Analysis of segregated boundary-domain integral equations for BVPs with non-smooth coefficients on Lipschitz domains

Segregated direct boundary-domain integral equations (BDIEs) based on a parametrix and associated with the Dirichlet and Neumann boundary value problems for the linear stationary diffusion partial differential equation with a variable Hölder-continuous coefficients on Lipschitz domains are formulated. The PDE right-hand sides belong to the Sobolev (Bessel potential) space Hs−2(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H^{s-2}(\Omega)$\end{document} or H˜s−2(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\widetilde{H}^{s-2}( \Omega)$\end{document}, 12<s<32\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{1}{2}< s<\frac{3}{2}$\end{document}, when neither strong classical nor weak canonical co-normal derivatives are well defined. Equivalence of the BDIEs to the original BVP, BDIE solvability, solution uniqueness/non-uniqueness, and the Fredholm property and invertibility of the BDIE operators are analysed in appropriate Sobolev spaces. It is shown that the BDIE operators for the Neumann BVP are not invertible; however, some finite-dimensional perturbations are constructed leading to invertibility of the perturbed (stabilised) operators.


Introduction
Many applications in science and engineering can be modeled by boundary-value problems (BVPs) for partial differential equations with variable coefficients. Reduction of the BVPs with arbitrarily variable coefficients to explicit boundary integral equations is usually not possible, since the fundamental solution needed for such reduction is generally not available in an analytical form (except for some special dependence of the coefficients on coordinates). Using a parametrix (Levi function) introduced in [24], [19] as a substitute of a fundamental solution, it is possible however to reduce such a BVP to a system of boundary-domain integral equations, BDIEs, (see e.g. [37,Sect. 18], [42,43], where the Dirichlet, Neumann and Robin problems for some PDEs were reduced to indirect BDIEs). However, many questions about their equivalence to the original BVP, solvability, solution uniqueness and invertibility of corresponding integral operators remained open for rather long time.
In [3,5,29,6,8], the 3D mixed (Dirichlet-Neumann) boundary value problem (BVP) for the stationary diffusion PDE with infinitely smooth variable coefficient on a domain with an infinitely smooth boundary and a square-integrable right-hand side was reduced to either segregated or united direct Boundary-Domain Integral or Integro-Differential Equations, some of the which coincide with those formulated in [28]. Such BVPs appear e.g. in electrostatics, stationary heat transfer and other diffusion problems for inhomogeneous media.
For a function from the Sobolev space H s (Ω), 1 2 < s < 3 2 , a classical co-normal derivative in the sense of traces may not exist. However, when this function satisfies a second order partial differential equation with a right-hand side from H s−2 (Ω), the generalised co-normal derivative can be defined in the weak sense, associated with the first Green identity and with an extension of the PDE right hand side to H s−2 (Ω) (see [26,Lemma 4.3], [30,Definition 3.1]). Since the extension is non-unique, the co-normal derivative operator appears to be also non-unique and non-linear in u unless a linear relation between u and the PDE right hand side extension is enforced. This creates some difficulties in formulating the boundary-domain integral equations.
These difficulties are addressed in this paper presenting formulation and analysis of direct segregated BDIE systems equivalent to the Dirichlet and Neumann boundary value problems, on Lipschitz domains, for the divergent-type PDE with a non-smooth Hölder-Lipschitz variable scalar coefficient and a general right hand side from H s−2 (Ω), extended when necessary to H s−2 (Ω). This needed a non-trivial generalisation of the third Green identity and its co-normal derivative for such functions, which essentially extends the approach implemented in [3,5,29,6,8] for the right hand side from L 2 (Ω), with smooth coefficient and domain boundary. Equivalence of the BDIEs to the original BVP, BDIE solvability, solution uniqueness/nonuniqueness, as well as Fredholm properties and invertibility of the BDIE operators are analysed in the Sobolev (Bessel potential) spaces. It is shown that the BDIE operators for the Neumann BVP are not invertible, and appropriate finite-dimensional perturbations are constructed leading to invertibility of the perturbed (stabilised) operators. Some preliminary results in this direction, for the infinitely smooth coefficient and domains, were presented in [32].
Employing this definition, Theorem 7.2 from Section 7 can be reformulated as follows.
If a ≡ 1, then A = ∆ and T ± (f ± ; u) become generalised normal derivatives denoted as T ± ∆ (f ± ; u). THEOREM 2.9 (Lemma 4.3 in [26]), Theorem 3.2 in [30], and Theorem 5.3 [31]). Under the hypotheses of Definition 2.8, the generalised co-normal derivatives T ± u(f ± ; u) are independent of (non-unique) choice of the operator γ −1 , we have the estimate 15) and the first Green identity holds in the following form for u ∈ H s (Ω ± ) such that r Ω ± Au = r Ω ±f ± for somẽ As follows from Definition 2.8, the generalized co-normal derivative is nonlinear with respect to u for fixedf ± , but still linear with respect to the couple (f ± , u), i.e., for any complex numbers α 1 and α 2 , (2.17) Let us also define some subspaces of H s (Ω ± ), cf. [15,11,30,31].
DEFINITION 2.10. Let s ∈ R and A * : H s (Ω ± ) → D * (Ω ± ) be a linear operator. For t ∈ R, we introduce the space and the corresponding inner product.
We will mostly use the operators A or ∆ as A * in the above definition. Note that since Au = a∆u+∇a·∇u, (Ω ± ), 1/2 < s < 3/2, we define the canonical co-normal derivatives T ± u ∈ H s− 3 2 (∂Ω) as If a ≡ 1, T ± u becomes canonical normal derivative denoted as T ± ∆ u.
THEOREM 2.13 (Theorem 3.9 in [30] and Theorem 6.6 in [31]). Under the hypotheses of Definition 2.12, the canonical co-normal derivatives T ± u are independent of (non-unique) choice of the operator γ −1 , the operators T ± : H s,− 1 2 (Ω ± ; A) → H s− 3 2 (∂Ω) are continuous, and the first Green identity holds in the following form, The canonical co-normal derivatives in Definition 2.12 are completely defined by the function u and operator A only and do not depend explicitly on the right hand sidesf ± , unlike the generalised co-normal derivatives defined in (2.16), while the operators T ± are linear in u. Note that the canonical co-normal derivatives coincides with the classical co-normal derivatives T ± u = T c± u if the latter do exist, sf. [31, Corollaries 6.11 and 6.14], which is generally not the case for the generalised conormal derivatives even for smooth functions u, unlessf ± =Ã Ω ± u is chosen.

