Introduction There is by now a very rich theory of boundary value problems for the Laplace operator, and more generally for second order divergence form operators−divA∇

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

LAYER POTENTIALS FOR GENERAL LINEAR ELLIPTIC SYSTEMS

ARIEL BARTON

Abstract. In this article we construct layer potentials for elliptic differential operators using the Babuˇska-Lax-Milgram theorem, without recourse to the fundamental solution; this allows layer potentials to be constructed in very general settings. We then generalize several well known properties of layer potentials for harmonic and second order equations, in particular the Green’s formula, jump relations, adjoint relations, and Verchota’s equivalence between well-posedness of boundary value problems and invertibility of layer potentials.

1. Introduction

There is by now a very rich theory of boundary value problems for the Laplace operator, and more generally for second order divergence form operators−divA∇.

The Dirichlet problem

−divA∇u= 0 in Ω, u=f on∂Ω, kuk_X≤Ckfk_D and the Neumann problem

−divA∇u= 0 in Ω, ν·A∇u=g on∂Ω, kuk_X≤Ckgk_N

are known to be well-posed for many classes of coefficientsAand domains Ω, and with solutions in many spacesX and boundary data in many boundary spaces D andN.

A great deal of current research consists in extending these well posedness results to more general situations, such as operators of order 2m≥4 (for example, [19, 25, 45, 47, 53, 54]; see also the survey paper [22]), operators with lower order terms (for example, [24, 30, 34, 55, 62]) and operators acting on functions defined on manifolds (for example, [46, 50, 51]).

Two very useful tools in the second order theory are the double and single layer potentials given by

D^Ω_Af(x) = Z

∂Ω

ν·A^∗(y)∇yE^L∗(y, x)f(y)dσ(y), (1.1) S_L^Ωg(x) =

Z

∂Ω

E^L∗(y, x)g(y)dσ(y) (1.2)

2010Mathematics Subject Classification. 35J58, 31B10.

Key words and phrases. Higher order differential equation; layer potentials; Dirichlet problem;

Neumann problem.

c

2017 Texas State University.

Submitted March 27, 2017. Published December 15, 2017.

1

where ν is the unit outward normal to Ω and where E^L(y, x) is the fundamental solution for the operatorL=−divA∇, that is, the formal solution toLE^L(·, x) = δx. These operators are inspired by a formal integration by parts

u(x) = Z

Ω

L^∗E^L∗(·, x)u

=− Z

∂Ω

ν·A^∗∇E^L∗(·, x)u dσ+ Z

∂Ω

E^L∗(·, x)ν·A∇u dσ+ Z

Ω

E^L∗(·, x)Lu which gives the Green’s formula

u(x) =−D^Ω_A(u

_∂Ω)(x) +S_L^Ω(ν·A∇u)(x) ifx∈Ω andLu= 0 in Ω at least for relatively well-behaved solutionsu.

Such potentials have many well known properties beyond the above Green’s formula, including jump and adjoint relations. In particular, by a clever argument of Verchota [63] and some extensions in [21, 23], given certain boundedness and trace results, well posedness of the Dirichlet problem in both Ω and its complement is equivalent to invertibility of the operator g 7→ S_L^Ωg

_∂Ω, and well posedness of the Neumann problem in both domains is equivalent to invertibility of the operator f 7→ν·A∇D_A^Ωf.

This equivalence has been used to solve boundary value problems in many papers, including [29, 32, 33, 63] in the case of harmonic functions (that is, the caseA=I and L=−∆) and [5, 14, 23, 35, 37, 38] in the case of more general second order operators under various assumptions on the coefficientsA. Layer potentials have been used in other ways in [4, 9, 21, 44, 48, 49, 56, 59, 65]. Boundary value problems were studied using a functional calculus approach in [6, 7, 8, 9, 10, 11, 12]; in [58]

it was shown that certain operators arising in this theory coincided with layer potentials.

Thus, it is desirable to extend layer potentials to more general situations. It is possible to proceed as in the homogeneous second order case, by constructing the fundamental solution, formally integrating by parts, and showing that the resulting integral operators have appropriate properties. In the case of higher order operators with constant coefficients, this has been done in [2, 27, 28, 52, 53, 64]. All three steps are somewhat involved in the case of variable coefficient operators (although see [15, 30] for fundamental solutions, for higher order operators without lower order terms, and for second order operators with lower order terms, respectively).

An alternative, more abstract construction is possible. The fundamental solution for various operators was constructed in [15, 30, 36] as the kernel of the Newton potential, which may itself be constructed very simply using the Lax-Milgram the- orem. It is possible to rewrite the formulas (1.1) and (1.2) for second order layer potentials directly in terms of the Newton potential, without mediating by the fun- damental solution, and this construction generalizes very easily. It is this approach that was taken in [18, 20].

In this paper we will provide the details of this construction in a very general context. Roughly, this construction is valid for all differential operatorsLthat may be inverted via the Babuˇska-Lax-Milgram theorem, and all domains Ω for which suitable boundary trace operators exist. We will also show that many properties of traditional layer potentials are valid in the general case.

The organization of this paper is as follows. The goal of this paper is to construct layer potentials associated to an operator L as bounded linear operators from a

spaceD2or N2 to a Hilbert spaceH2 given certain conditions onD2,N2 andH2. In Section 2 we will list these conditions and define our terminology. Because these properties are somewhat abstract, in Section 3 we will give an example of spaces H₂, D₂ and N₂ that satisfy these conditions in the case whereL is a higher order differential operator in divergence form without lower order terms.

This is the context of the paper [19]; we intend to apply the results of the present paper therein to solve the Neumann problem with boundary data inL²for operators with transversally independent self-adjoint coefficients.

In Section 4 of this paper we will provide the details of the construction of layer potentials. We will prove the higher order analogues for the Green’s formula, adjoint relations, and jump relations in Section 5. Finally, in Section 6 we will show that the equivalence between well posedness of boundary value problems and invertibility of layer potentials of [21, 23, 63] extends to the general case.

2. Terminology

We will construct layer potentialsD_B^Ω andS_L^Ωusing the following objects.

• Two Hilbert spacesH₁andH₂.

