Localized Boundary-Domain Singular Integral Equations of Dirichlet Problem for Self-adjoint Second Order Strongly Elliptic PDE Systems

The paper deals with the three-dimensional Dirichlet boundary value problem (BVP) for a second order strongly elliptic self-adjoint system of partial differential equations in the divergence form with variable coefficients and develops the integral potential method based on a localized parametrix. Using Green's representation formula and properties of the localized layer and volume potentials, we reduce the Dirichlet BVP to a system of localized boundary-domain integral equations (LBDIEs). The equivalence between the Dirichlet BVP and the corresponding LBDIE system is studied. We establish that the obtained localized boundary-domain integral operator belongs to the Boutet de Monvel algebra. With the help of the Wiener-Hopf factorization method we investigate corresponding Fredholm properties and prove invertibility of the localized operator in appropriate Sobolev (Bessel potential) spaces.


Introduction
We consider the Dirichlet boundary value problem (BVP) for a second-order strongly elliptic self-adjoint system of partial differential equations in the divergence form with variable coefficients and develop the generalized integral potential method based on a localized parametrix.
The BVP treated in the paper is well investigated in the literature by the variational method and also by the classical integral potential method, when the corresponding fundamental solution is available in explicit form (e.g. [1][2][3]) or when at least its properties are known to be good enough (see, e.g. [4,5] and references therein).
Our goal here is to develop a localized integral potential method for general second-order strongly elliptic self-adjoint systems of partial differential equations with variable coefficients. We show that a solution of the problem can be represented by explicit localized parametrix-based potentials and that the corresponding localized boundary-domain integral operator (LBDIO) is invertible, which is important for analysis of convergence and stability of localized boundary-domain integral equation (LBDIE)-based numerical methods for PDEs (e.g. [6][7][8][9][10][11][12][13]).
Using Green's representation formula and properties of the localized layer and volume potentials, we reduce the Dirichlet BVP to a system of LBDIEs. First, we establish the equivalence between the original BVP and the corresponding LBDIE system, which appeared to be quite non-trivial task and plays a crucial role in our analysis. Afterwards, we establish that the LBDIO of the system belongs to the Boutet de Monvel operator algebra. Employing the Vishik-Eskin theory, based on the Wiener-Hopf factorization method, we investigate corresponding Fredholm properties and prove invertibility of the localized operator in appropriate Sobolev (Bessel potential) spaces.
In the references [14][15][16][17][18][19][20], the traditional and localised boundary-domain integral equation methods have been developed for the case of scalar elliptic second-order partial differential equations with variable coefficients, and here, we extend the LBDIE method to PDE systems.

Formulation of the boundary value problems and localized Green's third identity
Consider a uniformly strongly elliptic second-order self-adjoint matrix partial differential operator where @ x D .@ 1 , @ 2 , @ 3 /, @ j D @ xj D @=@x j , a pq kj D a qp jk D a kq pj 2 C 1 , j, k, p, q D 1, 2, 3.
Here and in what follows, the Einstein summation by repeated indices from 1 to 3 is assumed if not otherwise stated.
We assume that the coefficients a pq kj are real and the quadratic form a pq kj .x/ Á kp Á qj is uniformly positive definite with respect to symmetric variables Á kp D Á pk 2 R, which implies that the principal homogeneous symbol of the operator A.x, @ x / with opposite sign, 3 3 is uniformly positive definite, which for the real symmetric coefficients a pq kj means there are positive constants c 1 and c 2 such that c 1 j j 2 j j 2 Ä N A.x, / Ä c 2 j j 2 j j 2 Here, a b :D a > b :D P 3 jD1 a j b j is the bilinear product of two-column vectors a, b 2 C 3 . Further, let D C be a bounded domain in R 3 with a simply connected boundary @ D S 2 C 1 , D [ S. Throughout the paper, n D .n 1 , n 2 , n 3 / denotes the unit normal vector to S directed outward the domain . Set :D R 3 n . By H r . / D H r 2 . / and H r .S/ D H r 2 .S/, r 2 R, we denote the Bessel potential spaces on a domain and on a closed manifold S without boundary, while D.R 3 / and D. / stand for C 1 functions with compact support in R 3 and in , respectively, and S.R 3 / denotes the Schwartz space of rapidly decreasing functions in R 3 . Recall that H 0 . / D L 2 . / is a space of square integrable functions in . For a vector u D .u 1 , u 2 , u 3 / > , the inclusion u D .u 1 , u 2 , u 3 / > 2 H r means that each component u j belongs to the space H r .
Let us denote by C u and u the traces of u on S from the interior and exterior of C , respectively. We also need the following subspace of H 1 . /, see, for example, [21], The Dirichlet BVP reads as follows: Find a vector function u D .u 1 , u 2 , u 3 / > 2 H 1, 0 . , A/ satisfying the differential equation Au D f in (2.4) and the Dirichlet boundary condition C u D ' 0 on S, (2.5) where ' 0 D .' 01 , ' 02 , ' 03 / > 2 H 1=2 .S/ and f D .f 1 , f 2 , f 3 / > 2 H 0 . / are given vector functions. Equation (2.4) is understood in the distributional sense, while the Dirichlet boundary condition (2.5) is understood in the usual trace sense. The classical co-normal derivative operators, T˙, associated with the differential operator A.x, @ x /, are well defined in terms of the gradient traces on the boundary S for a sufficiently smooth vector function v, say v 2 H 2 . /, as follows: The co-normal derivative operator defined in (2.6) can be extended by continuity to the space H 1, 0 . ; A/. The extension is inspired by Green's first identity (cf. [3,21,22]) as follows: is positive definite in the symmetric variables " qj . Therefore, Green's first identity (2.8) and Korn's inequality along with the Lax-Milgram lemma imply that the Dirichlet BVP (2.4)-(2.5) is uniquely solvable in the space H 1, 0 . ; A/ (e.g. [1][2][3]23]).

