Finite energy solutions of Maxwell’s equations and constructiveHodge decompositions on nonsmooth Riemannian manifolds ∗

Dorina Mitrea† and Marius Mitrea‡

July, 1999

1 Introduction

W. V. D. Hodge’s seminal work on harmonic integrals in the 1930’s (cf. [15]) has had a lastinginfluence and profound implications in analysis. One specific direction, potential theory in Rieman-nian manifolds (with boundary) involves the study of natural boundary problems for the HodgeLaplacian. When all structures involved are smooth these problems turn out to be regular elliptic;thus the powerful machinery of pseudodifferential operators and Calderon-Zygmund theory applies.

On the other hand, relaxing the smoothness hypotheses can fundamentally alter the very natureof the problems under discussion and new techniques have to be developed to deal with the emergentdifficulties. For example, in the presence of boundary irregularities the relevant operators may failto be pseudodifferential and, typically, can only be described in terms of singular integrals.

One approach which avoids some of these problems and which has been successfully employedby a number of authors in the 1950’s ([37], [38], [35], [36], [22], [5], [13], [14], [7], [1]) is basedon calculus of variations. The most relaxed smoothness assumptions are due to C. B. Morrey; herequires a metric tensor whose coefficients are only Lipschitz continuous but the boundary of thedomain has to be of class C1,1.

More recent developments include [28] where such classical issues have been tackled from themodern perspective of Calderon’s program. Originating in A. P. Calderon’s pioneering work in thelate 1950’s and early 1960’s, the trademark of this program is the systematic use of harmonic analy-sis techniques in order to obtain sharp results in PDE’s. A broader survey of related developments,up to the early 1990’s, is to be found in C. E. Kenig’s book [21].

The goal of this paper is to continue the line of research initiated in [28] with a special emphasison Hodge theory and Maxwell’s equations in Lipschitz subdomains of Riemannian manifolds, whichare two intimately interrelated topics.

The main question that we address is that of the effectiveness of the method of layer potentialsin combination with or as a substitute for the variational approach to global PDE’s (as arising in,e.g., the context of Hodge theory; cf. [36]). The attractive features of the integral equation methodare its constructiveness (which in some circumstances allows for a better understanding of the finerproperties of solutions) and its adaptability to situations when nonsmooth, non-Hilbert structures(such as Lipschitz boundaries, nonsmooth metric tensors, Lp data) are present.

∗1991Mathematics Subject Classification. Primary 31C12, 58A14, 42B20; Secondary 58J32, 31B10, 78A30Key words: Hodge decompositions, Lipschitz domains, Maxwell’s equations, Lp, layer potentials

†Partially supported by a University of Missouri Research Board Grant‡Partially supported by NSF grant DMS-9870018

1

In its classical form (cf., e.g., [41], [44] and the references therein) the Hodge decompositiontheorem states that if Ω is a smooth subdomain of a smooth, compact Riemannian manifold M,then any smooth differential form u in Ω can be decomposed as

u = dα + δβ + γ, (1.1)

where α, β, γ are smooth differential forms in Ω, two of which have a vanishing normal or tangentialcomponent on ∂Ω, and so that dγ = δγ = 0. Here d is the exterior differential operator and δ itsformal adjoint. One convenient approach to (1.1) is to let γ be the projection of u on the space of so-called harmonic fields and to take α := δGu, β := dGu. In this scenario, G is defined as the inverseof ∆, the Hodge-Laplacian, with suitable boundary conditions in Ω (such as relative or absoluteboundary conditions; cf. [41]). Lp estimates for α and β then follow from the observation thatGu solves a regular elliptic BVP (i.e., an elliptic boundary value problem satisfying the Lopatinskicondition). Such an approach is developed in, e.g., [41], [40].

If all structures involved are smooth then G is a pseudodifferential operator of order −2. Inparticular, with Hs,p denoting the usual class of Lp-based Sobolev spaces,

G, dG, δG : L2(Ω) −→ H1,2(Ω), (1.2)

dδG, δdG : L2(Ω) −→ L2(Ω), (1.3)

are bounded operators and, moreover, L2 can be replaced by Lp, 1 < p < ∞; cf. [37]. In thiscontext, a natural issue is to investigate the extent to which (1.2)–(1.3) remain valid if ∂Ω isallowed to have irregularities.

A first result which stands in sharp contrast with the situation for smooth domains describedabove is as follows. Assume that M is equipped with a Riemannian metric tensor with coefficientsin H2,r for some r > dimM. Also, let Ω ⊆ M be a Lipschitz domain, i.e. a domain whoseboundary can be described in local coordinates by means of graphs of Lipschitz functions. Underthese assumptions, we prove that the corresponding Green operator satisfies

G, dG, δG : L2(Ω) −→ H1/2,2(Ω) (1.4)

and the exponent 1/2 is optimal in the class of all Lipschitz domains. The Lp-analogue of (1.4) is

G, dG, δG : Lp(Ω) −→ B1/p,p∗p (Ω), 2− ε < p < 2 + ε, (1.5)

where ε = ε(Ω) > 0, Bs,qp (Ω) is the class of Besov spaces on Ω and, throughout the paper, p∗ :=

max p, 2. It should be pointed out, however, that with an additional convexity assumption on thedomain Ω, (1.2) remains valid even if ∂Ω is only Lipschitz. See §6 for a more precise statement.

