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

We consider the time-harmonic acoustic wave scattering by a bounded {\it anisotropic inhomogeneity} embedded in an unbounded {\it 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 {\it singular localised boundary-domain integral equations}. Fredholm properties of the corresponding {\it localised boundary-domain integral operator} are 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.


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 ∈ R is formulated mathematically as a transmission problem for a second-order elliptic partial differential equation with variable coefficients A 2 (x, x ) u(x) ≡ x k (a (2) k (x) x u(x))+ 2 2 (x) u(x) = 2 in the inhomogeneous anisotropic bounded region Ω + ⊂ R 3 and for a Helmholtz type equation with constant coefficients A 1 ( x )u(x) ≡ a (1) k x k x u(x) + 2 1 u(x) = 1 in the homogeneous anisotropic unbounded region Ω − = R 3 ∖Ω + . The material parameters a (q) k and q are not assumed to be continuous across the interface S = Ω − = Ω + 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 u and its co-normal derivative on S. The transmission problems for the Helmholtz equation, ie, when A 2 (x, ) = A 1 ( ) = Δ + 2 , which corresponds to a homogeneous isotropic media, are well studied in the case of smooth and Lipschitz interface (see Costabel and Stephan, 1 Kleinman and Martin, 2 Kress and Roach, 3 Torres and Welland, 4

and the references therein).
The special isotropic transmission problems when A 2 (x, x ) = Δ + 2 2 (x) and A 1 ( x ) = Δ + 2 is the Helmholtz operator are also well presented in the literature (see Colton and Kress, 5 Nédélec,6 and the references therein). The acoustic scattering problem in the whole space corresponding to a more general isotropic case, when a (2) k (x) = a(x) k , where kj is Kronecker delta and A 1 ( x ) = Δ + 2 , was analysed by the indirect boundary-domain integral equation method by Werner. 7,8 Applying the potential method based on the Helmholtz fundamental solution, 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 Martin,9 where the problem was reduced to a singular integro-differential equation in the inhomogeneous bounded region Ω + . Using the uniqueness and existence results obtained by Werner,7,8 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 nonlocal 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 Miranda 10 ).
Our goal here is to show that the acoustic transmission problems for anisotropic heterogeneous structures can be equivalently reformulated as systems of singular localised boundary-domain integral equations (LBDIEs) with the help of a localised harmonic parametrix based on the harmonic fundamental solution, which is a quasi-parametrix for the considered PDEs of acoustics, and to prove that the corresponding singular localised 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 Mikhailov 11 , Zhu et al 12,13 and Sladek et al 14 ). The main novelty of the paper is in application of the singular localised boundary-domain integral equations method to the problem of acoustic transmission through a penetrable, anisotropic, inhomogeneous obstacle.
The paper is organised as follows. First, after mathematical formulation of the problem, we introduce layer and volume potentials based on a localised 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 localised 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 (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 analyse also an alternative nonlocal 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 multilayer piecewise inhomogeneous anisotropic structures.
Finally, for the readers convenience, we collected necessary auxiliary material related to classes of localising functions, properties of localised potentials and anisotropic radiating potentials in three brief appendices.

FORMULATION OF THE TRANSMISSION PROBLEM
Let Ω + = Ω 2 be a bounded domain in R 3 with a simply connected boundary Ω 2 = S, and Ω − = Ω 1 ∶= R 3 ∖Ω 2 . For simplicity, we assume that S ∈ C ∞ if not otherwise stated. Throughout the paper, n = (n 1 , n 2 , n 3 ) denotes the unit normal vector to S directed outward the domain Ω 2 .
We assume that the propagation region of a time harmonic acoustic wave u tot is the whole space R 3 that consists of an inhomogeneous part Ω 2 and a homogeneous part Ω 1 . Acoustic wave propagation is governed by the uniformly elliptic second-order scalar partial differential equation and (x) are real-valued functions, ∈ R is a frequency parameter, while ∈ L 2,comp (R 3 ) is the volume force amplitude. Here and in what follows, the Einstein summation by repeated indices from 1 to 3 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, eg, Colton and Kress, 5 chapter 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 a kj and are constant in the homogeneous unbounded region Ω 1 and the following relations hold: where a (1) k and 1 are constants, while a (2) k and 2 are smooth function in Ω 2 , Moreover, the matrices a q = [a (q) k ] 3 k, =1 are uniformly positive definite, ie, there are positive constants c 1 and c 2 such that We do not assume that the coefficients a kj and are continuous across S in general, ie, the case a (2) k (x) ≠ a (1) k and 2 (x) ≠ 1 for x ∈ S is covered by our analysis. Further, let us denote For a function v sufficiently smooth in Ω 1 and Ω 2 , the classical co-normal derivative operators, T ± cq are well defined as here, the symbols + and − denote one-sided boundary trace operators on S from the interior and exterior domains, respectively. Their continuous right inverse operators, which are nonuniquely defined, are denoted by symbols ( ± ) −1 .
