Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 309, pp. 1–23. 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 − div A∇. The Dirichlet problem

− div A∇u = 0 in Ω, u = f on ∂Ω, kukX ≤ CkfkD and the Neumann problem

− div A∇u = 0 in Ω, ν · A∇u = g on ∂Ω, kukX ≤ CkgkN are known to be well-posed for many classes of coefficients A and domains Ω, and with solutions in many spaces X and boundary data in many boundary spaces D and N. 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 Z Ω ∗ L∗ DAf(x) = ν · A (y)∇yE (y, x)f(y) dσ(y), (1.1) ∂Ω Z Ω L∗ SL g(x) = E (y, x)g(y) dσ(y) (1.2) ∂Ω

2010 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 2 A. BARTON EJDE-2017/309 where ν is the unit outward normal to Ω and where EL(y, x) is the fundamental solution for the operator L = − div A∇, that is, the formal solution to LEL( · , x) = δx. These operators are inspired by a formal integration by parts Z u(x) = L∗EL∗ ( · , x) u Ω Z Z Z = − ν · A∗∇EL∗ (·, x)u dσ + EL∗ (· , x)ν · A∇u dσ + EL∗ (· , x)Lu ∂Ω ∂Ω Ω which gives the Green’s formula Ω Ω u(x) = −DA(u ∂Ω)(x) + SL (ν · A∇u)(x) if x ∈ Ω and Lu = 0 in Ω at least for relatively well-behaved solutions u. 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 Ω is equivalent to invertibility of the operator g 7→ SL g ∂Ω, and well posedness of the Neumann problem in both domains is equivalent to invertibility of the operator Ω f 7→ ν · A∇DAf. 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 case A = I and L = −∆) and [5, 14, 23, 35, 37, 38] in the case of more general second order operators under various assumptions on the coefficients A. 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 operators L that 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 EJDE-2017/309 LAYER POTENTIALS 3 space D2 or N2 to a Hilbert space H2 given certain conditions on D2, N2 and H2. 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 where L 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 in L2 for 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 Ω Ω We will construct layer potentials DB and SL using the following objects.

• Two Hilbert spaces H1 and H2. Ω C Ω C • Six (quasi)-normed vector spaces Hb1 , Hb1 , Hb2 , Hb2 , Db 1 and Db 2. Ω Ω Ω • Bounded sesquilinear forms B : H1 × H2 → C, B : H1 × H2 → C, and C C C Ω C B : H1 × H2 → C. (We will define the spaces Hj , Hj momentarily.) ˙ ˙ • Bounded linear operators Tr1 : H1 → Db 1 and Tr2 : H2 → Db 2. 1 Ω 2 Ω • Bounded linear operators (·) Ω : H1 → Hb1 and (·) Ω : H2 → Hb2 . When no ambiguity will arise we will suppress the superscript and refer to both

operators as Ω. j C • Bounded linear operators (·) : Hj → Hbj for j = 1, 2; we again often refer C to both operators as C. Ω C We will work not with the spaces Hbj , Hbj and Db j, but with the (normed) vector Ω C spaces Hj , Hj and Dj defined as follows.

Ω H = {F : F ∈ Hj}/ ∼ with norm kfk Ω = inf{kF kH : F = f}, (2.1) j Ω Hj j Ω C H = {F : F ∈ Hj}/ ∼ with norm kfk C = inf{kF kH : F = f}, (2.2) j C Hj j C ˙ ˙ ˙ ˙ Dj = {Trj F : F ∈ Hj}/ ∼ with norm kfkDj = inf{kF kHj : Trj F = f} (2.3) where ∼ denotes the equivalence relation f ∼ g if kf − 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 and v ∈ H2 we have that |B(w, v)| |B(u, w)| sup ≥ λkvkH2 , sup ≥ λkukH1 . w∈H1\{0} kwkH1 w∈H2\{0} kwkH2

Condition 2.2. If u ∈ H1 and v ∈ H2, then

Ω C B(u, v) = B (u Ω, v Ω) + B (u C, v C). 4 A. BARTON EJDE-2017/309

Condition 2.3. If ϕ, ψ ∈ Hj for j = 1 or j = 2, and if Tr˙ j ϕ = Tr˙ j ψ, then there is a w ∈ Hj such that ˙ ˙ ˙ w Ω = ϕ Ω, w C = ψ C, and Trj w = Trj ϕ = Trj ψ. We now introduce some further terminology. If X is a quasi-Banach space, we will let X∗ be the space of conjugate linear functionals on X. We define the conjugate linear operator L as follows. If u ∈ H2, let Lu be the ∗ element of H1 given by hϕ, Lui = B(ϕ, u). (2.4) ∗ Notice that L is bounded H2 → H1. Ω ˙ ∗ If u ∈ H2 , we let (Lu) Ω be the element of {ϕ ∈ H1 : Tr1 ϕ = 0} given by Ω ˙ hϕ, (Lu) Ωi = B (ϕ Ω, u) for all ϕ ∈ H1 with Tr1 ϕ = 0. (2.5)

If u ∈ H2, we will often use (Lu) as shorthand for (L(u )) . We will primarily Ω Ω Ω be concerned with the case (Lu) Ω = 0. We will let ∗ ∗ N2 = D1, N1 = D2 (2.6) denote the spaces of conjugate linear functionals on D1 and D2. We will now define Ω the Neumann boundary values of an element u of H2 that satisfies (Lu) Ω = 0. If ˙ ˙ Ω Tr1 ϕ = Tr1 ψ and (Lu) Ω = 0, then B (ϕ Ω − ψ Ω, u) = 0 by definition of (Lu) Ω. Ω ˙ ˙ Thus, B (ϕ Ω, u) depends only on Tr1 ϕ, not on ϕ. Thus, MBΩ u defined as follows is a well defined element of N2. ˙ ˙ Ω hTr1 ϕ, MBΩ ui = B (ϕ Ω, u) for all ϕ ∈ H1. (2.7) We can compute ˙ ˙ Ω ˙ ˙ Ω ˙ |hf, M Ω ui| ≤ kB k inf{kϕkH : Tr1 ϕ = f}kuk Ω = kB kkfkD kuk Ω B 1 H2 1 H2

Ω and so we have the bound k M˙ Ω ukN ≤ kB k kuk Ω . B 2 H2 Ω If (Lu) Ω 6= 0, then the conjugate linear operator given by ϕ 7→ B (ϕ Ω, u) Ω is still of interest. We will denote this operator LBΩ u; that is, if u ∈ H2 , then ∗ LBΩ u ∈ H1 is defined by Ω hϕ, LBΩ ui = B (ϕ Ω, u) for all ϕ ∈ H1. (2.8)

If u ∈ H2 then as before we will use LBΩ u as a shorthand for LBΩ (u Ω).

