer, and Newtonian volume potentials respectively,

 Vχg(y):=−∫SPχ(x−y)g(x)dSx,Wχg(y):=−∫S[T2(x,∂x)Pχ(x−y)]g(x)dSx,y∈R3∖S, (25) Pχh(y):=∫Ω2Pχ(x−y)h(x)dx,y∈R3. (26)

If the domain of integration in (24) and (26) is the whole space , we employ the notation

 Nχh(y):=v.p.∫R3[A2(x,∂x)Pχ(x−y)]h(x)dx,Pχh(y):=∫R3Pχ(x−y)h(x)dx, (27)

where the operator in the first integral in (27) is assumed to be extended to the whole . Some mapping properties of the above potentials needed in our analysis are collected in Appendix B.

In view of the following distributional equality

 ∂2∂xk∂xj1|x−y|=−4πδkj3δ(x−y)+v.p.∂2∂xk∂xj1|x−y|,

where is the Kronecker delta and is the Dirac distribution, we have (again in the distributional sense)

 A2(x,∂x)Pχ(x−y) =a(2)kj(x)∂2Pχ(x−y)∂xk∂xj+a(2)kj(x)∂xk∂Pχ(x−y)∂xj+ω2κ2(x)Pχ(x−y) =β(x)δ(x−y)+v.p.A2(x,∂x)Pχ(x−y), (28)

where

 v.p.A2(x,∂x)Pχ(x−y) =v.p.[−a(2)kj(x)4π∂2∂xk∂xj1|x−y|]+R(x,y)=v.p.[−a(2)kj(y)4π∂2∂xk∂xj1|x−y|]+˜R(x,y), (29)
 ˜R(x,y):=R(x,y)−a(2)kj(x)−a(2)kj(y)4π∂2∂xk∂xj1|x−y|.

Since , the functions and possess weak singularities of type as .

It is evident that if , then the terms in square brackets in formula (29) vanish and becomes a weakly singular kernel.

Using the integration by parts formula in (24), one can easily derive the following relation for

 Nχu2=−βu2−Wχγ+u2+Qχu2inΩ2, (30)

where

 Qχu2(y):=−∫Ω2a(2)kl(x)∂Pχ(x−y)∂xl∂u2(x)∂xkdx=∂ylPχ(a(2)kl∂ku2)(y),∀y∈Ω2. (31)

From Green’s third identity (22) and Theorem B we deduce

 βu2+Nχu2∈H1,0(Ω2,Δ)foru2∈H1,0(Ω2,A2), (32)

which, in turn, along with relations (30) and (31) implies

 Qχu2=∂ylPχ(a(2)kl∂ku2)∈H1,0(Ω2,Δ)foru∈H1,0(Ω2,A2).

In what follows, in our analysis we need the explicit expression of the principal homogeneous symbol of the singular integral operator , which due to (28) and (29) reads as

 S0(Nχ;y,ξ) =Fz→ξ⎛⎝−v.p.[a(2)kl(y)4π∂2∂zk∂zl1|z|]⎞⎠=−a(2)kl(y)4πFz→ξ(v.p.[∂2∂zk∂zl1|z|]) =−a(2)kl(y)4πF