, and H s (S) = H s 2 (S), s ∈ R, we denote the L 2 -based Bessel potential spaces on an open domain Ω ⊂ R 3 and on a closed manifold S without boundary, while (Ω) stands for the space of infinitely differentiable test functions with support in Ω. Recall that H 0 (Ω) = L 2 (Ω) is a space of square integrable functions in Ω. Let the symbol r Ω denote the restriction operator onto Ω.
Since the boundary traces of gradients, ± ( x j v(x)) are generally not well defined on functions from H 1 (Ω q ), the classical co-normal derivatives (6) are not well defined on such functions either, cf Mikhailov,14 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 H 1 (Ω 2 ) and H 1 loc (Ω 1 ) to which the classical co-normal derivatives can be continuously extended, cf, eg, Grisvard, 15 Costabel, 16 and Mikhailov 17 : We will also use the corresponding spaces with the Laplace operator Δ instead of A q .
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 H 1, 0 loc (Ω 1 ; A 1 ) and H 1,0 (Ω 2 ; A 2 ) giving the canonical co-normal derivative operators, T ± 1 and T + 2 , defined in the weak form as where are the right inverse operators to the trace operators ± , and the angular brackets ⟨·, ·⟩ S should be understood as duality pairing of H − 1 2 (S) with H 1 2 (S), which extends the usual bilinear L 2 (S) inner product.
The canonical co-normal derivatives T − 2 u and T + 1 u can be defined analogously for functions from the spaces H 1, 0 loc (Ω 1 ; A 2 ) and H 1,0 (Ω 2 ; A 1 ), respectively, provided that the variable coefficients a (2) k (x) and 2 (x) are continuously extended from Ω 2 to the whole space R 3 preserving the smoothness. It is evident that for functions from the space H 2 (Ω 2 ) and H 2 loc (Ω 1 ), the classical and canonical co-normal derivative operators coincide. Concerning the canonical and generalised co-normal derivatives in wider functional spaces, see Mikhailov. 17 For two times continuously differentiable function w in a neighbourhood of S, we employ also the notation T q (x, x )w ∶= a (q) k n k (x)( x w(x)), x ∈ S, to denote both the classical and the canonical co-normal derivatives. Recall that the definitions of the co-normal derivatives T ± q do not depend on the choice of the right inverse operators ( ± ) −1 , and the following Green's first and second identities hold (cf Mikhailov, 17 Theorem 3.9), By Z(Ω 1 ), we denote a subclass of complex-valued functions from H 1 loc (Ω 1 ) satisfying the Sommerfeld radiation conditions at infinity (see Vekua 18 and Colton and Kress 5 for the Helmholtz operator and Vainberg 19 and Jentsch et al 20 for the "anisotropic" operator A 1 defined by (5)). Denote by S the characteristic surface (ellipsoid) associated with the operator A 1 , For an arbitrary vector ∈ R 3 with | | = 1, there exists only one point ( ) ∈ S such that the outward unit normal vector n( ( )) to S at the point ( ) has the same direction as , ie, n( ( )) = . Note that ( − ) = − ( ) ∈ S and n( − ( )) = − . It can easily be verified that where a −1 1 is the matrix inverse to a 1 ∶= . Definition 1. A complex-valued function v belongs to the class Z(Ω 1 ) if there exists a ball B(R) of radius R centred at the origin such that v ∈ C 1 (Ω 1 ∖B(R)) and v satisfies the Sommerfeld radiation conditions associated with the operator where ( ) ∈ S corresponds to the vector = x∕|x| (ie, ( ) is given by (11) with = x∕|x|).
Note that due to the ellipticity of the operator A 1 ( x ), any solution to the constant coefficient homogeneous equation Conditions (12) are equivalent to the classical Sommerfeld radiation conditions for the Helmholtz equation if A 1 ( ) = Δ( ) + 2 , ie, if 1 = 1 and a (1) k = k , where kj is the Kronecker delta. There holds the following analogue of the classical Rellich-Vekua lemma (for details, see Jentsch et al 20 and Natroshvili et al 21 ).
where Σ R is the sphere with radius R centred at the origin. Then v = 0 in Ω 1 .