Remark 2.4. We observe that, for a given sesquilinear form B defined on H1 × Ω C H2, there are often many choices of forms B and B that satisfy Condition 2.2. Conversely, for a given form BΩ there may be many forms BC such that the operator B given by Condition 2.2 satisfies Condition 2.1. See Remark 3.1 for an example. The operator L depends only on B, and not on a particular choice of BΩ and C ˙ Ω B . By contrast, the quantities MBΩ u and LBΩ u depend on B and not on B (that is, not on BC). Ω Ω We also comment on the quantity (Lu) Ω. If u ∈ H2 , then by definition of H2 ˙ ˙ there is some U ∈ H2 with u = U Ω. If Tr1 ϕ = 0 = Tr1 0, then by Condition 2.3 EJDE-2017/309 LAYER POTENTIALS 5

there is some w ∈ H1 with w = ϕ and w = 0 = 0. Thus, by the definition Ω Ω C C (2.5) of (Lu) Ω and Condition 2.2, Ω C hϕ, (Lu) Ωi = B (ϕ Ω, u) = B(w, U) − B (w C,U C) = B(w, U) Ω and so (Lu) Ω may be viewed as depending either on B or on B .

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. Let m ≥ 1 be an integer, and let L be an elliptic differential operator of the form

m X α β Lu = (−1) ∂ (Aαβ∂ u) (3.1) |α|=|β|=m for some (possibly complex) bounded measurable coefficients A defined on Rd. Here d α and β are multiindices in N0, where N0 denotes the nonnegative integers. As is standard in the theory, we say that Lu = 0 in an open set Ω in the weak sense if Z X α β ∞ ∂ ϕAαβ∂ u = 0 for all ϕ ∈ C0 (Ω). (3.2) Ω |α|=|β|=m We impose the following ellipticity condition: we require that for some λ > 0, Z X α β m 2 ˙ 2 d < ∂ ϕ Aαβ∂ ϕ ≥ λk∇ ϕkL2( d) for all ϕ ∈ Wm(R ). (3.3) d R |α|=|β|=m R

Let Ω ⊂ Rd be a Lipschitz domain, and let C = Rd \ 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. ˙ 2 d • H1 = H2 = H is the homogeneous Wm(R ) of locally integrable functions ϕ (or rather, of equivalence classes of functions modulo polynomials of degree m − 1) with weak derivatives of order m, and such that the H-norm given m by kϕk = k∇ ϕk 2 d is finite. This space is a Hilbert space with inner product H L (R ) P R α α hϕ, ψi = d ∂ ϕ ∂ ψ. |α|=m R Ω C Ω ˙ 2 m 2 • Hb and Hb are the Sobolev spaces Hb = Wm(Ω) = {ϕ : ∇ ϕ ∈ L (Ω)} and C ˙ 2 m 2 Hb = Wm(C) = {ϕ : ∇ ϕ ∈ L (C)} with the expected norms. ˙ 2,2 • Db denotes the (vector-valued) Besov space B1/2(∂Ω) of locally integrable func- tions modulo constants with norm Z Z |f(x) − f(y)|2 1/2 2,2 kfkB˙ (∂Ω) = d dσ(x) dσ(y) . 1/2 ∂Ω ∂Ω |x − y| • In [17, 18, 19], Ω is assumed to have connected boundary, and Tr˙ is the linear operator defined on H by ˙ Ω m−1 Ω γ Tr u = Tr ∇ u Ω = {Tr ∂ u}|γ|=m−1, where TrΩ is the standard boundary of Sobolev spaces. 6 A. BARTON EJDE-2017/309

Given a suitable modification of the trace space Db , it is also possible to choose ˙ Ω γ ˙ Ω m−1 Tr u = {Tr ∂ u}|γ|≤m−1 or Tr u = (Tr u, ∂ν u, . . . , ∂ν u), where ν is the unit outward normal, so that the boundary derivatives of u of 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 on H × H given by Z X α β B(ψ, ϕ) = ∂ ψAαβ∂ ϕ. (3.4) d |α|=|β|=m R

The sesquilinear forms BΩ and BC are defined analogously to B, but with the integral over Rd replaced by an integral over Ω or C. We now discuss the conditions imposed in Section 2. The forms B, BΩ and BC Ω are clearly bounded and sesquilinear, and the restriction operators Ω : H → Hb , C C : H → Hb are bounded and linear. ˙ d The trace operator Tr is linear. If Ω = R+ 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 → Db , where W is the Pm k inhomogeneous Sobolev space with norm k∇ ϕk 2 d , was established in [42, k=0 L (R ) Chapter V]. Then boundedness of Tr˙ : H → Db follows by the Poincar´einequality. 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 Tr˙ is 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

BΩ(ϕ , u) = 0 for all ϕ ∈ H with Tr˙ ϕ = 0; that is, (Lu) as defined in Section Ω Ω 2 satisfies (Lu) Ω = 0. If u ∈ HΩ, then formally

m X α Ω β LBΩ u = (−1) ∂ (AαβE (∂ u)) |α|=|β|=m where EΩ denotes extension from Ω to Rd by zero. ˙ If m = 1, then by an integration by parts argument we have that MBΩ u = ν·A∇u, where ν is the unit outward normal to Ω, whenever u is 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˙ = Ω m−1 ˙ Ω m−1 Tr ∇ , with that of [3, 64] if Tr u = (Tr u, ∂ν u, . . . , ∂ν u), and with [28, 52, ˙ Ω γ 53] if Tr u = {Tr ∂ u}|γ|≤m−1. Remark 3.1. Each of the sesquilinear forms B and BΩ may be associated with more than one choice of coefficients Aαβ. Ω For example, let Abαβ satisfy Abαβ(x) = Aαβ(x) for all x ∈ Ω. Then B is unchanged if Aαβ is replaced by Abαβ, but B is not. EJDE-2017/309 LAYER POTENTIALS 7

Conversely, let Aeαβ = Aαβ +Mαβ, where Mαβ is a constant that satisfies Mαβ = −Mβα. A straightforward integration by parts argument shows that B (and thus Ω C L) is unchanged if Aαβ is replaced by Aeαβ. However, the operators B and B do take different values if Aαβ is replaced by Aeαβ. Thus, as mentioned in Remark 2.4, B may be associated with more than one form BΩ, and BΩ 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 HbΩ to Ω Ω ˙ 2 H, and so H = Hb = Wm(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 ⊂ Rd is a Lipschitz domain, then Tr˙ is a bounded operator H → Db . If Tr˙ u = TrΩ ∇m−1u, as in [17, 18, 19], then Tr˙ has a bounded right inverse. ˙ Ω m−1 ˙ Ω γ See [16]. If Tr u = (Tr u, ∂ν u, . . . , ∂ν u) or Tr u = {Tr ∂ u}|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−1 ∞ d Ω m−1 ∞ d {∇ ϕ ∂Ω : ϕ ∈ C0 (R )} or {(Tr ϕ, ∂ν ϕ, . . . , ∂ν ϕ): ϕ ∈ C0 (R )} is dense ˙ 2,2 in D. Thus, if m = 1 then D = Db = B1/2(∂Ω). If m ≥ 2 then D is a closed proper subspace of Db , as the different partial derivatives of a common function must satisfy certain compatibility conditions. In this case D is 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 1 such that B(v, uT ) = T (v). Furthermore, kuT kH2 ≤ λ kT kH1→C, where λ is as in Condition 2.1.

