Analysis of segregated boundary-domain

R E S E A R C H Open Access

Analysis of segregated boundary-domain

integral equations for BVPs with non-smooth coefficients on Lipschitz domains

Sergey E. Mikhailov^1*

*Correspondence:

[email protected]

1Department of Mathematics, Brunel University London, Uxbridge, UK

Abstract

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) spaceH^s–2() orH^s–2(),

1

2<s<³₂, 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.

MSC: 35J25; 31B10; 45K05; 45A05

Keywords: Partial differential equations; Non-smooth coefficients; Sobolev spaces;

Parametrix; Integral equations; Equivalence; Lipschitz domain; Invertibility

1 Introduction

Many applications in science and engineering can be modelled by boundary-value prob- lems (BVPs) for partial differential equations with variable coefficients. Reduction of the BVPs with arbitrarily variable coefficients to explicit boundary integral equations is usu- ally 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 [20, 25] as a substi- tute 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. [38, Sect. 18], [43, 44], where the Dirichlet, Neumann, and Robin problems for some PDEs were reduced toindirectBDIEs).

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, 6, 8, 30], the 3D mixed (Dirichlet–Neumann) boundary value problem (BVP) for the stationary diffusion PDEwith infinitely smooth variable coefficient on a domain with an

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infinitely smooth boundary and a square-integrable right-hand sidewas reduced to either segregated or united direct boundary-domain integral or integro-differential equations, some of which coincide with those formulated in [29]. Such BVPs appear, for example, in electrostatics, stationary heat transfer, and other diffusion problems for inhomogeneous media.

For a function from the Sobolev spaceH^s(),¹₂<s<³₂, a classical co-normal derivative in the sense of traces may not exist. However, the generalised co-normal derivative can be defined in the weak sense, associated with the first Green identity and with an extension of the corresponding second-order PDE right-hand side toH^s–2() (see [27, Lemma 4.3], [31, Definition 3.1]). Since the extension is non-unique, the co-normal derivative operator appears to be also non-unique and non-linear inuunless a linear relation betweenuand the PDE right-hand side extension is enforced. This creates some difficulties in formulat- ing the boundary-domain integral equations.

These difficulties are addressed in this paper presenting formulation and analysis of di- rect 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 fromH^s–2(), extended when necessary toH^s–2(). This needed a non-trivial generalisation of the third Green identity and its co-normal derivative for such functions, which essentially extends the ap- proach implemented in [3, 5, 6, 8, 30] for the right-hand side fromL2(), with smooth coefficient and smooth domain boundary. Equivalence of the BDIEs to the original BVP, BDIE solvability, solution uniqueness/non-uniqueness, and the 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 [33].

Note that our analysis is mainly aimed not at the boundary-value problems, the proper- ties of which are well known nowadays, but rather at the BDIE systems per se. The analysis is interesting not only in its own rights but is also to be used further on for analysis of con- vergence and stability of BDIE-based numerical methods for PDEs; see, for example, [16, 29, 34, 35, 46–48, 52, 53].

2 Spaces, co-normal derivatives and boundary value problems

Let=₊be a bounded openn-dimensional region ofRⁿ,n≥3, and let_–=Rⁿ\₊ denote the corresponding exterior domain. For simplicity, we assume that their common boundary∂is a simply connected closed Lipschitz surface. Let₀denote₊,_–orRⁿ. In what follows,D(0) :=C^∞_comp(₀) and D(0) :={r0g:g∈D(Rⁿ)}. Here and fur- ther on, r0 denotes the restriction operator on ₀; we will also use the equivalent notation g|₀ :=r₀g. Further, H^s(0) = H₂^s(0) and H^s(∂) =H₂^s(∂) are the Bessel potential spaces, where s is a real number (see, e.g., [18, 26, 27]). We recall that H^s coincide with the Sobolev–Slobodetski spaces W₂^s for non-negative s. By H^s(0) we denote the closure of D(₀) in H^s(Rⁿ). It is a subspace ofH^s(Rⁿ), and for Lipschitz domains, H^s(0) ={g : g ∈H^s(Rⁿ),suppg ⊂ 0}. By H^s(0) and H_•^s(0) we denote the spaces of restrictions on ₀ of distributions from H^s(Rⁿ) and H^s(0), respec-

tively:

H^s(₀) :=

r0g:g∈H^s Rⁿ

, H_•^s(0) :=r₀H^s(0) :=

r₀g:g∈H^s(0)

⊂H^s(0),

endowed by the corresponding infimum norms and the Hilbert structure defined with the help of orthogonal projections; see [27, p. 77] for H^s(0). Note that the spaceH_•^s(0) coincides with the one denoted as L^ps,z(0) in [41, Eq. (5.2)] and [40, Eq. (2.212)] for p= 2.

Let us introduce the subspaceH_∂^s ₀ :={g:g∈H^s(Rⁿ),suppg⊂∂₀}ofH^s(Rⁿ) (and of H^s(0)). ByH˚^s(0) we denote the closure ofD(0) inH^s(0).

Definition 2.1 LetE˚₀ denote the operator of extension of functionsg∈H^s(0),s≥0, to the wholeRⁿby zero outside₀. By, e.g., [27, Lemma 3.32 and Theorem 3.33] (see also [31, Theorem 2.7]) the operatorE˚₀ :H^s(0)→H^s(0) is continuous if 0≤s< ¹₂, and we extend it also to the range –¹₂ <s< ¹₂ defining it for –¹₂ <s< 0 as (cf. the proof of [31, Theorem 2.16])

E˚0g,v0:=g,E˚0v0, ∀g∈H^s(), ∀v∈H^–s(). (2.1) Remark2.2 Note the following known or easily deduced results:

1. There hold the continuous embeddingsH_•^s(0)→H˚^s(0)→H^s(0); see [42, Eq. (2.123)].

2. H_•^s(0) =H˚^s(0)for anys> 1/2such thats–¹₂ is non-integer; see, e.g., [27, Theorem 3.3].

3. H˚^s(0) =H^s(0)for anys≤1/2; see [31, Theorem 2.12].

4. H_•^s(0) =H˚^s(0) =H^s(0)for anys< 1/2such thats–¹₂ is non-integer; see, e.g., [31, Lemma 2.15].

5. For anys∈R, there evidently exists an extension fromH_•^s(₀)toH^s(₀), and for anys≥–1/2, this extension is unique; see, e.g., [27, Lemma 3.39], [31,

Theorem 2.10(i)].

6. By [31, Theorem 2.16], for anys∈(–1/2, 1/2), the extension from

H_•^s(₀) =H˚^s(₀) =H^s(₀)toH^s(₀)is unique and is given by the operatorE˚₀. Remark2.3 Due to Remark 2.2(5), fors≥–1/2, the space H_•^s(0) is isometrically iso- morphic to the space H^s(0), and sometimes these spaces are identified. Particularly, if g1,g2∈H_•^s(0), then denoting by g˜1,g˜2∈H^s(0) the unique distributions such that gi=r0g˜iin0, we havegiH•^s(0)=˜giH^s(0)and (g1,g2)H•^s(0)= (g˜1,g˜2)H^s(0). Moreover, if s∈(–1/2, 1/2), then by Remark 2.2(6),g˜i=E˚₀gihence implyinggiH_•^s(0)=E˚₀giH^s(0). There is no such isomorphism for s< –1/2 since in such a case the extension from H_•^s(0) toH^s(0) is not unique. However, due to the definition of the spaces, there is still an isometric isomorphism between the spaceH_•^s(0) and the quotient spaceH^s(0)/H_∂^s ₀. Definition of the spaceH_•^s(₀), Remark 2.2, and Remark 2.3 imply the following asser- tion.

Corollary 2.4 The following restriction operators are isomorphisms:

r₀:H^s(₀)→H_•^s(₀), –1

2≤s, (2.2)

r0:H^s(0)→H^s(0) =H_•^s(0), –1 2<s<1

2, (2.3)

r₀:H^s(0)/H_∂^s ₀→H_•^s(0), s< –1

2. (2.4)

The inverse to the operator(2.3)is r^–1₀=E˚0;see Definition2.1.