Remark 1. For x ∈ Σ R and = x∕|x|, we have n(x) = , and in view of (6) and (12) for a function v ∈ Z(Ω 1 ), we get Therefore, by (11) and the symmetry condition a kj = a jk , we arrive at the relation On the other hand, matrix a 1 is positive definite, cf (4), which implies positive definiteness of the inverse matrix a −1 1 . Hence, there are positive constants 0 and 1 such that the inequality 0 < 0 ⩽ (a −1 1 · ) − 1 2 ⩽ 1 < ∞ holds for all ∈ Σ 1 . Consequently, (13) for ≠ 0 is equivalent to the condition in the well-known Rellich-Vekua lemma in the theory of the Helmholtz equation, Vekua, 18 Rellich, 22 and Colton and Kress, 5 In the unbounded region Ω 1 , we have a total wave field u tot = u inc + u sc , where u inc is a wave motion initiating known incident field and u sc is a radiating unknown scattered field. It is often assumed that the incident field is defined in the whole of R 3 , being, for example, a corresponding plane wave that solves the homogeneous equation A 1 u inc = 0 in R 3 but does not satisfy the Sommerfeld radiation conditions at infinity. Motivated by relations (2), let us set 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: and the transmission conditions on the interface S, 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 (7) and (8).
If the interface continuity of u tot and its co-normal derivatives is assumed, then 0 = − u inc , 0 = T − 1 u inc . Remark 2. If the variable coefficients a kj and the function in (1) and (2) belong to C 2 (R 3 ) and u inc ∈ H 2 loc (R 3 ), then conditions (16) and (17) can be reduced to the homogeneous ones by introducing a new unknown functioñ For the functionũ, the above formulated transmission problem is reduced then to the following one: (2), then Equation (19) can be equivalently reduced to the Lippmann-Schwinger type integral equation (see, eg, Colton and Kress, 5 chapter 8).
In our analysis, even for C 2 (R 3 )-smooth coefficients, we always will keep the transmission conditions (16) and (17), which allow us to reduce the problem under consideration to the system of localised boundary-domain integral equations that live on the bounded domain Ω 2 and its boundary S (cf Nédélec, 6 chapter 2).
Let us prove the uniqueness theorem for the transmission problem. (14) - (17)

Theorem 1. The homogeneous transmission problem
Proof. Denote by B(R) a ball centred at the origin and having radius R, Σ R ∶ = B(R). We assume that R is a sufficiently large such that Ω 2 ⊂ B(R). Let a pair (u 1 , u 2 ) be a solution to the homogeneous transmission problem (14) - (17). Note that u 1 ∈ C ∞ (Ω 1 ) due to ellipticity of the constant coefficient operator A 1 . We can write the first Green identities for the domains Ω 2 and Ω 1 (R) ∶= Ω 1 ∩ B(R) (see (9) and (10)), Since the matrices a q = [a (q) k ] 3 k, =1 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 Whence by Lemma 1 we deduce that u 1 = 0 in Ω 1 . In view of (16) and (17) then we see that the function u 2 solves the homogeneous Cauchy problem in Ω 2 for the elliptic partial differential equation A 2 u 2 = 0 with variable coefficients a (2) k and 2 being C 2 (Ω 2 )-smooth functions, see (3). By the interior and boundary regularity properties of solutions to elliptic problems, we have u 2 ∈ C 2 (Ω 2 ) and therefore u 2 = 0 in Ω 2 due to the well-known uniqueness theorem for the Cauchy problem (see, eg, Landis, 23 Theorem 3; Calderon, 24 Theorem 6).
Remark 3. Due to the recent results concerning the Cauchy problem for scalar elliptic operators, one can reduce the smoothness of coefficients a (2) k and 2 to the Lipschitz continuity and require that Ω 2 is a Dini domain, see, eg, Theorem 2.9 in Tao et al. 25

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 localised boundary-domain integral equations. To this end, let us define a localised parametrix associated with the fundamental solution −(4 |x|) −1 of the Laplace operator, where is a cut-off function ∈ X 4 + , 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 Ω 2 (y, ) ∶ = Ω 2 ∖B( y, ) with B( y, ) being a ball centred at the point y ∈ Ω 2 with radius > 0. Substituting for v 2 (x) the parametrix P (x − y), by standard limiting arguments as → 0, one can derive Green's third identity for u ∈ H 1,0 (Ω 2 , A 2 ) (cf Chkadua et al 26 ), where  is a singular localised integral operator that is understood in the Cauchy principal value sense, V , W , and  are the localised single layer, double layer, and Newtonian volume potentials, respectively, Note that if P is replaced with the corresponding fundamental solution, then  u 2 = 0, = 1, and the third Green identity reduces to the familiar integral representation formula.