Parametrix and potential type operators on Lipschitz domains
Unless stated otherwise, we will henceforth assume that Ω = Ω + .
We will say, a function P (x, y) of two variables x, y ∈ R n is a parametrix (the Levi function) for the operator A(x, ∂ x ) in R n if (see, e.g., [24,19,37,18,43,42,28]) where δ(·) is the Dirac distribution and R(x, y) possesses a weak (integrable) singularity at x = y, i.e., Let ω n = 2 π n/2 Γ(n/2) denote the area of the unit sphere in R n . It is well known that function is the fundamental solution of the Laplace equation, i.e., ∆ x P ∆ (x, y) = ∆ y P ∆ (x, y) = δ(x − y).
Note that parametrix (3.4) and remainders (3.5), (3.7) are not smooth enough for the corresponding potential operators to be directly treated as in [26], which thus need some additional consideration.

13)
P g = 1 a P ∆ g, R g = − 1 a ∇ · P ∆ (g ∇a), R * g = −∇ · ∇a a P ∆ g . (3.14) Hence Employing relations (3.14) and the well known properties of the operator P ∆ as the pseudo-differential operator of order −2 together with Theorem 2.6, definitions (3.8)-(3.9) can be extended to g ∈ H s (R n ), g ∈ H s (Ω) and lower-smoothness coefficient a. For g ∈ H s (Ω) and g ∈ H s (Ω), the potentials P, R, R * defined on functions (or distributions) having support on Ω are understood as