• Six (quasi)-normed vector spacesHb^Ω₁,Hb^C₁,Hb^Ω₂,Hb^C₂,Db₁ andDb₂.

• Bounded sesquilinear forms B : H₁×H₂ →C, B^Ω : H^Ω₁ ×H^Ω₂ → C, and B^C:H^C₁×H^C₂ →C. (We will define the spaces H^Ω_j,H^C_j momentarily.)

• Bounded linear operators ˙Tr₁:H₁→Db₁ and ˙Tr₂:H₂→Db₂.

• Bounded linear operators (·)

1

Ω : H₁ → Hb^Ω₁ and (·)

2

Ω : H₂ → Hb^Ω₂. When no ambiguity will arise we will suppress the superscript and refer to both operators as

_Ω.

• Bounded linear operators (·)

j

C:Hj→Hb^C_j forj= 1, 2; we again often refer to both operators as

_C.

We will work not with the spacesHb^Ω_j,Hb^C_j andDbj, but with the (normed) vector spacesH^Ω_j,H^C_j andD_j defined as follows.

H^Ω_j ={F

_Ω:F ∈H_j}/∼ with normkfk_HΩ

j = inf{kFk_Hj :F

_Ω=f}, (2.1) H^C_j ={F

_C:F ∈Hj}/∼ with normkfk_HC

j = inf{kFkH_j :F

_C=f}, (2.2) D_j={Tr˙ _jF :F ∈H_j}/∼ with normkf˙kDj = inf{kFkHj : ˙Tr_jF = ˙f} (2.3) where∼denotes the equivalence relationf ∼g ifkf−gk= 0.

Throughout we will impose the following conditions on the given function spaces and operators.

Condition 2.1. B is coercive; that is, there is some λ > 0 such that for every u∈H1 andv∈H2 we have that

sup

w∈H1\{0}

|B(w, v)|

kwkH₁

≥λkvkH₂, sup

w∈H2\{0}

|B(u, w)|

kwkH₂

≥λkukH₁.

Condition 2.2. If u∈H₁ andv∈H₂, then B(u, v) =B^Ω(u

Ω, v

Ω) +B^C(u C, v

C).

Condition 2.3. If ϕ,ψ∈Hj forj= 1 orj= 2, and ifTr˙ jϕ= ˙Trjψ, then there is aw∈Hj such that

w _Ω=ϕ

_Ω, w _C=ψ

_C, and Tr˙ jw= ˙Trjϕ= ˙Trjψ.

We now introduce some further terminology.

If X is a quasi-Banach space, we will let X^∗ be the space of conjugate linear functionals onX.

We define the conjugate linear operator L as follows. Ifu∈H2, letLube the element ofH^∗₁ given by

hϕ, Lui=B(ϕ, u). (2.4) Notice thatLis boundedH2→H^∗₁.

Ifu∈H^Ω₂, we let (Lu)

_Ωbe the element of{ϕ∈H1: ˙Tr1ϕ= 0}^∗ given by hϕ,(Lu)

_Ωi=B^Ω(ϕ

_Ω, u) for allϕ∈H1with ˙Tr1ϕ= 0. (2.5) Ifu∈H2, we will often use (Lu)

_Ωas shorthand for (L(u _Ω))

_Ω. We will primarily be concerned with the case (Lu)

_Ω= 0.

We will let

N2=D^∗₁, N1=D^∗₂ (2.6)

denote the spaces of conjugate linear functionals onD1andD2. We will now define the Neumann boundary values of an elementuof H^Ω₂ that satisfies (Lu)

_Ω= 0. If Tr˙ 1ϕ= ˙Tr1ψand (Lu)

_Ω= 0, thenB^Ω(ϕ _Ω−ψ

_Ω, u) = 0 by definition of (Lu) _Ω. Thus,B^Ω(ϕ

_Ω, u) depends only on ˙Tr1ϕ, not onϕ. Thus, ˙M_BΩudefined as follows is a well defined element ofN2.

hTr˙ 1ϕ,M˙ _BΩui=B^Ω(ϕ

_Ω, u) for allϕ∈H1. (2.7) We can compute

|hf˙,M˙ _BΩui| ≤ kB^Ωkinf{kϕk_H1 : ˙Tr₁ϕ= ˙f}kuk_HΩ

2 =kB^Ωkkf˙k_D1kuk_HΩ 2

and so we have the boundkM˙ _BΩukN₂ ≤ kB^Ωk kuk_HΩ 2. If (Lu)

_Ω 6= 0, then the conjugate linear operator given by ϕ 7→ B^Ω(ϕ _Ω, u) is still of interest. We will denote this operator L_BΩu; that is, if u ∈ H^Ω₂, then L_BΩu∈H^∗₁is defined by

hϕ, L_BΩui=B^Ω(ϕ

_Ω, u) for allϕ∈H1. (2.8) Ifu∈H2then as before we will use L_BΩuas a shorthand forL_BΩ(u

_Ω).

Remark 2.4. We observe that, for a given sesquilinear formB defined on H₁× H₂, there are often many choices of forms B^Ωand B^C that satisfy Condition 2.2.

Conversely, for a given formB^Ωthere may be many formsB^Csuch that the operator Bgiven by Condition 2.2 satisfies Condition 2.1. See Remark 3.1 for an example.

The operatorL depends only onB, and not on a particular choice of B^Ω and B^C. By contrast, the quantities ˙M_BΩuand L_BΩudepend on B^Ω and not onB (that is, not onB^C).

We also comment on the quantity (Lu)

_Ω. If u∈H^Ω₂, then by definition of H^Ω₂ there is some U ∈H₂ withu=U

Ω. If ˙Tr₁ϕ= 0 = ˙Tr₁0, then by Condition 2.3

there is somew∈H1 withw _Ω=ϕ

_Ωandw _C= 0

_C= 0. Thus, by the definition (2.5) of (Lu)

_Ωand Condition 2.2, hϕ,(Lu)

_Ωi=B^Ω(ϕ

_Ω, u) =B(w, U)−B^C(w _C, U

_C) =B(w, U) and so (Lu)

_Ωmay be viewed as depending either onBor onB^Ω. 3. An example: higher order differential equations

In this section, we provide an example of a situation in which the terminology of Section 2 and the construction and properties of layer potentials of Sections 4 and 5 may be applied. We remark that this is the situation of [19], and that we will therein apply the results of this paper.