We construct layer potentials as follows. Let g˙ ∈ N2. Then the operator Tg˙ ϕ = hg˙ , Tr˙ 1 ϕi = hTr˙ 1 ϕ, g˙ i is a bounded linear operator on H1. By the Babuˇska-Lax- Ω Milgram theorem, there is a unique uT = SL g˙ ∈ H2 such that Ω ˙ B(ϕ, SL g˙ ) = hTr1 ϕ, g˙ i for all ϕ ∈ H1. (4.1) Ω We will let SL g˙ denote the single layer potential of g˙ . Observe that the dependence Ω of SL on 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 and C, and so Ω C SL g˙ = SLg˙ . (4.2) The double layer potential is somewhat more involved. We begin by defining the Newton potential. 8 A. BARTON EJDE-2017/309

∗ Let H be an element of H1. By the Babuˇska-Lax-Milgram theorem, there is a L unique element N H of H2 that satisfies L B(ϕ, N H) = hϕ, Hi for all ϕ ∈ H1. (4.3) We refer to N L as 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 Ω L ˙ SL g˙ = N (Tg˙ ) where hϕ, Tg˙ i = hTr1 ϕ, g˙ i. (4.4) ˙ We now return to the double layer potential. Let f ∈ D2. Then there is some ˙ F ∈ H2 such that Tr˙ 2 F = f. Let Ω ˙ Ω ˙ L ˙ ˙ DBf = DL,BΩ f = −F Ω + (N (LBΩ F )) Ω if Tr2 F = f. (4.5) Ω ˙ Ω Notice that DBf is an element of H2 , not of H2. Further observe that the single layer Ω ˙ ˙ potential SL depends only on Tr1 and B (equivalently on Tr1 and the operator L), Ω Ω Ω and not on the particular choice of B . The double layer potential DB = DL,BΩ , by contrast, depends on both L (or B) and BΩ. Ω ˙ We conclude this section by showing that DBf is well defined, that is, does not depend on the choice of F 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 C Ω Ω kB k C kB k Ω 1 ˙ Ω ˙ ˙ ˙ kDBfkH ≤ kfkD2 , kDBfkHC ≤ kfkD2 , kSL g˙ kH2 ≤ kg˙ kN2 . 2 λ 2 λ λ Proof. By Theorem 4.1, we have 1 1 kSΩg˙ k ≤ kT k ≤ k Tr˙ k kg˙ k . L H2 λ g˙ H1→C λ 1 H1→D1 D1→C ˙ By definition of D1 and N2, k Tr1 kH1→D1 = 1 and kg˙ kD1→C = kg˙ kN2 , and so Ω SL : 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 of L, if ϕ ∈ H1 then hϕ, LF i = B(ϕ, F ). By definition of N L, B(ϕ, N L(LF )) = hϕ, LF i. Thus, by coercivity of B, L F = N (LF ) for all F ∈ H2. (4.6) Ω C By definition of B , B and LBΩ F , Ω C hϕ, LF i = B(ϕ, F ) = B (ϕ Ω,F Ω) + B (ϕ C,F C) = hϕ, LBΩ F i + hϕ, LBC F i for all ϕ ∈ H1. Thus, LF = LBΩ F + LBC F and so L L L L − F + N (LBΩ F ) = −F + N (LF ) − N (LBC F ) = −N (LBC F ). (4.7) ˙ 0 In particular, suppose that f = Tr˙ 2 F = Tr˙ 2 F . By Condition 2.3, there is some 0 w ∈ H2 such that w Ω = F Ω and w C = F C. Then L L −F Ω + (N (LBΩ F )) Ω = −w Ω + (N (LBΩ w)) Ω L L 0 = −(N (LBC w)) Ω = −(N (LBC F )) Ω 0 L 0 = −F Ω + (N (LBΩ F )) Ω Ω ˙ ˙ and so DBf is well-defined, that is, depends only on f and not the choice of function ˙ F with Tr˙ 2 F = f. EJDE-2017/309 LAYER POTENTIALS 9

Furthermore, we have the alternative formula Ω ˙ L ˙ ˙ DBf = −(N (LBC F )) Ω if Tr2 F = f. (4.8) Thus, Ω ˙ L L kD fk Ω ≤ inf k(N (L C F )) k Ω ≤ inf kN (L C F )kH B H2 B Ω H2 B 2 Tr˙ 2 F =f˙ Tr˙ 2 F =f˙ Ω by definition of the H2 -norm. By Theorem 4.1 and the definition of N L, we have that

L 1 kN (L C F )k ≤ kL C F k . B H2 λ B H1→C C Since LBC F (ϕ) = B (ϕ C,F C), we have that C C kL C F kH → ≤ kB kkF k C ≤ kB kkF kH B 1 C C H2 2 and so Ω ˙ 1 C 1 C ˙ kD fk Ω ≤ inf kB kkF kH = kB kkfkD B H2 2 2 Tr˙ 2 F =f˙ λ λ 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). ˙ Ω ˙ Ω Lemma 5.1. Let f ∈ D2, g˙ ∈ N2, and let u = DBf or u = SL g˙ . Then Ω (Lu) Ω = 0. Ω ˙ Proof. Recall that (Lu) Ω = 0 if B (ϕ+ Ω, u) = 0 for all ϕ+ ∈ H1 with Tr1 ϕ+ = 0. ˙ ˙ If Tr1 ϕ+ = 0 = Tr1 0, then by Condition 2.3 there is some ϕ ∈ H1 with ϕ Ω = ϕ+, ˙ ϕ C = 0 and Tr1 ϕ = 0. By definition (4.1) of the single layer potential, L ˙ Ω Ω ˙ C Ω ˙ Ω Ω ˙ 0 = B(ϕ, S g) = B (ϕ Ω, SL g Ω) + B (ϕ C, SL g C) = B (ϕ+ Ω, SL g Ω) as desired. Ω Turning to the double layer potential, if ϕ ∈ H1, then by definition (4.5) of DB, C Ω formula (4.8) for DB and linearity of B , Ω Ω ˙ Ω  Ω L  B (ϕ Ω, DBf) = −B ϕ Ω,F Ω + B ϕ Ω, (N (LBΩ F )) Ω , C C ˙ C L  B (ϕ C, DBf) = −B ϕ C, (N (LBΩ F )) C . Subtracting and applying Condition 2.2, Ω Ω ˙ C C ˙ Ω  L  B (ϕ Ω, DBf) − B (ϕ C, DBf C) = −B ϕ Ω,F Ω + B ϕ, N (LBΩ F ) . By definition (4.3) of N L, L  B ϕ, N (LBΩ F ) = hϕ, LBΩ F i and by the definition (2.8) of LBΩ F , L  Ω B ϕ, N (LBΩ F ) = B (ϕ Ω,F Ω). 10 A. BARTON EJDE-2017/309