Going further, (1.3) continues to hold for arbitrary Lipschitz subdomains of the manifold Mas simple Hilbert space methods show, but the case p 6= 2 is considerably more challenging andsubtle. In this paper we prove that for any Lipschitz domain Ω ⊆ M there exists ε = ε(Ω) > 0 sothat

dδG, δdG : Lp(Ω) −→ Lp(Ω) (1.6)

are bounded for each 2 − ε < p < 2 + ε. While deciding the optimal range of validity for (1.6)in general remains an open problem for the moment, in the class of Lipschitz domains there arenecessary restrictions. For example, the recent work in [10], [33] shows that for any p /∈ [3/2, 3],there exists a Lipschitz domain Ω so that the aforementioned operators are not bounded in Lp(Ω),

2

even at the level of 1-forms (or vector fields). Note that, in particular, these counterexamples provethat the operators dδG and δdG are not of Calderon-Zygmund type if ∂Ω contains irregularities.

Our strategy for proving the results mentioned above is to express the Green operator G interms of explicit integral operators and inverses of boundary singular integral operators (SIO’s) ofCalderon-Zygmund type. The lack of an algebra structure or that of a symbolic calculus for suchSIO’s in the nonsmooth setting complicates matters and is responsible for some of the limitationsalluded to before.

A brief outline of the main steps involved in this program is as follows:

L2-Hodgetheory

⇒ theory of finite L2-energyMaxwell’s equations

⇒ theory of finite Lp-energyMaxwell’s equations

⇒ Lp-Hodgetheory.

We elaborate on this scheme below. To begin with, recall that at the level of differential forms,Maxwell’s system in Ω reads

(Maxwell)

dE − ikH = 0 in Ω,

δH + ikE = 0 in Ω,

ν ∧ E = prescribed on ∂Ω,

(1.7)

where k (the so-called wave number) is a fixed complex parameter, ν ∈ T ∗M is the unit conormalto ∂Ω and ∧ is the usual exterior product of forms. The (global) L2-energy of a solution (E, H) isgiven by

∫ ∫

Ω

[

|E|2 + |H|2]

dVol, (1.8)

where dVol is the volume element on M. See, e.g., [6], Vol. I, pp. 96-97 or Vol. III, p. 240, [2], p. 68and [41], Vol. I, p. 169. Its finiteness simply indicates that E, H ∈ L2(Ω). Finite Lp-energy solutionsof (1.7) are defined analogously, replacing the L2 norm by the Lp norm.

Now, the L2 Hodge theory on Lipschitz domains (itself a consequence of standard variationalprinciples) allows one to characterize the natural boundary space, call it X 2(∂Ω), for which (1.7)is well-posed in the L2 context, in an arbitrary Lipschitz domain Ω. A direct attack of this resultbased on integral methods would have required inverting a boundary operator of the form 1

2I +Mk(where I is the identity and Mk is a certain SIO, the so-called magnetostatic operator) in the spaceX 2(∂Ω), which is a difficult task to perform directly. However, having disposed first of the L2

well-posedness of (1.7) enables us to reverse the process and prove the former as a consequenceof the latter (‘reverse engineering’). The main tool we use to accomplish this goal, that is, theinvertibility of the operator 1

2I + Mk on X 2(∂Ω), is an ad-hoc adaptation and extension of whathas been occasionally called Rellich type estimates; see, e.g., [21] and the references therein for apresentation of this technique in more classical circumstances. Here we also make essential use ofinvertibility results for the operator 1

2I + Mk on (appropriate subspaces of) L2(∂Ω), first proved in[28] and which we recall in §2.

Next, we observe that there is actually a more general class of spaces X p(∂Ω)1<p<∞ on whichthe magnetostatic operator is well-defined, linear and bounded. At this stage, we would like to usesome well-known stability results (cf. [20] for a discussion) in order to conclude that

the operator 12I + Mk is invertible on X p(∂Ω) for |2− p| < ε. (1.9)

For this segment in our analysis to work, we need to prove that X p(∂Ω)1<p<∞ is a complexinterpolation scale, i.e.

3

[X p0(∂Ω),X p1(∂Ω)]θ = X p(∂Ω), 0 < θ < 1, 1/p := (1− θ)/p0 + θ/p1. (1.10)

Thus, we run into a so-called “subspace interpolation problem” which is not straightforward sincethere is no standard way to handle such an issue.

Having dealt with (1.10) we then retrace our steps (cf. the discussion in the L2 setting from theprevious paragraph) and, armed with (1.9), we now prove the well-posedness of Maxwell’s system(1.7) in the class of forms of finite Lp-energy in arbitrary Lipschitz domains if |p− 2| is small. Thisresult also turns out to be the right ingredient for developing a constructive Hodge theory in the Lp

context. In fact, the cycle is continued by returning to (1.7) and using Lp-Hodge theory in orderto solve the “full” Lp-Poisson problem associated with the Maxwell system.