Letm≥1 be an integer, and letLbe an elliptic differential operator of the form Lu= (−1)^m X

|α|=|β|=m

∂^α(Aαβ∂^βu) (3.1)

for some (possibly complex) bounded measurable coefficientsAdefined onR^d. Here αandβ are multiindices inN^d0, whereN0 denotes the nonnegative integers. As is standard in the theory, we say thatLu= 0 in an open set Ω in the weak sense if

Z

Ω

X

|α|=|β|=m

∂^αϕA_αβ∂^βu= 0 for allϕ∈C₀^∞(Ω). (3.2) We impose the following ellipticity condition: we require that for someλ >0,

< X

|α|=|β|=m

Z

R^d

∂^αϕ Aαβ∂^βϕ≥λk∇^mϕk²_L2(R^d) for allϕ∈W˙_m²(R^d). (3.3)

Let Ω⊂R^d be a Lipschitz domain, and letC=R^d\Ω denote the interior of its complement. Observe that∂Ω =∂C.

The following function spaces and linear operators satisfy the conditions of Sec- tion 2.

•H1=H2=His the homogeneous Sobolev space ˙W_m²(R^d) of locally integrable functions ϕ(or rather, of equivalence classes of functions modulo polynomials of degreem−1) with weak derivatives of order m, and such that the H-norm given bykϕkH=k∇^mϕk_L2(R^d)is finite. This space is a Hilbert space with inner product hϕ, ψi=P

|α|=m

R

R^d∂^αϕ ∂^αψ.

• Hb^Ω andHb^C are the Sobolev spacesHb^Ω= ˙W_m²(Ω) ={ϕ:∇^mϕ∈L²(Ω)} and Hb^C= ˙W_m²(C) ={ϕ:∇^mϕ∈L²(C)} with the expected norms.

•Db denotes the (vector-valued) Besov space ˙B^2,2_1/2(∂Ω) of locally integrable func- tions modulo constants with norm

kfkB˙^2,2_1/2(∂Ω)= Z

∂Ω

Z

∂Ω

|f(x)−f(y)|²

|x−y|^d dσ(x)dσ(y) 1/2

.

•In [17, 18, 19], Ω is assumed to have connected boundary, and ˙Tris the linear operator defined onHby

Tr˙ u= Tr^Ω∇^m−1u

_Ω={Tr^Ω∂^γu}_|γ|=m−1,

where Tr^Ωis the standard boundary trace operator of Sobolev spaces.

Given a suitable modification of the trace spaceD, it is also possible to chooseb Tr˙ u={Tr^Ω∂^γu}_|γ|≤m−1 or Tr˙ u= (Tr^Ωu, ∂_νu, . . . , ∂^m−1_ν u),

where ν is the unit outward normal, so that the boundary derivatives of uof all orders are recorded. See, for example, [3, 47, 53, 57, 60]. In this case,∂Ω need not be connected.

•B is the sesquilinear form onH×Hgiven by B(ψ, ϕ) = X

|α|=|β|=m

Z

R^d

∂^αψA_αβ∂^βϕ. (3.4)

The sesquilinear forms B^Ω and B^C are defined analogously to B, but with the integral overR^d replaced by an integral over Ω orC.

We now discuss the conditions imposed in Section 2. The formsB,B^ΩandB^C are clearly bounded and sesquilinear, and the restriction operators

_Ω: H→ Hb^Ω,

_C:H→Hb^Care bounded and linear.

The trace operator ˙Tris linear. If Ω =R^d+ is the half-space, then boundedness of ˙Tr: H→ D was established in [39, Section 5]; this extends to the case where Ω is the domain above a Lipschitz graph via a change of variables. If Ω is a bounded Lipschitz domain, then boundedness of ˙Tr : W → D, whereb W is the inhomogeneous Sobolev space with normPm

k=0k∇^kϕk_L2(R^d), was established in [42, Chapter V]. Then boundedness of ˙Tr:H→Db follows by the Poincar´e inequality.

By assumption, Condition 2.1 is valid. Because Ω is a Lipschitz domain, we have that∂Ω has Lebesgue measure zero, and so Condition 2.2 is valid. A straightforward density argument shows that if ˙Tris bounded, then Condition 2.3 is valid.

Thus, the given spaces and operators satisfy the conditions imposed at the be- ginning of Section 2.

We now comment on a few of the other quantities defined in Section 2. If u ∈ H, and if Lu = 0 in Ω in the weak sense of formula (3.2), then by density B^Ω(ϕ

_Ω, u) = 0 for allϕ∈Hwith ˙Trϕ= 0; that is, (Lu)

_Ωas defined in Section 2 satisfies (Lu)

_Ω= 0.

Ifu∈H^Ω, then formally

L_BΩu= (−1)^m X

|α|=|β|=m

∂^α(A_αβE^Ω(∂^βu))

whereE^Ω denotes extension from Ω toR^d by zero.

If m = 1, then by an integration by parts argument we have that ˙M_BΩu = ν·A∇u, whereνis the unit outward normal to Ω, wheneveruis sufficiently smooth.

The weak formulation of Neumann boundary values of formula (2.7) coincides with the formulation of higher order Neumann boundary data of [17, 18, 19] if ˙Tr = Tr^Ω∇^m−1, with that of [3, 64] if ˙Tru= (Tr^Ωu, ∂νu, . . . , ∂^m−1_ν u), and with [28, 52, 53] if ˙Tru={Tr^Ω∂^γu}_|γ|≤m−1.

Remark 3.1. Each of the sesquilinear forms B and B^Ω may be associated with more than one choice of coefficientsAαβ.

For example, let Abαβ satisfy Abαβ(x) = Aαβ(x) for all x ∈ Ω. Then B^Ω is unchanged ifA_αβ is replaced byAb_αβ, butB is not.