Thus, Ω Ω ˙ C C ˙ B (ϕ Ω, DBf) − B (ϕ C, DBf) = 0 for all ϕ ∈ H1. (5.1) ˙ In particular, as before if Tr1 ϕ+ = 0 then there is some ϕ with ϕ Ω = ϕ+ Ω, Ω Ω ˙ ϕ C = 0 and so B (ϕ Ω, DBf) = 0. This completes the proof.  Ω Lemma 5.2. If u ∈ H2 and (Lu) Ω = 0, then Ω ˙ Ω ˙ C ˙ C ˙ u = −DB(Tr2 U) + SL (MBΩ u) Ω, 0 = DB(Tr2 U) + SL(MBΩ u) C for any U ∈ H2 with U Ω = u. Proof. By definition (4.5) of the double layer potential, Ω ˙ L L −DB(Tr2 U) = U Ω − (N (LBΩ U)) Ω = u − (N (LBΩ u)) Ω and by formula (4.8) C ˙ L DB(Tr2 U) = −(N (LBΩ u)) C. L Ω ˙ It suffices to show that N (LBΩ u) = SL (MBΩ u). Let ϕ ∈ H1. By formulas (4.1) and (2.7), Ω ˙ ˙ ˙ Ω B(ϕ, SL (MBΩ u)) = hTr1 ϕ, MBΩ ui = B (ϕ Ω, u).

By formula (4.3) for the Newton potential and by the definition (2.8) of LBΩ u, L Ω B(ϕ, N (LBΩ u)) = hϕ, LBΩ ui = B (ϕ Ω, u). L Ω ˙ Thus, B(ϕ, N (LBΩ u)) = B(ϕ, SL (MBΩ u)) for all ϕ ∈ H1; by coercivity of B, L Ω ˙ we must have that N (LBΩ u) = SL (MBΩ u). This completes the proof.  ∗ Ω C ∗ Let B (ϕ, ψ) = B(ψ, ϕ) and define B∗ , B∗ analogously. Then B is a bounded and coercive operator H2 × H1 → C, and so we can define the double and single Ω Ω Ω layer potentials DB∗ : D1 → H1 , SL∗ : N1 → H1. We then have the following adjoint relations. Lemma 5.3. We have the adjoint relations Ω Ω hϕ˙ , M˙ Ω D f˙i = hM˙ Ω D ∗ ϕ˙ , f˙i, (5.2) B B B∗ B ˙ Ω ˙ Ω hγ˙ , Tr2 SL g˙ i = hTr1 SL∗ γ˙ , g˙ i (5.3) ˙ for all f ∈ D2, ϕ˙ ∈ D1, g˙ ∈ N2 and γ˙ ∈ N1. ˙ Ω Ω ˙ ˙ ˙ L ˙ ˙ If we let Tr2 DBf = − Tr2 F +Tr2 N (LBΩ F )) for any F ∈ H2 with Tr2 F = f, ˙ Ω Ω ˙ then Tr2 DBf does not depend on the choice of F , and we have the duality relations Ω Ω Ω hγ˙ , Tr˙ D f˙i = h−γ˙ + M˙ Ω S ∗ γ˙ , f˙i. (5.4) 2 B B∗ L Proof. By formula (4.1), ˙ Ω Ω Ω hTr1 SL∗ γ˙ , g˙ i = B(SL∗ γ˙ , SL g˙ i, ˙ Ω ∗ Ω Ω hTr2 SL g˙ , γ˙ i = B (SL g˙ , SL∗ γ˙ i and so formula (5.3) follows by definition of B∗. ˙ Let Φ ∈ H1 and F ∈ H2 with Tr˙ 1 Φ = ϕ˙ , Tr˙ 2 F = f. Then by formulas (2.7) and (4.5), Ω Ω Ω Ω Ω L ˙ ˙ Ω ˙ ˙ Ω hϕ, MB DBfi = B (Φ Ω, DBf) = −B (Φ Ω,F Ω) + B (Φ Ω, (N (LB F )) Ω) EJDE-2017/309 LAYER POTENTIALS 11 and so Ω Ω Ω L ˙ ˙ Ω ˙ Ω hϕ, MB DBfi = −B∗ (F Ω, Φ Ω) + B∗ ((N (LB F )) Ω, Φ Ω). By formula (2.8), Ω L L B ((N (L Ω F )) , Φ ) = hN (L Ω F ),L Ω Φi. ∗ B Ω Ω B B∗ ∗ ∗ L∗ By formula (4.3), if H ∈ H1 and ϕ ∈ H2 then B (ϕ, N H) = hϕ, Hi. Letting L ϕ = N (L Ω F ) and H = L Ω Φ, we see that B B∗ Ω L ∗ L L∗  B ((N (L Ω F )) , Φ ) = B N (L Ω F ), N (L Ω Φ) . ∗ B Ω Ω B B∗ Therefore,

Ω Ω ∗ L L∗ hϕ˙ , M˙ Ω D f˙i = −B (F , Φ ) + B (N (L Ω F ), N (L Ω Φ)). B B ∗ Ω Ω B B∗ By the same argument

Ω Ω L∗ L hf˙, M˙ Ω D ∗ ϕ˙ i = −B (Φ ,F ) + B(N (L Ω Φ), N (L Ω F )) B∗ B Ω Ω B∗ B ∗ Ω and by definition of B and B∗ formula (5.2) is proven. ˙ Ω Ω Finally, by definition of Tr2 DB, ˙ Ω Ω ˙ ˙ ˙ L hγ˙ , Tr2 DBfi = −hγ˙ , Tr2 F i + hγ˙ , Tr2 N (LBΩ F ))i. By the definition (4.1) of the single layer potential,

˙ L ∗ L Ω hγ˙ , Tr2 N (LBΩ F )i = B (N (LBΩ F ), SL∗ γ˙ ). By definition of B∗ and the definition (4.3) of the Newton potential,

∗ L Ω Ω B (N (LBΩ F ), SL∗ γ˙ ) = hSL∗ γ˙ ,LBΩ F i and by the definition (2.8) of LBΩ F , Ω Ω Ω ∗ ˙ Ω ∗ ˙ hSL γ,LB F i = B (SL γ Ω,F Ω). By the definition (2.7) of Neumann boundary values,

Ω Ω ˙ ˙ Ω B (F , S ∗ γ˙ ) = hTr F, M Ω (S ∗ γ˙ )i ∗ Ω L Ω 2 B∗ L Ω and so Ω Ω Ω hγ˙ , Tr˙ D f˙i = −hγ˙ , f˙i + hM˙ Ω (S ∗ γ˙ ), f˙i 2 B B∗ L Ω ˙ Ω Ω for any choice of F . Thus Tr2 DB is well-defined and formula (5.4) is valid.  We conclude this section with the jump relations for layer potentials. ˙ Ω Ω ˙ Lemma 5.4. Let Tr2 DB be as in Lemma 5.3. If f ∈ D2 and g˙ ∈ N2, then we have the jump and continuity relations ˙ Ω Ω ˙ ˙ C C ˙ ˙ Tr2 DBf + Tr2 DBf = −f, (5.5) Ω C ˙ Ω ˙ ˙ C ˙ ˙ MB (SL g Ω) + MB (SLg C) = g, (5.6) ˙ Ω ˙ ˙ C ˙ MBΩ (DBf) − MBC (DBf) = 0. (5.7) ˙ Ω Ω ˙ C C If there are bounded operators Tr2 : H2 → D2 and Tr2 : H2 → D2 such that ˙ ˙ Ω ˙ C Tr2 F = Tr2 (F Ω) = Tr2 (F C) for all F ∈ H2, then in addition ˙ Ω Ω ˙ ˙ C C ˙ Tr2 (SL g Ω) − Tr2 (SLg C) = 0. (5.8) 12 A. BARTON EJDE-2017/309