Definition 2.5 For a non-negative integer m and 0 <θ ≤1, let C^m,θ(0) denote the Hölder–Lipschitz space in the closed domain ₀. Similar to [32, Definition 3.1], g ∈ C+^μ(0) forμ≥0 means that

g∈L∞(0)whenμ= 0;

g∈C^μ–1,1(0)whenμis a positive integer;

g∈C^m,θ+(₀)for some> 0whenμ=m+θ, wheremis a non-negative integer, and 0 <θ< 1.

Employing this definition, Theorem 7.2 from Sect. 7 can be reformulated as follows.

Theorem 2.6 Let₀ be an open set inRⁿ,σ∈R,v∈H^σ(0),and g∈C^|σ+^|(0).Then g is a multiplier in H^σ(0),i.e.,gv∈H^σ(0)for every v∈H^σ(0),and the corresponding norm estimate holds.

Let us denote∂_j:=∂_xj:=∂/∂x_j(j= 1, 2, . . . ,n),∇= (∂₁,∂₂, . . . ,∂_n). Let

0 <amin≤a(x)≤amax<∞ for almost everyx∈_±. (2.5) We consider the scalar elliptic differential equation, which can be written in the following strong form ifuandaare sufficiently smooth:

Au(x) :=A(x,∇)u(x) :=∇ ·

a(x)∇u(x)

=f(x), x∈_±, (2.6)

whereuis an unknown function andf is a given function in_±.

Foru∈H^s(±), 1/2 <s< 3/2, anda∈C^|s–1|+ (±), the partial differential operatorAis understood in the sense of distributions:

Au,v_±:= –E_±(u,v), ∀v∈D(_±), (2.7)

where

E_±(u,v) :=a∇u,∇v _±:=

n i=1

a∂iu,∂_iv _±,

and the duality bracketsg,· _±denote value of a linear functional (distribution)gextend- ing the usualL2dual product. Ifs= 1, then

E_±(u,v) =

_±

a(x)∇u(x)· ∇v(x)dx.

Since the set D(±) is dense inH^2–s(_±), (2.7) defines, due to Theorem 2.6 (see, e.g., [32, Theorem 3.4]), the continuous linear operatorA:H^s(_±)→H^s–2(_±) = [H^2–s(_±)]^∗, where

Au,v_±:= –E_±(u,v), ∀u∈H^s(_±), v∈H^2–s(_±). (2.8) Let us also consider the operator Aˇ_±:H^s(_±)→H^s–2(_±) = [H^2–s(_±)]^∗ (see [31, Eq. (3.5)], [32, Eq. (5.1)]) defined by

ˇA_±u,v _±:= –Eˇ_±(u,v) := –E˚_±(a∇u),∇v

_±

= –E˚_±(a∇u),∇ve

Rⁿ= ∇ ·E˚_±(a∇u),ve

Rⁿ

= ∇ ·E˚_±(a∇u),v

_±, ∀u∈H^s(_±),v∈H^2–s(_±), (2.9) which is evidently continuous. Hereve∈H^2–s(Rⁿ) is such thatr_±ve=v. Evidently, weak definition (2.9) can be also written (in the strong-looking form) as

Aˇ_±u=∇ ·E˚_±r_±[a∇u]. (2.10)

For anyu∈H^s(_±), the functionalAˇ_±ubelongs toH^s–2(_±) and is a specific extension of the functionalAu∈H^s–2(_±); recall that the functionalAu∈H^s–2(_±) is defined on H^2–s(±), whereas the functionalAˇ_±uis defined onH^2–s(±).

Remark2.7 Note also that Definition 2.1 forE˚_±and definition (2.9) imply that ˇA_±u,v _±= –Eˇ_±(u,v) = –Eˇ_±(v,u) =u,Aˇ_±v _±,

∀u∈H^s(_±), v∈H^2–s(_±), 1/2 <s< 3/2.