If the domain of integration in (24) and (26) is the whole space R 3 , we employ the notation where the operator A 2 (x, x ) in the first integral in (27) is assumed to be extended to the whole R 3 . Some mapping properties of the above potentials needed in our analysis are collected in Appendix B.
In view of the following distributional equality, 2 x k x where kj is the Kronecker delta and (·) is the Dirac distribution, we have (again in the distributional sense) where possesses the strong Cauchy singularity as x → y. Thus, although P is a parametrix for the Laplace operator, it is not a parametrix for the operator A 2 , and we will call it instead a quasi-parametrix for A 2 . It is evident that if a (2) k (x) = a 2 (x) k , then the terms in square brackets in formula (29) vanish and v.p. A 2 (x, x )P (x − y) becomes a weakly singular kernel.
Using the integration by parts formula in (24), one can easily derive the following relation for where From Green's third identity (22) and Theorem 8, we deduce which, in turn, along with relations (30) and (31) implies In what follows, in our analysis, we need the explicit expression of the principal homogeneous symbol 0 (N ; , ) of the singular integral operator N , which due to (28) and (29) reads as where A 2 ( , ) = a (2) kl ( ) k l .
Here and in what follows,  and  −1 denote the distributional direct and inverse Fourier transform operators that for a summable function g read as Note that the principal homogeneous symbol 0 (N ; , ) is a rational homogeneous even function of order zero in . In view of Theorem 9 in the Appendix, the interior trace of equality (22) on S reads as where the functions and are defined by (23) and (B2),  + = +  ,  + = +  , while the operators  and  , generated by the direct values of the single and double layer potentials, are given by formulas (B1). Finally, we formulate a technical lemma that follows from formulas (30), (31), and Theorem 8.

Integral relations in the homogeneous unbounded domain
For any radiating solution , there holds Green's third identity (for details, see the references Colton and Kress, 5 Vekua, 18 Jentsch et al, 20 and Natroshvili et al 21 ) where Here, T 1 (x, x ) = a (1) k n k (x) x , n(x) is the outward unit normal vector to S at the point x ∈ S, and is a radiating fundamental solution of the operator A 1 (see, eg, Lemma 1.1 in Jentsch et al 20 ). If x belongs to a bounded subset of R 3 , then for sufficiently large |y|, we have the following asymptotic formula where = ( ) ∈ S corresponds to the direction = y∕|y| and is given by (11). The asymptotic formula (39) can be differentiated arbitrarily many times with respect to x and y. The mapping properties of these potentials and the boundary operators generated by them are collected in Appendix C. Evidently, the layer potentials V g and W g solve the homogeneous differential Equation (14), ie, while for 1 ∈ H 0 comp (Ω 1 ), the volume potential  1 ∈ H 2 loc (R 3 ) solves the following nonhomogeneous equation (see Lemma 5(i)) The exterior trace and co-normal derivative of the third Green identity (35) on S read as (see Lemma 5(ii)) where the integral operators  ,  ,  ′ , and  are defined in Appendix C by formulas (C1) -(C4). Note that the operators  , 2 −1 I − , 2 −1 I + ′ , and  involved in (42) and (43)  . Therefore, to obtain Dirichlet-to-Neumann or Neumann-to-Dirichlet mappings for arbitrary values of the frequency parameter , we apply the ideas of the so-called combined-field integral equations, cf Burton and Miller, 29 Brakhage and Werner, 30 Colton and Kress, 5, 27 Leis, 31 and Panich. 32 Multiply Equation (42) by −i with some fixed positive and add to Equation (43) to obtain where  g ∶= ( In view of Lemma 6, from (44) we derive the following analogue of the Steklov-Poincaré type relation for arbitrary where
Let us prove the following equivalence theorem.
From uniqueness Theorem 1 and the equivalence Theorem 2, the following assertion follows directly.