Conversely, letAeαβ=Aαβ+Mαβ, whereMαβis a constant that satisfiesMαβ=

−Mβα. A straightforward integration by parts argument shows thatB (and thus L) is unchanged ifAαβis replaced byAeαβ. However, the operatorsB^ΩandB^Cdo take different values ifA_αβ is replaced byAe_αβ.

Thus, as mentioned in Remark 2.4, B may be associated with more than one form B^Ω, and B^Ω may be associated with more than one form B, that satisfy Condition 2.2.

For many classes of domains there is a bounded extension operator from Hb^Ω to H, and soH^Ω=Hb^Ω= ˙W_m²(Ω) with equivalent norms. (If Ω is a Lipschitz domain then this is a well known result of Calder´on [26] and Stein [61, Theorem 5, p. 181];

the result is true for more general domains, see for example [41].)

As mentioned above, if Ω ⊂ R^d is a Lipschitz domain, then ˙Tr is a bounded operatorH→D.b

If ˙Tru= Tr^Ω∇^m−1u, as in [17, 18, 19], then ˙Tr has a bounded right inverse.

See [16]. If Tr˙ u = (Tr^Ωu, ∂_νu, . . . , ∂_ν^m−1u) or Tr˙ u = {Tr^Ω∂^γu}_|γ|≤m−1, as in [3, 47, 53, 57, 60], and if Ω is bounded, then Tr˙ has a bounded right inverse even if ∂Ω is not connected; see [42] or [47, Proposition 7.3]. Thus, in either of these cases, the norm in D is comparable to the Besov norm. Furthermore, {∇^m−1ϕ

_∂Ω :ϕ∈ C₀^∞(R^d)} or {(Tr^Ωϕ, ∂νϕ, . . . , ∂^m−1_ν ϕ) :ϕ∈ C₀^∞(R^d)} is dense inD. Thus, ifm= 1 thenD=Db = ˙B^2,2_1/2(∂Ω). Ifm≥2 then Dis a closed proper subspace ofD, as the different partial derivatives of a common function must satisfyb certain compatibility conditions. In this caseDis the Whitney-Sobolev space used in many papers, including [1, 17, 25, 47, 52, 53, 54].

4. Construction of layer potentials

We will now use the Babuˇska-Lax-Milgram theorem to construct layer potentials.

This theorem may be stated as follows.

Theorem 4.1 ([13, Theorem 2.1]). Let H1 and H2 be two Hilbert spaces, and let B be a bounded sesquilinear form on H1×H2 that is coercive in the sense that Condition 2.1 is valid.

Then for every linear functional T defined on H1 there is a unique uT ∈ H2

such that B(v, u_T) =T(v). Furthermore, ku_Tk_H2 ≤ _λ¹kTk_H1→C, where λ is as in Condition 2.1.

We construct layer potentials as follows. Let ˙g∈N₂. Then the operatorTg˙ϕ= hg,˙ Tr˙ 1ϕi=hTr˙ 1ϕ,gi˙ is a bounded linear operator on H1. By the Babuˇska-Lax- Milgram theorem, there is a uniqueuT =S_L^Ωg˙ ∈H2 such that

B(ϕ,S_L^Ωg) =˙ hTr˙ ₁ϕ,gi˙ for allϕ∈H₁. (4.1) We will letS_L^Ωg˙ denote the single layer potential of ˙g. Observe that the dependence ofS_L^Ωon the parameter Ω consists entirely of the dependence of the trace operator on Ω, and the connection between Tr˙ 1 and Ω is given by Condition 2.3. This condition is symmetric about an interchange of Ω andC, and so

S_L^Ωg˙ =S_L^Cg.˙ (4.2)

The double layer potential is somewhat more involved. We begin by defining the Newton potential.

LetH be an element of H^∗₁. By the Babuˇska-Lax-Milgram theorem, there is a unique elementN^LH ofH2that satisfies

B(ϕ,N^LH) =hϕ, Hi for allϕ∈H1. (4.3) We refer toN^Las the Newton potential. In some applications, it is easier to work with the Newton potential rather than the single layer potential directly; we remark that

S_L^Ωg˙ =N^L(Tg˙) where hϕ, Tg˙i=hTr˙ 1ϕ,gi.˙ (4.4) We now return to the double layer potential. Let ˙f ∈D2. Then there is some F ∈H2such that ˙Tr2F = ˙f. Let

D^Ω_Bf˙=D_L,B^Ω Ωf˙ =−F

_Ω+ (N^L(L_BΩF))

_Ω if ˙Tr₂F = ˙f. (4.5) Notice thatD^Ω_Bf˙is an element ofH^Ω₂, not ofH₂. Further observe that the single layer potentialS_L^Ωdepends only on ˙Tr1andB(equivalently on ˙Tr1and the operatorL), and not on the particular choice ofB^Ω. The double layer potentialD_B^Ω =D^Ω_L,BΩ, by contrast, depends on bothL(orB) andB^Ω.

We conclude this section by showing thatD_B^Ωf˙ is well defined, that is, does not depend on the choice ofF in formula (4.5). We also establish that layer potentials are bounded operators.

Lemma 4.2. The double layer potential is well defined. Furthermore, we have the bounds

kD^Ω_Bf˙k_HΩ

2 ≤kB^Ck

λ kf˙k_D2, kD^C_Bf˙k_HC

2 ≤ kB^Ωk

λ kf˙k_D2, kS_L^Ωgk˙ _H2 ≤ 1 λkgk˙ _N2. Proof. By Theorem 4.1, we have

kS_L^Ωgk˙ H2≤ 1

λkTg˙kH1→C≤ 1

λkTr˙ ₁kH1→D1kgk˙ D1→C.

By definition of D₁ and N₂, kTr˙ ₁k_H1→D1 = 1 and kgk˙ _D1→C = kgk˙ _N2, and so S_L^Ω:N₂→H₂ is bounded with operator norm at most 1/λ.