From the trace theorem (see, e.g., [11, 12, 26, 27]) foru∈H^s(_±), 1/2 <s< 3/2, it follows thatγ^±u∈H^s–12(∂), whereγ^±=γ_∂^± is the trace operator on∂from_±. Ifγ⁺u=γ^–u, then we will sometimes write justγu. Let alsoγ^–1:=γ_r^–1:H^s–12(∂)→H^s(Rⁿ) denote a (non-unique) continuous right inverse to the trace operatorγ, i.e.,γ γ^–1w=wfor any w∈H^s–12(∂). Hence alsoγ^±γ^–1w=wfor anyw∈H^s–12(∂).

Foru∈H^s(±),s>³₂, anda∈C(±), we denote byT^c±the corresponding classical (strong) co-normal derivative operators on∂in the sense of traces:

T^c±u(x) :=a(x)ν(x)·γ^±∇u(x) =a(x)∂νu(x), x∈∂, (2.11) whereν(x) =ν⁺(x) is the outward to₊unit normal vector at the pointx∈∂, and we will sometimes writeT^cu(x) ifT^c+u(x) =T^c–u(x). However, the classical co-normal derivative is, generally, not well defined ifu∈H^s(_±), 1/2 <s< 3/2, (see an example in [33, Ap- pendix A] of a function fromH¹(), where the classical normal derivative does not exist at boundary points).

Inspired by the first Green identity for smooth functions, we can definethe generalised co-normal derivative(cf., e.g., [27, Lemma 4.3]), [31, Definition 3.1], [32, Definition 5.2]).

Definition 2.8 Let 1/2 <s< 3/2,u∈H^s(_±),a∈C^|s–1|+ (_±), andr_±Au=r_±f˜_±for some f˜_±∈H^s–2(_±). Then thegeneralised co-normal derivatives T^±(f˜_±;u)∈H^s–32(∂) are de- fined in the weak form as

±T^±(f˜_±;u),w

∂:= f˜_±,γ^–1w

_±+Eˇ_±

u,γ^–1w

= f˜±–Aˇ_±u,γ^–1w

_±, ∀w∈H^32–s(∂), (2.12) i.e.,

T^±(f˜_±,u) :=± γ^–1∗

(f˜_±–Aˇ_±u). (2.13)

If a≡1, thenA=, andT^±(f˜_±;u) becomegeneralised normal derivativesdenoted as T^±(˜f_±;u).

The operator (γ^–1)^∗:H^–t(Rⁿ)→H^–t+12(∂) is dual toγ^–1:H^t–12(∂)→H^t(Rⁿ) and is defined as(γ^–1)^∗ψ,w ∂:=ψ,γ^–1w Rⁿfor anyw∈H^t–12,ψ∈H^–t(Rⁿ), 1/2 <t< 3/2. In (2.13) it was employed fort= 2 –s.

Theorem 2.9(Lemma 4.3 in [27], Theorem 3.2 in [31], and Theorem 5.3 in [32]) Un- der the hypotheses of Definition2.8,the generalised co-normal derivatives T^±u(f˜_±;u)are independent of(non-unique)choice of the operatorγ^–1,and we have the estimate

T^±(f˜±;u)

H^{s– 32}(∂)≤C1uH^s(±)+C2˜f±H^s–2(_±) (2.14) and the first Green identity in the form

±T^±(f˜_±;u),γ^±v

∂=˜f_±,v _±+Eˇ_±(u,v)

=˜f_±–Aˇ_±u,v_±, ∀v∈H^2–s(_±). (2.15) As follows from Definition 2.8, the generalized co-normal derivative is nonlinear with re- spect toufor fixedf˜±but still linear with respect to the couple (f˜±,u), i.e., for any complex numbersα₁andα₂,

α₁T^±(f˜_1±;u1) +α₂T⁺(f˜_2±;u2) =T^±(α1f˜_1±;α₁u1) +T^±(α2f˜_2±;α₂u2)

=T^±(α1f˜_1±+α2f˜_2±;α1u1+α2u2).