˙ Ω In the absence of an operator Tr2 , the continuity relation ˙ Ω ˙ C Tr2 SL g˙ − Tr2 SLg˙ = 0 (5.9) is valid (and follows immediately from formula (4.2)). Existence of the operator ˙ Ω ˙ Tr2 is equivalent to the condition that Tr2 u = 0 whenever u Ω = 0. This condition d d ˙ is natural if Ω ⊂ R is an open set, C = R \ Ω and Tr2 denotes a trace operator ˙ Ω restricting functions to the boundary ∂Ω. Observe that if such operators Tr2 and ˙ C Tr2 exist, then by the definition (4.5) of the double layer potential and by the ˙ Ω Ω ˙ Ω Ω ˙ ˙ Ω Ω ˙ definition of Tr2 DB in Lemma 5.3, Tr2 (DBf) = (Tr2 DB)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- Ω ˙ Ω ˙ C cause SL g˙ ∈ H2 and by the definition of Tr2 , Tr2 . ˙ Ω Ω The jump relation (5.5) follows from the definition of Tr2 DB and by using ˙ C C formula (4.7) to rewrite Tr2 DB. We observe that by the definition (2.7) of Neumann boundary values and the Ω C definitions (2.3) and (2.6) of D1 and N2, if u ∈ H2 and v ∈ H2 with (Lu) = 0 Ω and (Lv) C = 0, then ˙ ˙ ˙ ˙ ˙ Ω C MBΩ u + MBC v = ψ if and only if hTr1 ϕ, ψi = B (ϕ Ω, u) + B (ϕ C, v) for all ϕ ∈ 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 Ω ˙ Ω ˙ (Lub ) Ω = 0, TrcX u = f, kukX ≤ CkfkDX , and the Neumann problem Ω ˙ Ω ˙ (Lub ) Ω = 0, McX u = g, kukX ≤ CkgkNX for some constant C and some solution space X and spaces of Dirichlet and Neu- mann boundary data DX and NX. For example, if Lb is a second-order differential operator, then as in [31, 40, 43, 44] we might wish to establish well-posedness with ˙ p p Ω p DX = W1 (∂Ω), NX = L (∂Ω) and X = {u : Ne(∇u) ∈ L (∂Ω)}, where Ne is the modified nontangential maximal function introduced in [43]. Ω Ω If X = H2 , 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- Ω Ω tials, originally defined as bounded operators DB : D2 → H2 and SL : N2 → H2, Ω Ω may be extended to operators Db : DX → X and Sb : 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. EJDE-2017/309 LAYER POTENTIALS 13

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 a u ∈ H2 that satisfies ˙ ˙ ˙ (Lu) Ω = 0, Tr2 u = f, kukH2 ≤ CkfkD2 (6.1) Ω or the Neumann problem of finding a u ∈ H2 that satisfies

(Lu) = 0, M˙ Ω u = g˙ , kuk Ω ≤ Ckg˙ kN . (6.2) Ω B H2 2 Under some modest additional assumptions on the operators L and BΩ, a stan- dard argument involving Theorem 4.1 yields unique solvability of these problems. ˚ 0 Lemma 6.1. Let Hj = {ϕ ∈ Hj : Tr˙ j ϕ = 0}. Suppose that there is a λ > 0 such that |B(w, v)| 0 |B(u, w)| 0 sup ≥ λ kvkH2 , sup ≥ λ kukH1 (6.3) kwkH kwkH w∈˚H1\{0} 1 w∈˚H2\{0} 2 ˚ ˚ ˙ for all u ∈ H1 and v ∈ H2. Then there is a C such that, for each f ∈ D2, there is a function u ∈ H2 such that the problem (6.1) is valid.

Furthermore, if u1 and u2 are two solutions to this problem then u1 Ω = u2 Ω. Ω Thus, there is a unique u ∈ H2 such that ˙ ˙ H ˙ (Lu) Ω = 0, Tr2 U = f for some U ∈ 2 with U Ω = u, kukH2 ≤ CkfkD2 . ˙ Ω In particular, if operators Tr2 as in Lemma 5.4 exist, then there exists a unique Ω solution u ∈ H2 to the problem Ω ˙ ˙ (Lu) = 0, Tr˙ u = f, kuk Ω ≤ CkfkD . Ω 2 H2 2

If the condition (3.3) is valid, or more generally if H1 = 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 a u ∈ H2 that is a joint solution both to the problem (6.1) and to the problem ˙ ˙ ˙ (Lu) C = 0, Tr2 u = f, kukH2 ≤ CkfkD2 . ˙ ˙ ˚ Because f ∈ D2, there is some F ∈ H2 such that Tr˙ 2 F = 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 that B(ϕ, w) = B(ϕ, F ) for each ˚ ϕ ∈ H1. Let u = F − w. Then u is the unique element of H2 that satisfies ˙ ˚ Tr˙ 2 u = Tr˙ 2 F − Tr˙ 2 w = f and B(ϕ, u) = 0 for all ϕ ∈ 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 all ϕ ∈ H1. Thus, u is the the unique element of H2 that satisfies ˙ ˙ Tr2 u = f and (Lu) = 0 = (Lu) . Ω C We now turn to uniqueness. Let u be as before. Suppose that (Lu1) Ω = 0 and ˙ ˙ Tr2 u1 = f. Then by Condition 2.3, there is some w ∈ H2 such that w = u1 Ω Ω and w C = u C. But then (Lw) Ω = (Lu1) Ω = 0 and (Lw) C = (Lu) C = 0, and ˙ ˙ ˙ ˙ Tr2 w = Tr2 u1 = Tr2 u = f, and so w = u. In particular u1 Ω = w Ω = u Ω, as desired.  14 A. BARTON EJDE-2017/309