16)
Pg := r Ω PE Ω g, Rg := r Ω RE Ω g, R * g := r Ω R * EΩ g, To prove mapping properties of the parametrix-based volume potential operators in Sobolev spaces, we first provide some well-known results for the classical Newtonian volume potential associated with the Laplace operator.
Let Ω be a bounded Lipschitz domain in R n . The following operators are continuous (3.20) is such that r Ωf0 = r Ωf , then there exist constants C 0 , C 1 > 0 such that (3.27) , a ∈ C s + (Ω); (3.40) is such that r Ωf0 = r Ω APf , then there exist constants C 0 , C 1 > 0 such that where we have taken into account that ∆P ∆ g = g. The first term in the right hand side of (3.45) belongs to H − 1 2 • (Ω), while, since a ∈ C 3 2 + (Ω), a > 0, the second term belongs to H 1 2 (Ω) and can be extended by zero to H 0 (Ω) ⊂ H − 1 2 (Ω), which completes the proof of continuity for operator (3.30).
Continuity of the operator (3.31), follows from the second relation (3.13) together with Theorem 2.6 and continuity of operator (3.18). Indeed, let us take arbitrary µ ∈ D(R n ), let B µ be a ball such that supp µ ⊂ B µ and let µ 1 ∈ D(R n ) be such that µ 1 = 1 in B µ . Then for any g ∈ H s−1 (R n ), we have, where c i are some positive constants (depending on µ, µ 1 and a), and we took into account that C This implies continuity of (3.31).
Continuity of operator (3.35) is implied by the last relation in (3.13), continuity of operator (3.18) and Theorem 2.6 in the chain of inequalities analogous to (3.46). Similarly, continuity of operator (3.36) is implied by the last relation in (3.14), continuity of operator (3.19) and Theorem 2.6. Continuity of operator (3.37) is implied by continuity of (3.36) since a ∈ C 3 2 + (Ω) implies that there exists ǫ > 0 such that a ∈ C 1,1/2+ǫ (Ω), and we can take σ ∈ ( 3 2 , min{s + 1, 3 2 + ǫ}). The parametrix-based single and the double layer surface potential operators are defined as where the integrals are understood as duality forms if ψ and ϕ are not integrable. Particularly, for ψ ∈ where γ * ψ and T c * ϕ are well defined for any ψ ∈ D * (∂Ω) and for any ϕ ∈ L 1 (∂Ω), a ∈ L ∞ (∂Ω), in the sense of distribution as which evidently implies that supp γ * ψ ⊂ ∂Ω and supp T c * ϕ ⊂ ∂Ω. Moreover, are the continuous operators adjoint, respectively, to the continuous trace operator γ : and to the continuous classical conormal derivative operator T c : H s+1 loc (R n ) → H s− 1 2 (∂Ω); for the continuity of T c and T c * , it is also assumed that a ∈ C s− 1 2 + (∂Ω). When a = 1, formulas (3.49), (3.50) define the corresponding harmonic potentials that we denote as V ∆ and W ∆ , respectively. From definitions (3.49), (3.50), similar to (3.14), we have, cf. [3], We will mainly need the restrictions of the layer potentials to Ω, i.e., r Ω V , r Ω W , but will often omit the restriction operator r Ω if this is clear from the context.
The mapping properties as well as jump relations for the single and double layer potentials are well known for the case a = const. Employing (3.14)-(3.88), they were extended to the case of infinitely smooth boundary and variable coefficient a(x) in [3,5]. Before proving the corresponding properties for the parametrix-based potentials on Lipschitz domains, we collect below the following well-know mapping and jump properties for the harmonic potentials on Lipschitz domains.
Let Ω be a bounded Lipschitz domain in R n .
Let us similarly prove the second jump relation in (3.74). Let ϕ ∈ H s− 1 2 (∂Ω). From (3.52), we obtain, in the sense of distributions, where we have taken into account that ∆P ∆ T c * ϕ = T c * ϕ. Then for any test function φ ∈ D(R n ) we obtain, Hence from the second Green identity (2.23) for v = W ϕ and u = φ, we have, Here we employed that r Ω ± T c * ϕ = 0 since supp T c * ϕ ⊂ ∂Ω. Let us also take into account that γ + φ = Then summing up (3.81) for Ω and Ω − , we obtain which implies the second jump relation in (3.74).
Theorem 3.4(iii) and the first relation in (3.73) imply the following assertion.

The third Green identity and integral relations
We will apply in this section some limiting procedures to obtain the parametrix-based third Green identities.
For some functionsf , Ψ, Φ, let us consider a more general "indirect" integral relation, associated with The following two lemmas extend Lemma 4.1 from [3], where the corresponding assertion was proved for f ∈ L 2 (Ω), s = 1, a ∈ C ∞ (Ω) and the infinitely smooth boundary.
Let us now discuss the trace and two forms of co-normal derivative associated with equation (4.7).
Equation (5.1) is understood in the distributional sense (2.9) and the Dirichlet boundary condition (5.2) in the trace sense. The following uniqueness assertion is well known for s = 1 and follows from the first Green identity; hence it also holds true for 1 ≤ s < 3/2.