Parametrix-based operators and integral identities
As it has already been mentioned, our goal here is to develop the LBDIE method for the Dirichlet BVP (2.4)-(2.5). Let F .x/ :D 1=OE 4 jxj denote the scalar fundamental solution of the Laplace operator, D @ 2 1 C@ 2 2 C@ 2 3 . Let us define a localized matrix parametrix for the the matrix operator I as where P .x/ Á P .x/ :D .x/ F .x/ is a scalar function of the vector argument x, I is the unit 3 3 matrix and is a localizing function (Appendix A) Throughout the paper, we assume that condition (2.10) is satisfied if not otherwise stated. Note that the function can have a compact support, which is useful for numerical implementations, but generally this is not necessary, and the class X k C include also the functions not compactly supported but sufficiently fast decreasing at infinity, see [24] and Appendix A for details.
For sufficiently smooth vector functions u and v, say u, v 2 C 2 . /, there holds Green's second identity Denote by B.y, "/ a ball centred at point y, with radius " > 0, and let †.y, "/ :D @B.y, "/. Let us take as v.x/, successively, the columns of the matrix P.x y/, where y is an arbitrarily fixed interior point in , and write the identity (2.11) for the region " :D n B.y, "/ with " > 0 such that B.y, "/ . Keeping in mind that P > .x y/ D P.x y/ and OEA.x, @ x /P.x y/ > D OEA.x, @ x /P.x y/, we arrive at the equality, Z (2.12) The normal vector on †.y, "/ is directed inward " . Let the operator N defined as be the Cauchy principal value singular integral operator, which is well defined if the limit in the right-hand side exists. The similar operator with integration over the whole space R 3 is denoted as Note that @ 2 @x k @x j where ı kj is the Kronecker delta, and ı. / is the Dirac distribution, the left-hand side in (2.15) is also understood in the distributional sense, while the second summand in the right-hand side is a Cauchy-integrable function. Therefore, in view of (2.9) and taking into account that .0/ D 1, we can write the following equality in the distributional sense OEA.x, @/P.x y/ pq , (2.16) whereˇ.
The definition of N can be extended to smaller r as  Here, the densities g and h are three-dimensional vector functions. Introducing the following localized scalar Newtonian volume potential with h 0 being a scalar density function, we evidently obtain, for any vector function h D .h 1 , h 2 , h 3 / > . We will also need the localized vector Newtonian volume potential similar to (2.31) but with integration over the whole space R 3 , Mapping properties of potentials (2.29)-(2.33) are investigated in [15,24] and provided in Appendix B. We refer to relation (2.28) as Green's third identity. Because of the density of D. / in H 1, 0 . ; A/ ([22, Theorem 3.12]) and the mapping properties of the potentials, Green's third identity (2.28) is valid also for u 2 H 1, 0 . ; A/. In this case, the co-normal derivative T C u is understood in the sense of definition (2.7). In particular, (2.28) holds true for solutions of the previously formulated Dirichlet BVP (2.4)-(2.5).
On the other hand, applying the first Green identity (2.8) on " to u 2 H 1 . / and to P.x y/, as v.x/, and taking the limit as " ! 0, one can easily derive another, more general form of the third Green identity, where for the p-th component of the vector Q u.y/, we have Using the properties of localized potentials described in Appendix B (Theorems B.1 and B.4) and taking the trace of Equation (2.28) on S, we arrive at the relation for u 2 H 1, 0 . C ; A/, where the localized boundary integral operators V and W are generated by the localized single and double layer potentials and are defined in (B1) and (B2), the matrix is defined by (B17), while Now, we prove the following technical lemma.
By Theorems B.1 and B.2, it follows that the right-hand side function in the equality belongs to the space We have Clearly, R .x y/ D O.jx yj 2 / as x ! y and by (2.40) and (2.41), one can establish that for arbitrary scalar test function 2 D. /, there holds the relation (e.g. [26]) It is easy to see that [24] R : Consequently, Hence, the embedding Au 2 H 0 . / follows from (2.38) due to (2.39) and (2.44).
Actually, the continuity of operator in (2.44) and identity (2.45) in the proof of Lemma 2.2 imply by (2.34) the following assertion.