Lemma 6.2. Suppose that there is a λ0 > 0 such that Ω Ω |B (w, v)| 0 |B (u, w)| 0 sup ≥ λ kvkHΩ , sup ≥ λ kukHΩ (6.4) Ω kwk Ω 2 Ω kwk Ω 1 w∈H1 \{0} H1 w∈H2 \{0} H2 Ω Ω for all u ∈ H1 and v ∈ H2 . ˚ ˙ ˙ ˙ ˙ Let D1 = {Tr1 ϕ : ϕ ∈ H1, ϕ Ω = 0}. Suppose that g ∈ N2 and that hf, gi = 0 ˙ ˚ for all f ∈ D1. Then there is a C independent of g˙ such that there is exactly one Ω function u ∈ H2 such that the problem (6.2) is valid. ˙ ˙ Ω Recall that hTr1 ϕ, MBΩ ui = B (ϕ Ω, u) for all ϕ ∈ H1; thus, the given con- ˙ Ω dition on g˙ is necessary. If operators Tr1 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 Z X α β m 2 ˙ 2 d < ∂ ϕ Aαβ ∂ ϕ ≥ λk∇ ϕkL2(Ω) for all ϕ ∈ Wm(R ). |α|=|β|=m Ω ˙ ˙ ˚ Proof of Lemma 6.2. Let g˙ ∈ N2 with hf, g˙ i = 0 for all f ∈ D1. Let Tg˙ be the Ω ˙ ˙ operator on H1 given by Tg˙ ϕ = hTr1 Φ, gi for any Φ ∈ H1 with Φ Ω = ϕ. Then Tg˙ is bounded and well defined. Ω Ω By Theorem 4.1, there is a unique u ∈ H2 such that B (ϕ, u) = Tg˙ ϕ for all Ω Ω ˙ ϕ ∈ H1 . By definition of Tg˙ , we have that B (Φ , u) = hTr1 Φ, g˙ i for any Φ ∈ H1. Ω ˙ Ω ˙ By the definitions (2.5) and (2.7), we have that (Lu) Ω = 0 and MB u = g. Ω Ω ˙ Ω ˙ Conversely, if (Lu1) Ω = 0 and MB u1 = g, then B (ϕ, u1) = Tg˙ ϕ for all ϕ ∈ H1 , and so u1 = u and the solution is unique.  6.2. From invertibility to well posedness. In this section we will need the following objects. Ω • Quasi-Banach spaces Y , DX and NX. Ω Ω Ω Ω • Linear operators TrcX : Y → DX and McX : Y → NX. Ω Ω Ω Ω • Linear operators Db : DX → Y and Sb : NX → Y . For the sake of the applications, we will introduce the following notation. Definition 6.3. We will let XΩ be any superspace of YΩ, that is, any quasi-Banach Ω Ω Ω space with X ⊇ Y and with kukXΩ = kukYΩ for any u ∈ Y . Ω We will let (Lb · ) Ω be any operator defined on X such that (Lub ) Ω = 0 if and Ω Ω Ω only if u ∈ Y . Thus, Y = {u ∈ X :(Lub ) Ω = 0}; we will routinely use this expression for YΩ. Such a superspace and operator must exist. For example, we could take XΩ = Ω Ω Ω Y , and given an X ⊇ Y we could let (Lb · ) Ω be the (nonlinear) indicator function of XΩ \ YΩ.

Ω Ω Remark 6.4. In the situation of Section 6.1, Y = {u ∈ H2 :(Lu) Ω = 0}, Ω Ω and so the use of the space X = H2 and operator (Lb · ) Ω = (L · ) Ω is very natural. As discussed above, in the situation of [31, 40, 43, 44], the use of the space XΩ = {u : Ne(∇u) ∈ Lp(Ω)} and the operator L given by formula (3.2) is equally natural. EJDE-2017/309 LAYER POTENTIALS 15

This section could be written strictly in terms of the space YΩ; however, we have Ω chosen to use the auxiliary space X and operator (Lb · ) Ω because of their natural roles in the applications. Ω Ω Ω Remark 6.5. Inherent in the requirements that Db : DX → Y and Sb : NX → Ω ˙ Ω ˙ ˙ Y is the requirement that if g ∈ NX then (Lb(Sb g)) Ω = 0, and if f ∈ DX then Ω ˙ (Lb(Db f)) Ω = 0. Ω C Remark 6.6. Recall that SL = SL is defined in terms of a “global” Hilbert space Ω Ω Ω ˙ Ω ˙ H2. If X = H2 , then we have in mind the example Sb g = SL g Ω. In the general case, we do not assume the existence of a global quasi-Banach space X whose restrictions to Ω lie in XΩ, and thus we will let SbΩg˙ be an element of XΩ without assuming a global extension. ˙ Ω ˙ Ω Ω In applications it is often useful to define Tr2 , MBΩ , L, DB and SL in terms Ω of some Hilbert spaces Hj, Hj and to extend these operators to operators with domain or range XΩ by density or some other means. See, for example, [19]. We Ω Ω Ω Ω will not assume that the operators TrcX, McX, Lb, Db and Sb arise by density; we will merely require that they satisfy certain properties similar to those established in Section 5. Ω Ω Specifically, we will often use the following conditions; observe that if X = H2 Ω Ω ˙ Ω for some H2 as in Section 2, and if TrcX is the operator Tr2 of Lemma 5.4, then these properties are valid. Ω Ω Ω Condition 6.7. TrcX is bounded Y → DX; that is, TrcX is a bounded operator Ω from {u ∈ X :(Lub ) Ω = 0} to DX. Ω Ω Condition 6.8. McX is a bounded operator {u ∈ X :(Lub ) Ω = 0} → NX. Ω Ω Condition 6.9. The single layer potential Sb is bounded NX → Y ; equivalently, Ω Ω Sb is bounded NX → X . Ω Ω Condition 6.10. The double layer potential Db is bounded DX → X . Ω Ω Condition 6.11. If u ∈ Y , that is, if u ∈ X and (Lub ) Ω = 0, then we have the Green’s formula Ω Ω Ω Ω u = −Db (TrcX u) + Sb (McX u). The following theorem is straightforward to prove and is the core of the classic method of layer potentials. Ω Ω Ω Ω Ω Theorem 6.12. Let X , (Lb · ) Ω, DX, NX, TrcX, McX, Db , and Sb be as given at the beginning of this section. Ω Ω ˙ Suppose that TrcX Sb : NX → DX is surjective. Then for every f ∈ DX, there is some u such that Ω ˙ Ω (Lub ) Ω = 0, TrcX u = f, u ∈ X . (6.5) Ω Ω Suppose in addition that Condition 6.9 is valid, and that TrcX Sb : NX → DX has a ˙ bounded right inverse, that is, there is a constant C0 such that if f ∈ DX, then there ˙ ˙ is some pre-image g˙ of f with kg˙ kNX ≤ C0kfkDX . Then there is some constant C1 ˙ depending on C0 and the implicit constant in Condition 6.9 such that if f ∈ DX, then there is some u ∈ XΩ such that Ω ˙ Ω ˙ (Lub ) Ω = 0, TrcX u = f, kukX ≤ C1kfkDX . (6.6) 16 A. BARTON EJDE-2017/309

Ω Ω Suppose that McX Db : DX → NX is surjective. Then for every g˙ ∈ NX, there is some u such that Ω ˙ Ω (Lub ) Ω = 0, McX u = g, u ∈ X . (6.7) Ω Ω If Condition 6.10 is valid and McX Db : DX → NX has a bounded right inverse, then there is some constant C1 depending on the bound on that inverse and the implicit Ω constant in Condition 6.10 such that if g˙ ∈ NX, then there is some u ∈ X such that Ω ˙ Ω ˙ (Lub ) Ω = 0, McX u = g, kukX ≤ C1kgkNX . (6.8) Thus, surjectivity of layer potentials implies existence of solutions to boundary value problems. We may also show that injectivity of layer potentials implies uniqueness of solu- tions to boundary value problems. This argument appeared first in [23] and is the converse to an argument of [63].