ANALYSIS OF THE LBDIO
Let us rewrite the LBDIE system (50) -(55) in a more convenient form for our further purposes whereE =E Ω 2 denotes the extension operator by zero from Ω 2 onto Ω 1 , N is a pseudodifferential operator given in (27), N + = + N , and  + = +  . Note that for a function u 2 ∈ H 1 (Ω 2 ), we have u 2 +  u 2 = ( I + N )Eu 2 in Ω 2 . It can easily be seen that if the unknowns (u 2 , 2 , 2 ) are determined from the first three equations of system (64) -(69), then the unknowns ( 1 , 1 , u 1 ) are determined explicitly from the last three equations of the same system. Therefore, the main task is to investigate the matrix integral operator generated by the left hand side expressions in (64) -(66).
Let us rewrite the first three equations of the LBDIE system (64) -(69) in matrix form Let us introduce the spaces Recall that for ∈ X 4 + , the principal homogeneous symbol 0 (N ; , ) of the operator N given by (33) is a rational homogeneous function of order zero in . Therefore, applying the inclusion (32) and the mapping properties of the pseudodifferential operators with rational type symbols (see, eg, Hsiao and Wendland, 34 Theorem 8.4.13) and using Theorems 8 and 10 we deduce that the operators are continuous for ∈ X 4 + . Now, we prove the main theorem of this section. (73) Evidently, the triangular matrix operator is also invertible due to Lemma 6, from (73), it follows that the block-triangular matrix operator

and consequently, operator (72) is invertible if and only if the following operator is invertible
Further, we apply the Vishik-Eskin approach, developed in Eskin, 35 and establish that operator (74) is invertible. The proof is performed in four steps.
Step 1. Here, we show that the operator is Fredholm with zero index. In view of (33), the principal homogeneous symbol of the operator I + N can be written as Since the symbol 0 (D 11 ; , ) given by (77) is an even rational homogeneous function of order 0 in it follows that its factorisation index equals to zero (see Eskin, 35 §6 ). Moreover, the operator I + N possesses the transmission property. Therefore, we can apply the theory of pseudodifferential operators satisfying the transmission property to deduce that operator (76) is Fredholm (see Eskin, 35 Theorem 11.1 and Lemma 23.9; Boutet de Monvel 36 ). To show that IndD 11 = 0, we use the fact that the operators D 11 and D 11,t , where are homotopic. Evidently D 11,0 = I and D 11,1 = D 11 . In view of (33) and (77), for all t ∈ [0, 1], for all ∈ Ω 2 , and for all ∈ R 3 ∖{0}, and consequently the operator D 11,t is elliptic. Since 0 (D 11,t ; , ) is rational, even, and homogeneous of order zero in , we conclude that the operator D 11,t ∶ H 1 (Ω 2 ) → H 1 (Ω 2 ) is continuous Fredholm operator for all t ∈ [0, 1]. Therefore IndD 11,t is the same for all t ∈ [0, 1]. On the other hand, due to the equality D 11,0 = I, we get IndD 11 = IndD 11,1 = IndD 11,t = IndD 11,0 = 0.
Step 2.Now we show that the operator D defined by (74) and (75) is Fredholm. To this end, we apply the local principle (see, eg, Eskin, 35 §19 and §22). Let U j be an open neighbourhood of a fixed point̃∈ R 3 and let ( ) 0 , ( ) 0 ∈ (U ) be such that supp ( ) 0 ∩supp ( ) 0 ≠ ∅ contains some open neighbourhood U ′ ⊂ U of the point y 0 . Consider the operator ( ) 0 D ( ) 0 .We separate two possible cases: (1)̃∈ Ω 2 and (2)̃∈ S. In the first case, wheñ∈ Ω 2 , we can choose a neighbourhood U of the point̃such that U ⊂ Ω 2 . Then the operator where D 11 is defined by (76). As we have already shown in Step 1, this operator is Fredholm with zero index. In the second case, wheñ∈ S, we need to check that the Šapiro-Lopatinskiȋ type condition for the operator D is fulfilled, ie, we have to show that the so-called boundary symbol that is constructed by means of the principal homogeneous symbols of the pseudodifferential operators involved in (75) is nonsingular (see Eskin,35 §12). To write the boundary symbol function explicitly, we assume that the symbols are "frozen" at the point̃∈ S considered as the origin O ′ of some local coordinate system. Denote byã (2) kl (̃) the corresponding "frozen" coefficients of the principal part of the differential operator A 2 (y, y ) subjected to a translation and an orthogonal transformation related to the local co-ordinate system. If the matrix of the transformation of the original co-ordinate system Oy 1 y 2 y 3 to the new one O Evidently, the matrixã 2 (̃) = [ã (2) is positive definite and for arbitrarỹ∈ S, we havẽ due to (78) and (B2). Further, let us note that the layer potentials can be represented by means of the volume potential (see, eg, Chkadua et al 26 ) is the adjoint operator to the trace operator , ie, and H −t S does not contain nonzero elements, when t ⩽ 1 2 (see Lemma 3.39 in McLean, 37 Theorem 2.10(i) in Mikhailov 17 ). In view of (79) and (80), the operator D 12 in (75) can be represented as and its principal homogeneous symbol due to the above formulas and Remark 6 in Appendix C can be written as since the principal homogeneous symbol of the operator P reads as 0 (P; ) = − z→ [(4 |z|) −1 ] = −| | −2 .