BDIE formulations and equivalence to the Dirichlet problem
where ϕ 0 is the known right hand side of the Dirichlet boundary condition (5.2), and ψ ∈ H s− 3 2 (∂Ω) is a new unknown function that will be regarded as formally segregated from u. Thus we will look for the unknown couple (u, ψ) ∈ H s (Ω) × H s− 3 2 (∂Ω).
BDIE system (D2 ∆ ). Let a ∈ C s + (Ω). To obtain a segregated BDIE system of the second kind, we will use equation (4.4) in Ω and equation (4.30) on ∂Ω. Then we arrive at the following BDIE system (D2 ∆ ), where Due to the mapping properties of the operators involved in (5.11) we have F D2∆ ∈ H s (Ω) × H s− 3 2 (∂Ω).
BDIE system (D2). Let the coefficient be smoother than in the first two cases, a ∈ C 3 2 + (Ω). Now we will use equation (4.4) in Ω and equation (4.31) on ∂Ω. Then we arrive at another BDIE system of the second kind, (D2), which is similar to the corresponding system in [32], where

Properties of BDIE system operators for the Dirichlet problem
BDIE systems (D1), (D2 ∆ ) and (D2) can be written as respectively. Here U D := (u, ψ) ⊤ ∈ H s (Ω) × H s− 3 2 (∂Ω), while F D1 , F D2∆ and F D2 are given by (5.5), (5.8) and (5.11), respectively. Note that 20) are continuous due to the mapping properties of the operators constituting them, see Section 3, while for the right hand sides of the BDIE systems we have the following inclusions Proof. The operators continuity has been proved above already.
To prove the Fredholm property of operator (5.18), let us consider the operator As a result of compactness properties of the operators R and γ + R given by (3.33) and (3.40) in Theorem 3.2), the operator D 1 0 is a compact perturbation of operator (5.18). The operator D 1 0 is an upper triangular matrix operator with the following scalar diagonal invertible operators where the invertibility of the operator V is implied by invertibility of operator V ∆ in (7.4) and by the first relation in (3.55). This implies that is an invertible operator. Thus (5.18) is a Fredholm operator with zero index.
The operator is a compact perturbation of operator (5.19). Indeed the operators R : H s (Ω) → H s (Ω) is compact due to 3. This implies that the operator is invertible and hence operator (5.19) is Fredholm with zero index.
Theorems 5.4 and 5.2 imply the following assertion.

Segregated BDIE systems for the Neumann Problem
Let us consider the Neumann Problem: Find a function u ∈ H s (Ω) satisfying equations A u = r Ωf in Ω, (6.1) Equation (6.1) is understood in the distribution sense (2.9), and the Neumann boundary condition (6.2) in the weak sense (2.16). The following assertion is well-known and can be proved, e.g., using the variational settings and the Lax-Milgram lemma.   (Ω), the homogeneous Neumann problem, associated with (6.1)-(6.2), also admits only one linearly independent solution u 0 = 1 in H s (Ω).

BDIE formulations and equivalence to the Neumann problem
We will explore different possibilities of reducing the Neumann problem (6.1)-(6.2) to a BDIE system. Let where ψ 0 is the known right hand side of the Neumann boundary condition (6.2), and ϕ ∈ H s− 1 2 (∂Ω) is a new unknown function that will be regarded as formally segregated from u. Thus we will look for the unknown couple (u, ϕ) ∈ H s (Ω) × H s− 1 2 (∂Ω).
BDIE system (N1). Let the coefficient be smoother than in the previous case, a ∈ C 3 2 + (Ω). Now, using equation (4.4) in Ω and equation (4.31) on ∂Ω, we arrive at the following BDIE system (N1) of two equations for the couple of unknowns, (u, ϕ), which is similar to the corresponding system in [32], in Ω, (6.7) Due to the mapping properties of the operators involved in (6.9) we have F N 1 ∈ H s (Ω) × H s− 3 2 (∂Ω).
BDIE system (N2). Let again a ∈ C s + (Ω). If we use equation (4.4) in Ω and equation (4.29) on ∂Ω, we arrive for the couple (u, ϕ) at the following BDIE system (N2) of two equations of the second kind, which is also similar to the corresponding system in [32], in Ω, (6.10) Due to the mapping properties of the operators involved in (6.12), we have F N 2 ∈ H s (Ω) × H s− 1 2 (∂Ω).  (Ω)), then the couple solves the other two BDE systems, while u solves the Neumann problem (6.1)-(6.2) and γ + u = ϕ.
Proof. The operators continuity is already proved above.
Let us consider operator (6.14). Due to estimate (2.7) and Theorem 7.3, the operator L 0 : H s− 1 2 (∂Ω) → H s− 3 2 (∂Ω) is a Fredholm operator with zero index. Therefore the operator is also Fredholm with zero index. Operator (6.14) is a compact perturbation of N 1 0 since the operators are compact, due to Theorem 3.2, as has been shown in the compactness proof for operator (5.21). Thus operator (6.14) is Fredholm with zero index as well.