Corollary 2.3
If 2 X 3 , then the following operator is bounded,ˇC

Localized boundary-domain integral equation formulation of the Dirichlet problem and the equivalence theorem
S/ and f 2 H 0 . /. As we have derived earlier, there holds relations (2.28) and (2.36), which now can be rewritten in the form where :D T C u 2 H 1 2 .S/ and is defined by (B17). One can consider these relations as an LBDIE system with respect to the unknown vector functions u and . Now, we prove the following equivalence theorem. Further, because u 2 H 1, 0 . , A/, we can write Green's third identity (2.28), which in view of (3.4) can be rewritten as From (3.1) and (3.5), it follows that Hence, by Lemma 6.3 in [24], we have

Symbols and invertibility of a domain operator in the half-space
In what follows in our analysis, we need the explicit expression of the principal homogeneous symbol matrix S.N /.y, / of the singular integral operator N , which due to (2.13), (2.14) and (2.18) reads as while the Fourier transform operator F is defined as Here, we have applied that F z! .4 jzj/ 1 D j j 2 (e.g. [27]).
As we see, the entries of principal homogeneous symbol matrix S.N /.y, / of the operator N are even rational homogeneous functions in of order 0. It can easily be verified that both the characteristic function of the singular kernel in (2.18) and the symbol (4.1) satisfy the Tricomi condition, that is, their integral averages over the unit sphere vanish (cf. [26]). Relation is an even rational homogeneous matrix function of order 0 in and due to (2.2) it is positive definite, OES.B/.y, / N c 1 j j 2 for all y 2 , 2 R 3 n f0g and 2 C 3 .
Consequently, B is a strongly elliptic pseudo-differential operator of zero order (i.e. Cauchy-type singular integral operator) and the partial indices of factorization of the symbol (4.4) equal to zero (cf. [28][29][30]). We need some auxiliary assertions in our further analysis. To formulate them, let Q y 2 S D @ be some fixed point and consider the frozen symbol S. Q B/.Q y, / Á S. Q B/. /, where Q B denotes the operator B written in chosen local co-ordinate system. Further, let b Q B denote the pseudo-differential operator with the symbol Then, the frozen principal homogeneous symbol matrix S. Q B/. / is also the principal homogeneous symbol matrix of the operator b Q B. It can be factorized with respect to the variable 3 as Here, ‚ .˙/ . 0 , 3 / :D 3˙i j 0 j are the 'plus' and 'minus' factors of the symbol ‚. / :D j j 2 , and Q A .˙/ . 0 , 3 / are the 'plus' and 'minus' polynomial matrix factors of the first order in 3 of the positive definite polynomial symbol matrix Q A. 0 , 3 / Á Q A.e y, 0 , 3 / corresponding to the frozen differential operator A.Q y, @ x / at the point Q y 2 S [31][32][33], that is, with det Q It is easy to see that the factor matrices Q A .˙/ . 0 , 3 / have the following structure: which are well defined at any 2 R 3 for a bounded smooth function h. 0 , / satisfying the relation h. 0 , Let V E C be the extension operator by zero from R 3 C onto the whole space R 3 and r C :D r : H s .R 3 / ! H s .R 3 C / be the restriction operator to the half-space R 3 C . First, we prove the following assertion. Lemma 4.1 Let s 0 and 2 X k C with integer k 2. The operator Moreover, for f 2 H s .R 3 C /, the unique solution of the equation for u 2 H s .R 3 C / can be represented in the form u D r C u C , where / be a solution of this equation, and let us denote where The Fourier transform of (4.14) leads to the following relation Because of (4.5), we have the following factorization .  In accordance with Lemma 5.4 in [27], we conclude that the representation of the vector function F .g/. / in the form (4.22) is unique in view of inclusions (4.21), which in turn leads to the following relations: Now, from (4.18), (4.20) and the first equation in (4.23), it follows that u C 2 Q H 0 .R 3 C / is representable in the form .