We now turn to the double layer potential. We will begin with a few properties of the Newton potential. By definition ofL, ifϕ∈H1 thenhϕ, LFi=B(ϕ, F). By definition ofN^L,B(ϕ,N^L(LF)) =hϕ, LFi. Thus, by coercivity ofB,

F =N^L(LF) for allF ∈H2. (4.6) By definition ofB^Ω, B^CandL_BΩF,

hϕ, LFi=B(ϕ, F) =B^Ω(ϕ _Ω, F

_Ω) +B^C(ϕ _C, F

_C) =hϕ, L_BΩFi+hϕ, L_BCFi for allϕ∈H1. Thus,LF =L_BΩF+L_BCF and so

−F+N^L(L_BΩF) =−F+N^L(LF)− N^L(L_BCF) =−N^L(L_BCF). (4.7) In particular, suppose that ˙f = ˙Tr2F = ˙Tr2F⁰. By Condition 2.3, there is some w∈H2 such thatw

_Ω=F

_Ωandw _C=F⁰

_C. Then

−F

_Ω+ (N^L(L_BΩF))

_Ω=−w

_Ω+ (N^L(L_BΩw)) _Ω

=−(N^L(L_BCw))

_Ω=−(N^L(L_BCF⁰)) _Ω

=−F⁰

_Ω+ (N^L(L_BΩF⁰)) _Ω

and soD^Ω_Bf˙is well-defined, that is, depends only on ˙f and not the choice of function F with ˙Tr₂F= ˙f.

Furthermore, we have the alternative formula D^Ω_Bf˙=−(N^L(L_BCF))

Ω if ˙Tr₂F = ˙f. (4.8) Thus,

kD^Ω_Bf˙k_HΩ

2 ≤ inf

Tr˙ ₂F= ˙f

k(N^L(L_BCF)) _Ωk_HΩ

2 ≤ inf

Tr˙ ₂F= ˙f

kN^L(L_BCF)kH₂

by definition of theH^Ω₂-norm.

By Theorem 4.1 and the definition ofN^L, we have that kN^L(L_BCF)kH₂ ≤ 1

λkL_BCFkH₁→C. SinceL_BCF(ϕ) =B^C(ϕ

_C, F

_C), we have that kL_BCFkH1→C≤ kB^CkkF

_Ck_HC

2 ≤ kB^CkkFkH2

and so

kD^Ω_Bf˙k_HΩ

2 ≤ inf

Tr˙ ₂F= ˙f

1

λkB^CkkFkH₂ = 1

λkB^Ckkf˙kD₂

as desired.

5. Properties of layer potentials

We will begin this section by showing that layer potentials are solutions to the equation (Lu)

_Ω = 0 (Lemma 5.1). We will then prove the Green’s formula (Lemma 5.2), the adjoint formulas for layer potentials (Lemma 5.3), and conclude this section by proving the jump relations for layer potentials (Lemma 5.4).

Lemma 5.1. Let f˙ ∈ D2, g˙ ∈ N2, and let u = D^Ω_Bf˙ or u = S_L^Ωg˙

_Ω. Then (Lu)

_Ω= 0.

Proof. Recall that (Lu)

_Ω= 0 ifB^Ω(ϕ+

_Ω, u) = 0 for allϕ+∈H1with ˙Tr1ϕ+= 0.

If ˙Tr₁ϕ₊= 0 = ˙Tr₁0, then by Condition 2.3 there is someϕ∈H₁withϕ

_Ω=ϕ₊, ϕ

C= 0 and ˙Tr₁ϕ= 0.

By definition (4.1) of the single layer potential, 0 =B(ϕ,S^Lg) =˙ B^Ω(ϕ

_Ω,S_L^Ωg˙

_Ω) +B^C(ϕ _C,S_L^Ωg˙

_C) =B^Ω(ϕ_{+ Ω},S_L^Ωg˙

_Ω) as desired.

Turning to the double layer potential, ifϕ∈H1, then by definition (4.5) ofD_B^Ω, formula (4.8) forD^C_Band linearity ofB^Ω,

B^Ω(ϕ

_Ω,D_B^Ωf) =˙ −B^Ω ϕ _Ω, F

_Ω

+B^Ω ϕ

_Ω,(N^L(L_BΩF)) _Ω

, B^C(ϕ

_C,D_B^Cf) =˙ −B^C ϕ

_C,(N^L(L_BΩF)) _C

. Subtracting and applying Condition 2.2,

B^Ω(ϕ

_Ω,D^Ω_Bf˙)−B^C(ϕ C,D^C_Bf˙

C) =−B^Ω ϕ _Ω, F

_Ω

+B ϕ,N^L(L_BΩF) . By definition (4.3) ofN^L,

B ϕ,N^L(L_BΩF)

=hϕ, L_BΩFi and by the definition (2.8) ofL_BΩF,

B ϕ,N^L(L_BΩF)

=B^Ω(ϕ Ω, F

Ω).

Thus,

B^Ω(ϕ

_Ω,D^Ω_Bf˙)−B^C(ϕ

_C,D^C_Bf˙) = 0 for allϕ∈H1. (5.1) In particular, as before if Tr˙ ₁ϕ₊ = 0 then there is some ϕ with ϕ

_Ω = ϕ_{+ Ω}, ϕ

_C= 0 and soB^Ω(ϕ

_Ω,D_B^Ωf˙) = 0. This completes the proof.

Lemma 5.2. If u∈H^Ω₂ and(Lu)

_Ω= 0, then u=−D^Ω_B( ˙Tr2U) +S_L^Ω( ˙M_BΩu)

_Ω, 0 =D_B^C( ˙Tr2U) +S_L^C( ˙M_BΩu) _C

for any U ∈H2 with U _Ω=u.

Proof. By definition (4.5) of the double layer potential,

−D_B^Ω( ˙Tr2U) =U

_Ω−(N^L(L_BΩU))

_Ω=u−(N^L(L_BΩu)) _Ω and by formula (4.8)

D^C_B( ˙Tr2U) =−(N^L(L_BΩu)) _C. It suffices to show thatN^L(L_BΩu) =S_L^Ω( ˙M_BΩu).

Letϕ∈H₁. By formulas (4.1) and (2.7),

B(ϕ,S_L^Ω( ˙M_BΩu)) =hTr˙ 1ϕ,M˙ _BΩui=B^Ω(ϕ _Ω, u).