Let us also define some subspaces ofH^s(_±); see [11, 15, 31, 32].

Definition 2.10 Lets∈R, and letA_∗:H^s(_±)→D^∗(_±) be a linear operator. Fort∈R, we introduce the space

H^s,t(_±;A_∗) :=

g:g∈H^s(_±),A_∗g∈H_•^t(_±)

endowed with the normgH^s,t(±;A∗):= (g²_Hs(±)+A∗g²_H•t(_±))^1/2and the correspond- ing inner product.

Definition 2.11 Let₀be either₊ or_–. By Remark 2.3, ifg∈H^s,t(0;A_∗) for some s∈Randt≥–¹₂, then there exists auniquedistributionf˜∈H^t(₀) such thatr₀f˜=A_∗g, and hencef˜=A˜_∗0g, where A˜_∗0 :=r^–1₀A_∗. The operatorA˜_∗0 :H^s,t(0;A_∗)→H^t(0), which is continuous by Corollary 2.4, is called thecanonicalextension of the operator A_∗:H^s,t(0;A_∗)→H_•^t(0), and moreover, if –¹₂<t<¹₂, thenA˜_∗0=E˚₀A_∗.

We will mostly use the operatorsAorasA_∗ in the definition. Note that sinceAu= au+∇a· ∇u, for 1/2 <s< 3/2, we haveH^s,–12(₀;A) =H^s,–12(₀;) ifa∈C

3

+2(₀), with equivalent norms.

Let us now define thecanonicalconormal derivative; see [32, Definition 6.5].

Definition 2.12 Foru∈H^s,–12(±;A) anda∈C^|s–1|+ (±), 1/2 <s< 3/2, we define the canonical co-normal derivatives T^±u∈H^s–32(∂) as

±T^±u,w

∂:= A˜_±u,γ^–1w

_±+Eˇ_±

u,γ^–1w

= A˜_±u–Aˇ_±u,γ^–1w

_±

= γ^–1∗

(A˜_±u–Aˇ_±u),w

∂ ∀w∈H^32–s(∂), (2.16) i.e,

T^±u:=± γ^–1∗

(A˜_±u–Aˇ_±u). (2.17)

Ifa≡1, thenT^±ubecomes thecanonical normal derivativedenoted asT^±u.

Theorem 2.13(Theorem 3.9 in [31] and Theorem 6.6 in [32]) Under the hypotheses of Definition2.12,the canonical co-normal derivatives T^±u are independent of(non-unique) choice of the operatorγ^–1,the operators T^±:H^s,–12(_±;A)→H^s–32(∂)are continuous, and the first Green identity holds in the form

±T^±u,γ^±v

∂= ˜A_±u,v _±+Eˇ_±(u,v)

= ˜A_±u–Aˇ_±u,v _±, ∀v∈H^2–s(±). (2.18) The canonical co-normal derivatives in Definition 2.12 are completely defined by the func- tionuand operatorAonly and do not depend explicitly on the right-hand sidesf˜±, un- like the generalised co-normal derivatives defined in (2.15), whereas the operatorsT^±are linear inu. Note that the canonical co-normal derivatives coincide with the classical co- normal derivativesT^±u=T^c±uif the latter do exist (see [32, Corollaries 6.11 and 6.14]), which is generally not the case for the generalised conormal derivatives even for smooth functionsu, unlessf˜_±=A˜_±uis chosen. Thus the canonical conormal derivative is a con- tinuous extension of the classical conormal derivative.

Let 1/2 <s< 3/2 anda∈C^|s–1|+ (_±). Ifu∈H^s,–12(_±;A), then Definitions 2.8 and 2.12 imply that the generalised co-normal derivative for arbitrary extensionsf˜±∈H^s–2(±) of the distributionsr_±Aucan be expressed as

T^±(f˜_±;u) =T^±u± γ^–1_∗

(f˜_±–A˜_±u). (2.19)