(4.24)
Evidently, for the solution u 2 H 0 .R 3 C / of Equation (4.13), then we get the following representation Note that the representation (4.25) does not depend on the choice of the extension f . Indeed, let f 1 2 H 0 .R 3 / be another extension (cf. [27], Lemma 5.2). Here, Â C denotes the multiplication operator by the Heaviside step function Â.x 3 / that is equal to 1 for x 3 > 0 and vanishes for x 3 < 0. Therefore, is a solution of Equation (4.13) for any f 2 H 0 .R 3 C /. To this end, let us first note that for the vector function under the restriction operator in (4.26), the following embedding holds Indeed, by Lemma 5.2 in [27], we have and (4.27) follows from Theorem 4.4, Lemmas 20.2 and 20.5 in [27]. From (4.26) and (4.27), we obtain (4.28) By the relation [27]), we get from equality (4.28), (cf. [27], Theorems 4.4, 5.1, Lemmas 20.2, 20.5 ), we easily derive then representation (4.28) of u C can be rewritten as . Therefore, using (4.29) and in view of (4.11), from Theorem 10.1, Lemmas 4.4, 20.2, and 20.5 in [27], we finally derive with some positive constants c 1 and c 2 , hence (4.30) follows.

Lemma 4.2
Let the factor matrix Q A .C/ . 0 , / be as in (4.7), and a .C/ and c .C/ ij be as in (4.8) and (4.9), respectively. Then, the following equality holds Here is a contour in the lower complex half-plane enclosing all the roots of the polynomial det Q A .C/ . 0 , / with respect to .

Proof
Note that det Q A .C/ . 0 , / is a third order polynomial in , while p .C/ ij . 0 , / is a second-order polynomial in defined in (4.9). Let R be a circle centred at the origin and having sufficiently large radius R. By the Cauchy theorem, then we derive where Q ij . 0 , / D O.j j 2 / as j j ! 1.

It is clear that lim
Therefore, by passing to the limit in (4.32) as R ! 1, we obtain Let us introduce new coordinates r D j 0 j, ! D 0 =j 0 j and denote Then, we have For further use, let us introduce the auxiliary operator … 0 defined as The operator … 0 can be extended to the class of functions g. 0 , 3 / that are rational in 3 with the denominator not vanishing for real non-zero D . 0 , 3 / 2 R 3 n f0g, homogeneous of order m 2 Z :D f0,˙1,˙2, : : : g in and infinitely differentiable with respect to for 0 ¤ 0. Then, one can show that (cf. Appendix C in [15] ) where r R C denotes the restriction operator onto R C D .0, C1/ with respect to x 3 , is a contour in the lower complex half-plane in , orientated anticlockwise and enclosing all the poles of the rational function g. 0 , /. It is clear that if g. 0 , / is holomorphic in in the lower complex half-plane (Im < 0/, then … 0 .g/. 0 / D 0.

Invertibility of the Dirichlet localized boundary-domain integral operator
We would like to prove the following assertion.

Theorem 5.1
Let the localizing function 2 X 1 C and r > 1 2 . Then, the operator is invertible.
We will reduce the theorem proof to several lemmas. Proof Because (4.4) is a rational function in , we can apply the theory of pseudo-differential operators with symbol satisfying the transmission conditions [25,[27][28][29]34]. Now, with the help of the local principle (Lemma 23.9 in [27]) and where t 2 OE0, 1, are homotopic. Note that B D B 1 . The principal homogeneous symbol of the operator B t has the form S.B t /.y, / Dˇ.y/ C t S.N/.y, / D .1 t/ˇ.y/ C tS.B/.y, /.
It is easy to see that the symbol S.B t /.y, / is positive definite, OES.B t /.y, / N D .1 t/ OEˇ.y/ N C t OES.B/.y, / N cj j 2 for all y 2 , ¤ 0, 2 C 3 and t 2 OE0, 1, where c is some positive number. Because S.B t /.y, / is rational, even, and homogeneous of order zero in , we conclude, as earlier, that the operator is Fredholm for all s 0 and for all t 2 OE0, 1. Therefore, Ind B t is the same for all t 2 OE0, 1. On the other hand, due to the equality B 0 D r I, we get

Lemma 5.3
Let 2 X 1 . The operator D given by (5.3) is Fredholm.