Now (6.28) can be written in the form
where is a harmonic function in Ω due to (6.29). The trace of equation (6.35) gives which coincides with (6.27).
Let s = 1 first. Then Theorems 6.1 and 6.3 imply that BDIE system (6.41)-(6.42) and hence (6.39)-(6.40) is solvable if and only if where we took into account that u 0 = 1 in R n . Thus the functional g * 1∆ defined by (6.38) generates the necessary and sufficient solvability condition of equation N 1∆ U = (F 1 , F 2 ) ⊤ . Hence g * 1∆ is a basis of the cokernel of N 1∆ (and thus the kernel of the operator N 1∆ * adjoint to N 1∆ ), for s = 1.
Let us now choose any s ∈ ( 1 2 , 3 2 ). By Theorem 6.4, operator (6.14) and thus its adjoint are Fredholm with zero index. We already proved that at s = 1 the kernel of the adjoint operator is spanned over g * 1∆ .
For any fixed coefficient a ∈ C σ + (Ω), the operator is continuous for any s ′ ∈ ( 1 2 , σ] and particularly for s ′ = s and s ′ = 1. Then Lemma 7.5 implies that the co-kernel of operator (6.44) will be the same for s ′ = s and s ′ = 1 and is spanned over g * 1∆ .

Now (6.49) can be written in the form
which coincides with (6.48).
Let s = 1 first. Then Theorems 6.1 and 6.3 imply that BDIE system (6.62)-(6.63) and hence (6.60)-(6.61) is solvable if and only if where we took into account that u 0 = 1 in R n . Thus the functional g * 1 defined by (6.59) generates the necessary and sufficient solvability condition of equation N 1 U = (F 1 , F 2 ) ⊤ . Hence g * 1 is a basis of the cokernel of N 1 (and thus the kernel of the operator adjoint to N 1 ), for s = 1.
Let now s ∈ ( 1 2 , 3 2 ). By Theorem 6.4, operator (6.15) and thus its adjoint are Fredholm with zero index. We already proved that at s = 1 the kernel of the adjoint operator is spanned over g * 1 . Then Lemma 7.5 implies that the kernel will be the same for any s ∈ ( 1 2 , 3 2 ).
To find the cokernel of operator (6.16), we will need some auxiliary assertions. Lemma 6.10 and Theorem 6.11 were proved in [32, Lemma 6.4 and Theorem 6.5] for the infinitely smooth coefficient a and boundary ∂Ω. Below we only slightly modified them for the non-smooth coefficients and Lipschitz boundary.
Let s = 1 first. Then Theorems 6.1 and 6.3 imply that BDIE system (6.76)-(6.77) is solvable if and only Thus the functional g * 2 defined by (6.73) generates the necessary and sufficient solvability condition of equation N 2 U = (F 1 , F 2 ) ⊤ . Hence g * 2 is a basis of the cokernel of operator (6.16) for s=1.
Let us now choose any s ∈ ( 1 2 , 3 2 ). By Theorem 6.4, operator (6.16) and thus its adjoint are Fredholm with zero index. We already proved that at s = 1 the kernel of the adjoint operator is spanned over g * 2 . For any fixed coefficient a ∈ C σ + (Ω), the operator is continuous for any s ′ ∈ ( 1 2 , σ] and particularly for s ′ = s and s ′ = 1. Then Lemma 7.5 implies that the co-kernel of operator (6.79) will be the same for s ′ = s and s ′ = 1 and is spanned over g * 2 .
This proves that condition (6.3) is sufficient.
For the functional g * 2 given by (6.73) in Theorem 6.12, since the operator V −1 ∆ : H  On the other hand g 0 (U 0 ) = 1. Hence Theorem 7.4 extracted from [27], implies the theorem claims.

Auxiliary assertions
We provide below some auxiliary results used in the main text.
Note that since D(Ω) ⊂ H hypotheses does always exist.
In all cases c is a positive constant independent of g, v or Ω 0 . are Fredholm with zero index.

of equation
Ax = y, (7.12) then the unique solution x of equation is a solution of equation (7.12) such that h * i (x) = 0 (i = 1, ..., k). (7.14) (iii) Vice versa, if x is a solution of equation (7.13) satisfying conditions (7.14), then conditions (7.11) are satisfied for the right-hand side y of equation (7.13) and x is a solution of equation (7.12) with the same right-hand side y.
Note that more results about finite-dimensional operator perturbations are available in [27].