By formula (4.3) for the Newton potential and by the definition (2.8) ofL_BΩu, B(ϕ,N^L(L_BΩu)) =hϕ, L_BΩui=B^Ω(ϕ

_Ω, u).

Thus, B(ϕ,N^L(L_BΩu)) = B(ϕ,S_L^Ω( ˙M_BΩu)) for all ϕ∈ H1; by coercivity of B, we must have thatN^L(L_BΩu) =S_L^Ω( ˙M_BΩu). This completes the proof.

LetB^∗(ϕ, ψ) =B(ψ, ϕ) and defineB^Ω_∗,B^C_∗ analogously. ThenB^∗is a bounded and coercive operator H₂×H₁ → C, and so we can define the double and single layer potentialsD^Ω_B∗:D1→H^Ω₁,S_L^Ω∗:N1→H1.

We then have the following adjoint relations.

Lemma 5.3. We have the adjoint relations hϕ,˙ M˙ _BΩD^Ω_Bf˙i=hM˙ _BΩ

∗ D_B^Ω∗ϕ,˙ fi,˙ (5.2) hγ,˙ Tr˙ 2S_L^Ωgi˙ =hTr˙ 1S_L^Ω∗γ,˙ gi˙ (5.3) for allf˙∈D2,ϕ˙ ∈D1,g˙ ∈N2 andγ˙ ∈N1.

If we letTr˙ ^Ω₂ D^Ω_Bf˙ =−Tr˙ 2F+ ˙Tr2N^L(L_BΩF))for anyF ∈H2withTr˙ 2F = ˙f, thenTr˙ ^Ω₂ D_B^Ωf˙ does not depend on the choice ofF, and we have the duality relations

hγ,˙ Tr˙ ^Ω₂ D^Ω_Bf˙i=h−γ˙ + ˙M_BΩ

∗ S_L^Ω∗γ,˙ f˙i. (5.4) Proof. By formula (4.1),

hTr˙ 1S_L^Ω∗γ,˙ gi˙ =B(S_L^Ω∗γ,˙ S_L^Ωgi,˙ hTr˙ 2S_L^Ωg,˙ γi˙ =B^∗(S_L^Ωg,˙ S_L^Ω∗γi˙ and so formula (5.3) follows by definition ofB^∗.

Let Φ ∈H1 and F ∈H2 with ˙Tr1Φ = ˙ϕ, ˙Tr2F = ˙f. Then by formulas (2.7) and (4.5),

hϕ,˙ M˙ _BΩD^Ω_Bf˙i=B^Ω(Φ

Ω,D_B^Ωf˙) =−B^Ω(Φ Ω, F

Ω) +B^Ω(Φ

Ω,(N^L(L_BΩF)) Ω)

and so

hϕ,˙ M˙ _BΩD^Ω_Bf˙i=−B^Ω_∗(F _Ω,Φ

_Ω) +B^Ω_∗((N^L(L_BΩF)) _Ω,Φ

_Ω).

By formula (2.8),

B^Ω_∗((N^L(L_BΩF)) _Ω,Φ

_Ω) =hN^L(L_BΩF), L_BΩ

∗Φi.

By formula (4.3), if H ∈ H^∗₁ and ϕ ∈ H2 then B^∗(ϕ,N^L∗H) = hϕ, Hi. Letting ϕ=N^L(L_BΩF) andH =L_BΩ

∗Φ, we see that B^Ω_∗((N^L(L_BΩF))

_Ω,Φ

_Ω) =B^∗ N^L(L_BΩF),N^L∗(L_BΩ

∗Φ) . Therefore,

hϕ,˙ M˙ _BΩD_B^Ωfi˙ =−B^Ω_∗(F _Ω,Φ

_Ω) +B^∗(N^L(L_BΩF),N^L∗(L_BΩ

∗Φ)).

By the same argument hf˙,M˙ _BΩ

∗D_B^Ω∗ϕi˙ =−B^Ω(Φ _Ω, F

_Ω) +B(N^L∗(L_BΩ

∗Φ),N^L(L_BΩF)) and by definition ofB^∗ andB^Ω_∗ formula (5.2) is proven.

Finally, by definition of ˙Tr^Ω₂ D^Ω_B,

hγ,˙ Tr˙ ^Ω₂ D^Ω_Bf˙i=−hγ,˙ Tr˙ 2Fi+hγ,˙ Tr˙ 2N^L(L_BΩF))i.

By the definition (4.1) of the single layer potential,

hγ,˙ Tr˙ 2N^L(L_BΩF)i=B^∗(N^L(L_BΩF),S_L^Ω∗γ).˙ By definition ofB^∗ and the definition (4.3) of the Newton potential,

B^∗(N^L(L_BΩF),S_L^Ω∗γ) =˙ hS_L^Ω∗γ, L˙ _BΩFi and by the definition (2.8) ofL_BΩF,

hS_L^Ω∗γ, L˙ _BΩFi=B^Ω(S_L^Ω∗γ˙ _Ω, F

_Ω).

By the definition (2.7) of Neumann boundary values, B^Ω_∗(F

Ω,S_L^Ω∗γ˙

Ω) =hTr˙ ₂F,M˙ _BΩ

∗(S_L^Ω∗γ˙ Ω)i and so

hγ,˙ Tr˙ ^Ω₂D^Ω_Bf˙i=−hγ,˙ f˙i+hM˙ _BΩ

∗(S_L^Ω∗γ˙ Ω),fi˙

for any choice ofF. Thus ˙Tr^Ω₂ D^Ω_Bis well-defined and formula (5.4) is valid.

We conclude this section with the jump relations for layer potentials.