Due to the Vishik-Eskin approach, now we have to construct the following matrix associated with the principal homogeneous symbols of the operators involved in D at the local co-ordinate system introduced above where R 11 (̃, ) is the principal homogeneous symbol of the operator D 11 = I + N , R 12 (̃, ) is the principal homogeneous symbol of operator (81) and is given by (82), R 21 (̃, ) is the principal homogeneous symbol of the operator N , is the principal homogeneous symbol of the boundary operator D 22 , which due to (75), (B4), (B5), and (C5) is written as Below, we drop the arguments̃and when it does not lead to misunderstanding. Now, we show that the Šapiro-Lopatinskiȋ type condition for the operator D is satisfied, ie, the boundary symbol (see Eskin,35 §12, formulas (12.25), (12.27)) associated with the operator D does not vanish for 3 ) denote the "plus" and "minus" factors, respectively, in the factorisation of the symbol R 11 ( ′ , 3 ) with respect to the variable 3 in the complex 3 plane, while Π + is a Cauchy type integral operator and Π ′ is the operator defined on the set of rational functions where − is a contour in the lower complex half-plane orientated counterclockwise and enclosing all poles of the rational function g with respect to 3 . Denote the roots of the equation A 2 ( ) ≡ã (2) kl k l = 0 with respect to 3 by ( where we assume that 2 > 0. Then Since Δ( ) = | | 2 = Δ (+) ( )Δ (−) ( ) with Δ (±) ( ) ∶= 3 ±i | ′ |, we get the following factorisation of the symbol R 11 ( ), Using formulas (84) -(86) and (88) -(91), we rewrite (87) as where With the help of residue theorem, by direct calculations, we find Therefore, from (93) in view of (95) -(97) and (90), we get Now, we evaluate the function S (2) 3l l . Since and are roots of the quadratic equation kl k l =ã (2) kl k l = 0, we have Again by direct calculations, we find .
Step 3. Here, we prove that the index of the operator D equals to zero. To this end, let us consider the operator with t ∈ [0, 1], and establish that it is homotopic to the operator D.
Evidently, D 1 = D and D t ∶H 1 (Ω 2 ) × H − 1 2 (S) → H 1 (Ω 2 ) × H 1 2 (S) is continuous. First, we show that for the operator D t , the Šapiro-Lopatinskiȋ condition is satisfied for all t ∈ [0, 1]. The counterpart of the matrix (83) now reads as where R 11 , R 12 , and R 21 are defined by formulas (84), (82), and (85), respectively, while in accordance with (104) and (86), The corresponding boundary symbol associated with the Šapiro-Lopatinskiȋ condition, the counterpart of (87), has the form and due to the inequalities (102) and (103), we have Thus, the Šapiro-Lopatinskiȋ condition for the operator D t is satisfied for all t ∈ [0, 1]. Therefore, as in the case of the operator D, it follows that the operator is Fredholm and has the same index for all t ∈ [0, 1]. On the other hand, the upper triangular matrix operator D 0 has zero index since one of the operators in the main diagonal, is Fredholm with zero index as it has been shown in Step 1. Consequently, IndD = IndD 1 = IndD t = IndD 0 = 0.
Theorem 3 and Corollaries 2 and 3 imply the following assertion.
Let us first prove the following equivalence theorem.
(i) The first part of the theorem follows from the derivation of variational Equation (119).

APPENDIX A: CLASSES OF CUT-OFF FUNCTIONS
Here, we present some classes of localising cut-off functions (for details, see Chkadua et al 33 ).