Proof
To investigate Fredholm properties of the operator D, we apply the local principle (cf. e.g. [27,35], 19 and 22). Because of this principle, we have to show first that the operator D is locally Fredholm at an arbitrary 'frozen' interior point Q y 2 , and secondly that the so called generalized Šapiro-Lopatinskiȋ condition for the operator D holds at an arbitrary 'frozen' boundary point Q y 2 S. To obtain the explicit form of this condition, we proceed as follows. Let Q U be a neighbourhood of a fixed point Q y 2 , and let Q 0 , and consider the operator Q 0 D Q ' 0 . We consider separately two possible cases, case (1): Q y 2 , and case (2):e y 2 S.
Case (1). Ife y 2 , then we can choose a neighbourhood Q U such that Q U . Therefore, the operator Q 0 D Q ' 0 has the same Fredholm properties as the operator Q 0 B Q ' 0 (see the similar arguments in the proof of Theorem 22.1 in [27]). Then by Lemma 5.2, we conclude that Q 0 D Q ' 0 is a locally Fredholm operator at interior points of . Case (2). If Q y 2 S, then at this point we have to 'freeze' the operator Q 0 D Q ' 0 , which means that we can choose a neighbourhood Q U sufficiently small such that at the local co-ordinate system with the origin at the point Q y and the third axis coinciding with the normal vector at the point Q y 2 S, the following decomposition holds is a bounded operator with small norm, while , is defined in the upper half-space R 3 C and possesses the following mapping property where ! D 0 j 0 j , D . 0 , n /, 0 D . 1 , ..., n 1 /. The generalized Šapiro-Lopatinskiȋ condition is related to the invertibility of the operator (5.5). Indeed, let us write the system corresponding to the operator b Q D: Note that the operator r C b Q B V E is a singular integral operator with even rational elliptic principal homogeneous symbol. Then, due to Lemma 4.1, the operator is invertible, we can determine Q u from Equation (5.6) and write It is easy to see that In view of the relation (e.g.
where the operator is dual to the trace operator . When the surface S coincides with R 2 D @R 3 C , then we have Q D e .e y 0 /˝ı 3 with ı 3 being the one-dimensional Dirac distribution in the Q y 3 direction. Then, we arrive at the equality With the help of these relations Equation (5.9) can be rewritten in the following form with e being a homogeneous function of order 1 given by the equality generated by the left-hand side expression in (5.10) is invertible. In particular, it follows that the system of Equation This condition is called the Šapiro-Lopatinskiȋ condition (cf. [27], Theorems 12.2 and 23.1, and also formulas (12.27) and (12.25)). Let us show that in our case the Šapiro-Lopatinskiȋ condition holds. To this end, let us note that the principal homogeneous symbols S. e N/, S. Q B/, S. Q P/ and S. Q V/ of the operators N, B, P, and V in the chosen local co-ordinate system involved in formula (5.12) read as where Q denotes the matrixˇwritten in chosen local co-ordinate system. Rewrite (5.12) in the form 14) (5.17) Now, from (5.14) with the help of (5.17), we derive Quite similarly, from (5.15) with the help of (5.17), we get Therefore, due to (5.13), (5.16), (5.18) and Lemma 4.2, we have

Proof
For t 2 OE0, 1, let us consider the operator with B t DˇC t N and establish that it is homotopic to the operator D D D 1 . We have to check that for the operator D t the Šapiro-Lopatinskiȋ condition is satisfied for all t 2 OE0, 1. Indeed, in this case the Šapiro-Lopatinskiȋ condition reads as and c for all 0 ¤ 0 and t 2 OE0, 1. Then, it is clear that for all 0 ¤ 0 and for all t 2 OE0, 1, which implies that for the operator D t the Šapiro-Lopatinskiȋ condition is satisfied. Therefore, the operator

APPENDIX A. Classes of localizing functions.
Here, we present the classes of localizing functions used in the main text (see [24] for details).
Evidently, we have the following imbeddings: X k1 X k2 and X k1 C X k2 C for k 1 > k 2 . The class X k C is defined in terms of the sinetransform. The following lemma from [24] provides an easily verifiable sufficient condition for non-negative non-increasing functions to belong to this class.
One can observe that 1k 2 X k C for k 1, while 2 2 X 1 C due to Lemma A.2.

APPENDIX B. Properties of localized potentials.
Here, we collect some assertions describing mapping properties of the localized potentials. The proofs coincide with or are similar to the ones in [24] and [15,Appendix B] (see also [1], Chapter 8 and the references therein). Let us introduce the boundary operators generated by the localized layer potentials associated with the localized parametrix P.x y/ Á P .