Lemma 5.4. Let Tr˙ ^Ω₂ D_B^Ω be as in Lemma 5.3. If f˙ ∈ D₂ and g˙ ∈ N₂, then we have the jump and continuity relations

Tr˙ ^Ω₂D^Ω_Bf˙+ ˙Tr^C₂D_B^Cf˙ =−f˙, (5.5) M˙ _BΩ(S_L^Ωg˙

_Ω) + ˙M_BC(S_L^Cg˙

_C) = ˙g, (5.6)

M˙ _BΩ(D_B^Ωf)˙ −M˙ _BC(D_B^Cf) = 0.˙ (5.7) If there are bounded operators Tr˙ ^Ω₂ : H^Ω₂ → D₂ and Tr˙ ^C₂ : H^C₂ → D₂ such that Tr˙ ₂F = ˙Tr^Ω₂(F

_Ω) = ˙Tr^C₂(F

_C) for allF ∈H₂, then in addition Tr˙ ^Ω₂(S_L^Ωg˙

_Ω)−Tr˙ ^C₂(S_L^Cg˙

_C) = 0. (5.8)

In the absence of an operator ˙Tr^Ω₂, the continuity relation

Tr˙ 2S_L^Ωg˙ −Tr˙ 2S_L^Cg˙ = 0 (5.9) is valid (and follows immediately from formula (4.2)). Existence of the operator Tr˙ ^Ω₂ is equivalent to the condition that ˙Tr2u= 0 wheneveru

_Ω= 0. This condition is natural if Ω⊂R^d is an open set,C=R^d\Ω and ˙Tr2 denotes a trace operator restricting functions to the boundary∂Ω. Observe that if such operators ˙Tr^Ω₂ and Tr˙ ^C₂ exist, then by the definition (4.5) of the double layer potential and by the definition of ˙Tr^Ω₂D^Ω_B in Lemma 5.3, ˙Tr^Ω₂(D^Ω_Bf˙) = ( ˙Tr^Ω₂ D_B^Ω) ˙f and so there is no ambiguity of notation.

Proof of Lemma 5.4. The continuity relation (5.8) follows from formula (4.2) be- causeS_L^Ωg˙ ∈H2 and by the definition of ˙Tr^Ω₂, ˙Tr^C₂.

The jump relation (5.5) follows from the definition of Tr˙ ^Ω₂ D_B^Ω and by using formula (4.7) to rewrite ˙Tr^C₂D_B^C.

We observe that by the definition (2.7) of Neumann boundary values and the definitions (2.3) and (2.6) ofD₁ and N₂, ifu∈H^Ω₂ and v ∈H^C₂ with (Lu)

Ω = 0 and (Lv)

_C= 0, then

M˙ _BΩu+ ˙M_BCv= ˙ψ if and only ifhTr˙ ₁ϕ,ψi˙ =B^Ω(ϕ

_Ω, u) +B^C(ϕ _C, v) for allϕ∈H1.

Therefore, the continuity relation (5.7) follows from formula (5.1), and the jump relation (5.6) follows from formula (4.2) and from the definition (4.1) of the single

layer potential.

6. Layer potentials and boundary value problems

We now discuss boundary value problems. We routinely wish to establish exis- tence and uniqueness of solutions to the Dirichlet problem

(bLu)

_Ω= 0, Trc^Ω_Xu= ˙f, kukX^Ω ≤Ckf˙kDX, and the Neumann problem

(bLu)

_Ω= 0, Mc^Ω_Xu= ˙g, kuk_XΩ ≤Ckgk˙ N_X

for some constantC and some solution spaceXand spaces of Dirichlet and Neu- mann boundary data DX andNX. For example, ifLb is a second-order differential operator, then as in [31, 40, 43, 44] we might wish to establish well-posedness with DX = ˙W₁^p(∂Ω), NX =L^p(∂Ω) and X^Ω={u:N(∇u)e ∈L^p(∂Ω)}, where Ne is the modified nontangential maximal function introduced in [43].

If X^Ω = H^Ω₂, DX = D2 and NX = N2, then under some modest additional assumptions, a brief and fairly standard argument involving the Babuˇska-Lax- Milgram theorem yields well posedness. We will provide these arguments in Section 6.1.

In more general spaces, the method of layer potentials states that if layer poten- tials, originally defined as bounded operatorsD^Ω_B :D₂ →H₂ and S_L^Ω :N₂ →H₂, may be extended to operatorsDb^Ω:DX→Xand Sb^Ω: NX→X, and if certain of the properties of layer potentials of Section 5 are preserved by that extension, then well posedness of boundary value problems are equivalent to certain invertibility properties of layer potentials.

In Sections 6.2 and 6.3 we will make this notion precise.

As in Sections 2, 4 and 5, we will work with layer potentials and function spaces in a very abstract setting.

6.1. Boundary value problems via the Babuˇska-Lax-Milgram theorem.

Consider the Dirichlet problem of finding au∈H₂ that satisfies (Lu)

_Ω= 0, Tr˙ 2u= ˙f, kukH₂ ≤Ckf˙kD₂ (6.1) or the Neumann problem of finding au∈H^Ω₂ that satisfies

(Lu)

_Ω= 0, M˙ _BΩu= ˙g, kuk_HΩ

2 ≤Ckgk˙ _N2. (6.2) Under some modest additional assumptions on the operatorsLandB^Ω, a stan- dard argument involving Theorem 4.1 yields unique solvability of these problems.

Lemma 6.1. Let ˚H_j ={ϕ∈H_j: ˙Tr_jϕ= 0}. Suppose that there is aλ⁰>0 such that

sup

w∈˚H1\{0}

|B(w, v)|

kwk_H1 ≥λ⁰kvkH₂, sup

w∈˚H2\{0}

|B(u, w)|

kwk_H2 ≥λ⁰kukH₁ (6.3) for all u∈˚H₁ andv∈˚H₂. Then there is aC such that, for each f˙ ∈D₂, there is a function u∈H₂ such that the problem (6.1)is valid.

Furthermore, if u₁ andu₂ are two solutions to this problem thenu₁

_Ω=u_{2 Ω}. Thus, there is a unique u∈H^Ω₂ such that

(Lu)

_Ω= 0, Tr˙ 2U = ˙f for someU ∈H2 withU

_Ω=u, kukH₂ ≤Ckf˙kD₂. In particular, if operators Tr˙ ^Ω₂ as in Lemma 5.4 exist, then there exists a unique solution u∈H^Ω₂ to the problem

(Lu)

Ω= 0, Tr˙ ^Ω₂u= ˙f, kuk_HΩ

2 ≤Ckf˙kD2.