Ω Ω Ω Ω Ω Theorem 6.13. Let X , (Lb ·) Ω, DX, NX, TrcX, McX, Db , and Sb be as given at the beginning of this section. Suppose that Condition 6.11 is valid. Suppose furthermore that the operator Ω Ω ˙ TrcX Sb : NX → DX is one-to-one. Then for each f ∈ DX, there is at most one solution u to the Dirichlet problem Ω ˙ Ω (Lub ) Ω = 0, TrcX u = f, u ∈ X . Ω Ω If Conditions 6.7, 6.9, 6.10 and 6.11 are all valid, and if TrcX Sb : NX → DX has a bounded left inverse, that is, there is a constant C0 such that the estimate Ω Ω kg˙ kNX ≤ C0k TrcX Sb g˙ kDX is valid for all g˙ ∈ NX, then there is some constant Ω Ω X Ω C1 such that every u ∈ with (Lub ) Ω = 0 satisfies kukX ≤ C1k TrcX ukDX (that is, if u satisfies the Dirichlet problem (6.5) then u must satisfy the Dirichlet problem (6.6)). Ω Ω Similarly, if Condition 6.11 is valid and the operator McX Db : DX → NX is one-to-one, then for each g˙ ∈ NX, there is at most one solution u to the Neumann problem Ω ˙ Ω (Lub ) Ω = 0, McX u = g, u ∈ X . Ω Ω If Conditions 6.8, 6.9, 6.10 and 6.11 are all valid, and if McX Db : DX → NX has Ω a bounded left inverse, then there is some constant C1 such that every u ∈ X with Ω Ω (Lub ) Ω = 0 satisfies kukX ≤ C1k McX ukDX . Proof. We present the proof only for the Neumann problem; the argument for the Dirichlet problem is similar. Throughout the proof we will let C denote a constant whose value may change from line to line. Ω Ω ˙ Ω Suppose that u, v ∈ X with (Lub ) Ω = (Lvb ) Ω = 0 in Ω and McX u = g = McX v. By Condition 6.11, Ω Ω Ω Ω Ω Ω Ω u = −Db (TrcX u) + Sb (McX u) = −Db (TrcX u) + Sb g˙ , Ω Ω Ω Ω Ω Ω Ω v = −Db (TrcX v) + Sb (McX v) = −Db (TrcX v) + Sb g˙ .

Ω Ω Ω Ω Ω Ω Ω Ω In particular, McX Db (TrcX u) = McX Db (TrcX v). If McX Db is one-to-one, then Ω Ω TrcX u = TrcX v. Another application of Condition 6.11 yields that u = v. EJDE-2017/309 LAYER POTENTIALS 17

Ω Ω Now, suppose that McX Db has a bounded left inverse; this implies that for ˙ ˙ Ω Ω ˙ Ω any f ∈ DX we have the estimate kfkDX ≤ C0k McX Db fkNX . Let u ∈ X with Ω Ω (Lub ) Ω = 0; we want to show that kukX ≤ Ck McX ukDX . By Condition 6.11, and because XΩ is a quasi-Banach space, Ω Ω Ω Ω kukXΩ ≤ CkDb (TrcX u)kXΩ + CkSb (McX u)kXΩ . By Conditions 6.9 and 6.10, Ω Ω kukX ≤ Ck TrcX ukDX + Ck McX ukNX . Ω Ω Applying our estimate on McX Db , we see that Ω Ω Ω Ω kukX ≤ Ck McX Db TrcX ukNX + Ck McX ukNX . Ω Ω Ω Ω By Condition 6.11, Db (TrcX u) = Sb (McX u) − u, and so Ω Ω Ω Ω kukX ≤ Ck McX Sb McX ukNX + Ck McX ukNX . An application of Conditions 6.8 and 6.9 completes the proof.  6.3. From well posedness to invertibility. We are now interested in the con- verse results. That is, we have shown that results for layer potentials imply results for boundary value problems; we would like to show that results for boundary value problems imply results for layer potentials. Notice that the above results were built on the Green’s formula (that is, Condi- tion 6.11). The converse results will be built on jump relations, as in Lemma 5.4. Recall that jump relations treat the interplay between layer potentials in a domain and in its complement; thus we will need to impose conditions in both domains. In this section we will need the following spaces and operators. U W • Quasi-Banach spaces Y , Y , DX and NX. As in Section 6.2, we will let U U W W Y = {u ∈ X :(Lub ) U = 0} and Y = {v ∈ X :(Lvb ) W = 0} for some U W superspaces X , X and operators (Lb ·) U,(Lb · ) W. U U U U W W • Linear operators TrcX : Y → DX, McX : Y → NX, TrcX : Y → DX, and W W McX : Y → NX. U U W W U U • Linear operators Db : DX → Y , Db : DX → Y , Sb : NX → Y and W W Sb : NX → Y . In the applications U is an open set in Rd or in a smooth manifold, and W = Rd \ U is the interior of its complement. The space XW is then a space of functions defined in W and is thus a different space from XU. However, we emphasize that we have defined only one space DX of Dirichlet boundary values and one space NX of Neumann boundary values; that is, the traces from both sides of the boundary must lie in the same spaces. We will often use the following conditions. Note the similarity between Con- ditions 6.7–6.11 and Conditions 6.14–6.18; Conditions 6.14–6.18 state that Condi- tions 6.7–6.11 hold for both Ω = U and Ω = W. U U W Condition 6.14. TrcX is bounded {u ∈ X :(Lub ) U = 0} → DX, and TrcX is W bounded {v ∈ X :(Lvb ) W = 0} → DX. U U W Condition 6.15. McX is bounded {u ∈ X :(Lub ) U = 0} → NX, and McX is W bounded {v ∈ X :(Lvb ) W = 0} → NX. 18 A. BARTON EJDE-2017/309

U U W W Condition 6.16. Sb is bounded NX → X , and Sb is bounded NX → X . U U W W Condition 6.17. Db is bounded DX → X , and Db is bounded DX → X . U W Condition 6.18. If u ∈ X and (Lub ) U = 0, and if v ∈ X and (Lvb ) W = 0, then U U U U W W W W u = −Db (TrcX u) + Sb (McX u) and v = −Db (TrcX v) + Sb (McX v).

Condition 6.19. If g˙ ∈ NX, then we have the continuity relation U U W W TrcX(Sb g˙ ) − TrcX (Sb g˙ ) = 0. ˙ Condition 6.20. If f ∈ DX, then we have the continuity relation U U ˙ W W ˙ McX(Db f) − McX (Db f) = 0.