If the condition (3.3) is valid, or more generally ifH₁=H₂ and Condition 2.1 is strengthened to the condition|B(u, u)| ≥λkuk², then the condition (6.3) is valid.

Proof of Lemma 6.1. We will in fact produce au∈H2that is a joint solution both to the problem (6.1) and to the problem

(Lu)

_C= 0, Tr˙ 2u= ˙f, kukH₂ ≤Ckfk˙ D₂.

Because ˙f ∈D2, there is some F ∈H2 such that ˙Tr2F = ˙f. Observe that ˚Hj

is a Hilbert space and that the operator T given by T ϕ = B(ϕ, F) is bounded.

By Theorem 4.1, there is a unique w∈˚H₂ such thatB(ϕ, w) = B(ϕ, F) for each ϕ ∈ ˚H1. Let u = F −w. Then u is the unique element of H2 that satisfies Tr˙ 2u= ˙Tr2F−Tr˙ 2w= ˙f andB(ϕ, u) = 0 for allϕ∈˚H1. By Conditions 2.2 and 2.3 and the definition (2.5) of (Lu)

Ω, (Lu)

Ω = 0 and (Lu)

C = 0 if and only if B(ϕ, u) = 0 for allϕ∈˚H₁. Thus,uis the the unique element ofH₂ that satisfies Tr˙ ₂u= ˙f and (Lu)

_Ω= 0 = (Lu) _C.

We now turn to uniqueness. Letube as before. Suppose that (Lu1)

_Ω= 0 and Tr˙ 2u1 = ˙f. Then by Condition 2.3, there is some w ∈H2 such that w

_Ω=u1

_Ω and w

_C=u

_C. But then (Lw)

_Ω = (Lu1)

_Ω= 0 and (Lw)

_C = (Lu)

_C= 0, and Tr˙ ₂w= ˙Tr₂u₁ = ˙Tr₂u= ˙f, and so w =u. In particular u₁

_Ω =w _Ω =u

_Ω, as

desired.

Lemma 6.2. Suppose that there is a λ⁰>0 such that sup

w∈H^Ω₁\{0}

|B^Ω(w, v)|

kwk_HΩ 1

≥λ⁰kvk_HΩ

2, sup

w∈H^Ω₂\{0}

|B^Ω(u, w)|

kwk_HΩ 2

≥λ⁰kuk_HΩ

1 (6.4)

for allu∈H^Ω₁ andv∈H^Ω₂. Let D˚₁={Tr˙ ₁ϕ:ϕ∈H₁, ϕ

_Ω= 0}. Suppose that g˙ ∈N₂ and thathf˙,gi˙ = 0 for all f˙ ∈D˚1. Then there is a C independent of g˙ such that there is exactly one function u∈H^Ω₂ such that the problem (6.2) is valid.

Recall that hTr˙ ₁ϕ,M˙ _BΩui= B^Ω(ϕ

_Ω, u) for all ϕ∈ H₁; thus, the given con- dition on ˙g is necessary. If operators ˙Tr^Ω₁ parallel to those in Lemma 5.4 exist, then ˚D₁={0}and so solutions to the Neumann problem exist for all ˙g∈N₂. In the case of the operators of Section 3, the condition (6.4) does not follow from the condition (3.3); this condition must be replaced by the condition

< X

|α|=|β|=m

Z

Ω

∂^αϕ Aαβ∂^βϕ≥λk∇^mϕk²_L2(Ω) for allϕ∈W˙_m²(R^d).

Proof of Lemma 6.2. Let ˙g ∈ N₂ with hf˙,gi˙ = 0 for all ˙f ∈ ˚D₁. Let Tg˙ be the operator onH^Ω₁ given byTg˙ϕ=hTr˙ 1Φ,gi˙ for any Φ∈H1with Φ

_Ω=ϕ. ThenTg˙

is bounded and well defined.

By Theorem 4.1, there is a unique u ∈ H^Ω₂ such that B^Ω(ϕ, u) = Tg˙ϕ for all ϕ∈H^Ω₁. By definition ofTg˙, we have thatB^Ω(Φ

_Ω, u) =hTr˙ ₁Φ,gi˙ for any Φ∈H₁. By the definitions (2.5) and (2.7), we have that (Lu)

_Ω = 0 and ˙M_BΩu = ˙g.

Conversely, if (Lu1)

_Ω= 0 and ˙M_BΩu1= ˙g, thenB^Ω(ϕ, u1) =Tg˙ϕfor allϕ∈H^Ω₁,

and sou1=uand the solution is unique.

6.2. From invertibility to well posedness. In this section we will need the following objects.

• Quasi-Banach spacesY^Ω,D_XandN_X.

• Linear operatorsTrc^Ω_X:Y^Ω→DXandMc^Ω_X:Y^Ω→NX.

• Linear operatorsDb^Ω:DX→Y^ΩandSb^Ω:NX→Y^Ω.

For the sake of the applications, we will introduce the following notation.

Definition 6.3. We will letX^Ωbe any superspace ofY^Ω, that is, any quasi-Banach space withX^Ω⊇Y^Ω and withkuk_XΩ =kuk_YΩ for anyu∈Y^Ω.

We will let (bL·)

_Ωbe any operator defined on X^Ω such that (bLu)

_Ω= 0 if and only if u∈ Y^Ω. Thus, Y^Ω = {u ∈ X^Ω : (Lu)b

_Ω = 0}; we will routinely use this expression forY^Ω.

Such a superspace and operator must exist. For example, we could takeX^Ω= Y^Ω, and given an X^Ω ⊇ Y^Ω we could let (Lb·)

_Ω be the (nonlinear) indicator function ofX^Ω\Y^Ω.

Remark 6.4. In the situation of Section 6.1, Y^Ω = {u ∈ H^Ω₂ : (Lu)

_Ω = 0}, and so the use of the space X^Ω = H^Ω₂ and operator (bL·)

_Ω = (L·)

_Ω is very natural. As discussed above, in the situation of [31, 40, 43, 44], the use of the space X^Ω={u:Ne(∇u)∈L^p(Ω)} and the operator Lgiven by formula (3.2) is equally natural.