Condition 6.21. If g˙ ∈ NX, then we have the jump relation U U W W McX(Sb g˙ ) + McX (Sb g˙ ) = g˙ . ˙ Condition 6.22. If f ∈ DX, then we have the jump relation U U ˙ W W ˙ ˙ TrcX(Db f) + TrcX (Db f) = −f. We now move from well posedness of boundary value problems to invertibility of layer potentials. The following theorem uses an argument of Verchota from [63]. Theorem 6.23. Assume that Conditions 6.19 and 6.21 are valid. Suppose that, ˙ for any f ∈ DX, there is at most one solution u or v to each of the two Dirichlet problems U ˙ U (Lub ) U = 0, TrcX u = f, u ∈ X , W ˙ W (Lvb ) W = 0, TrcX v = f, v ∈ X . U U Then TrcX Sb : NX → DX is one-to-one. If in addition Condition 6.15 is valid and there is a constant C0 such that every U W u ∈ X and v ∈ X with (Lub ) U = 0 and (Lvb ) W = 0 satisfies U W U W kukX ≤ C0k TrcX ukDX , kvkX ≤ C0k TrcX vkDX , U U then there is a constant C1 such that the bound kg˙ kNX ≤ C1k TrcX Sb g˙ kDX is valid for all g˙ ∈ NX. Similarly, assume that Conditions 6.20 and 6.22 are valid. Suppose that for any g˙ ∈ NX, there is at most one solution u or v to each of the two Neumann problems U ˙ U (Lub ) U = 0, McX u = g, u ∈ X , W ˙ W (Lvb ) W = 0, McX v = g, v ∈ X . U U Then McX Db : DX → NX is one-to-one. U If Condition 6.14 is valid and there is a constant C0 such that every u ∈ X and W v ∈ X with (Lub ) U = 0 and (Lvb ) W = 0 satisfies U W U W kukX ≤ C0k McX ukDX , kvkX ≤ C0k McX vkDX , ˙ U U ˙ then there is a constant C1 such that the bound kfkDX ≤ C1k McX Db fkNX is valid ˙ for all f ∈ DX. EJDE-2017/309 LAYER POTENTIALS 19

Proof. As in the proof of Theorem 6.13, we will consider only the relationship between the Neumann problem and the double layer potential. ˙ ˙ U ˙ U ˙ U U Let f, h ∈ DX and let u = Db f, w = Db h. Then u ∈ X and w ∈ X . If U U ˙ U U ˙ U U McX Db f = McX Db h, then McX u = McX w. Because there is at most one solution U U ˙ to the U-Neumann problem, we must have that u = w, and in particular TrcX Db f = U U ˙ TrcX Db h. W W ˙ W W ˙ By Condition 6.20, we have that McX Db f = McX Db h. By uniqueness of solutions to the W-Neumann problem, W W ˙ W W ˙ TrcX Db f = TrcX Db h. By Condition 6.22, we have that ˙ U U ˙ W W ˙ U U ˙ W W ˙ ˙ f = − TrcX Db f − TrcX Db f = − TrcX Db h − TrcX Db h = h U U and so McX Db is one-to-one. Now assume the stronger condition, that is, that C0 < ∞. Because DX is a ˙ quasi-Banach space, if f ∈ DX then by Condition 6.22, ˙ U U ˙ W W ˙ kfkDX ≤ Ck TrcX Db fkDX + Ck TrcX Db fkDX . U U ˙ U U ˙ By definition of Db , Db f ∈ X with (Lb(Db f)) U = 0. By Condition 6.14, U U U ˙ ˙ U k TrcX Db fkDX ≤ C2kDb fkX for some C2. Thus, U W ˙ ˙ U ˙ W kfkDX ≤ CC2kDb fkX + CC2kDb fkX .

By definition of C0, U U U W W W ˙ U ˙ ˙ W ˙ kDb fkX ≤ C0k McX Db fkNX and kDb fkX ≤ C0k McX Db fkNX . W W ˙ U U ˙ By Condition 6.20, McX Db f = McX Db f and so ˙ U U ˙ kfkDX ≤ 2CC2C0k McX Db fkNX as desired.  Finally, we consider the relationship between existence and surjectivity. The following argument appeared first in [21]. Theorem 6.24. Assume that Conditions 6.18, 6.19, and 6.22 are valid. Suppose ˙ that, for any f ∈ DX, there is at least one pair of solutions u and v to the pair of Dirichlet problems U W ˙ U W (Lub ) U = (Lvb ) W = 0, TrcX u = TrcX v = f, u ∈ X , v ∈ X . (6.9) U U Then TrcX Sb : NX → DX is onto. Suppose in addition that Condition 6.15 is valid, and that there is some C0 < ∞ ˙ such that, if f ∈ DX, then there is some pair of solutions u and v to the problem (6.9) with

U ˙ W ˙ kukX ≤ C0kfkDX , kvkX ≤ C0kfkDX . ˙ Then there is a constant C1 such that for any f ∈ DX, there is a g˙ ∈ NX such that U U ˙ ˙ TrcX Sb g˙ = f and kg˙ kNX ≤ C1kfkDX . 20 A. BARTON EJDE-2017/309

Similarly, assume that Conditions 6.18, 6.20, and 6.21 are valid. Suppose that for any g˙ ∈ NX, there is at least one pair of solutions u and v to the pair of Neumann problems U W ˙ U W (Lub ) U = (Lvb ) W = 0, McX u = McX v = g, u ∈ X , v ∈ X . (6.10) U U Then McX Db : DX → NX is onto. If in addition Condition 6.14 is valid, and if there is some C0 < ∞ such that, if g˙ ∈ NX, then there is some pair of solutions u and v to the problem (6.10) with

U W kukX ≤ C0kg˙ kNX , kvkX ≤ C0kg˙ kNX , (6.11) ˙ then there is a constant C1 such that for any g˙ ∈ NX, there is an f ∈ DX such that U U ˙ ˙ McX Db f = g˙ and kfkDX ≤ C1kg˙ kNX . Proof. As usual we present the proof for the Neumann problem. Choose some g˙ ∈ NX and let u and v be the solutions to the problem (6.10) assumed to exist. (If C0 < ∞ we further require that the bound (6.11) be valid.) ˙ U ˙ W By definition of TrcX, f = TrcX u and h = TrcX v exist and lie in DX. By Condition 6.18, U W 2g˙ = McX u + McX v U U ˙ U W W ˙ W = McX(−Db f + Sb g˙ ) + McX (−Db h + Sb g˙ ).

U W By Conditions 6.20 and 6.21 and linearity of the operators McX, McX , we have that U U ˙ U U U U ˙ U U 2g˙ = − McX Db f + McX Sb g˙ − McX Db h + g˙ − McX Sb g˙ U U ˙ ˙ = g˙ − McX Db (f + h).

U U Thus, McX Db is surjective. If C0 < ∞, then because DX is a quasi-Banach space and by Condition 6.14, ˙ ˙ kf + hkDX ≤ CC0kg˙ kNX for some constant C, as desired. 

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Ariel Barton Department of Mathematical , 309 SCEN, University of Arkansas, Fayetteville, AR 72701, USA E-mail address: [email protected]