The plan of the paper is as follows. Section 2 contains basic definitions and notation, as wellas a collection of preliminary results. Among other things, here we recall some of the main resultsfrom [32], [28] regarding the nature of the diagonal singularity of the Schwartz kernel of (∆−V )−1

(where V is a suitable scalar potential used to counteract the natural topological obstructions toinverting ∆) and the Cauchy-like operators on Lipschitz submanifolds of codimension one in Massociated with this kernel. This is an extension to the variable coefficient case of the Euclideantheory from [3].

Because of our low regularity assumptions we are led to considering some special boundaryspaces of differential forms, well adapted to the problems we plan to solve. We do this in Section 3,where we discuss the space X p(∂Ω), with Ω Lipschitz subdomain of M and 1 < p < ∞, and provethat this is an interpolation scale for the complex method.

The study of the interplay between Maxwell’s equations and Hodge theory is initiated in Section4. Here we discuss finite energy solutions for Maxwell’s equations (1.7). This is done by reducingthe original system to a boundary integral equation involving the magnetostatic operator in thespace X p(∂Ω). In this scenario, a key result is (1.9).

Section 5 contains an extensive discussion of the Green operators associated with the Hodge-Laplacian and the Hodge-Dirac operator. Our approach is constructive, i.e. based on explicitintegral representation formulas. The main novel point is to analyze the effect of nonsmoothstructures on the mapping properties of these operators; cf. also the previous comments.

Finally, Section 6 is reserved for discussing a pleiad of decompositions of Lp differential formsin arbitrary domains. We do this in a rather unified way and, again, the constructive aspect issystematically emphasized.

Acknowledgments. It is a pleasure to thank Rene Grognard for interesting discussions pertainingto the physical interpretation of some problems discussed in this paper.

2 Basic definitions, notation and preliminary results

Let M be a smooth, oriented, connected, compact, boundaryless manifold of real dimension m.We equip M with a metric tensor

g =∑

j,k

gjkdxj ⊗ dxk, (2.1)

whose coefficients are Lipschitz continuous. As is customary, take (gjk)j,k to be the matrix inverseto (gjk)j,k, and set g := det [(gjk)j,k]. Also, let dVol denote the corresponding volume element, sothat, locally, dVol =

√g dx.

4

Denote by TM the tangent bundle to M and by Λ`TM its `-th exterior power. Sections in thislatter vector bundle are `-differential forms and can be described in local coordinates (x1, ..., xm)as u =

∑

|I|=` uI dxI . Here the sum is performed over ordered `-tuples I = (i1, ..., i`), 1 ≤ i1 < i2 <· · · < i` ≤ m and, for each such I, dxI := dxi1 ∧ · · · ∧ dxi` . Also, the wedge stands for the usualexterior product of forms, while |I| denotes the cardinality of I.

The Hermitian structure in the fibers on TM extends naturally to T ∗M by setting 〈dxj , dxk〉x :=gjk(x). The latter further induces a Hermitian structure on Λ`TM by selecting ωI|I|=` to be anorthonormal frame in Λ`TM provided ωj1≤j≤m is an orthonormal frame in T ∗M (locally). Wedenote by 〈·, ·〉 the corresponding (pointwise) inner product.

Going further, introduce the Hodge star operator as the unique vector bundle morphism ∗ :Λ`TM→ Λm−lTM such that

u ∧ (∗u) = |u|2 dVol, ∀u ∈ Λ`TM. (2.2)

Here we regard dVol as an m-form on M, making use of the orientation we assume M has. Then,define the interior product between a 1-form α and an `-form u (i.e. contraction of u by α) bysetting

α ∨ u := (−1)(`−1)m ∗ (α ∧ ∗u). (2.3)

Let d stand for the (exterior) derivative operator and denote by δ its formal adjoint (with respectto the metric introduced above). In particular, if Ω is a reasonable subdomain of M with outwardunit conormal ν ∈ T ∗M, surface measure dσ, and u ∈ C1(Ω,Λ`TM), v ∈ C1(Ω, Λ`+1TM), then

∫ ∫

Ω〈du, v〉 dVol−

∫ ∫

Ω〈u, δv〉 dVol =

∫

∂Ω〈ν ∧ u, v〉 dσ =

∫

∂Ω〈u, ν ∨ v〉 dσ. (2.4)

For further reference some basic properties of these objects are summarized in the following lemma.

Lemma 2.1 For arbitrary one-forms α, β, and any `-form u, (m − `)-form v, and (` + 1)-formw, the following are true:

(1) ∗ ∗ u = (−1)`(m−`) u;

(2) 〈u, ∗v〉 = (−1)`(m−`)〈∗u, v〉 and 〈∗u, ∗v〉 = 〈u, v〉;

(3) α ∧ (α ∧ u) = 0 and α ∨ (α ∨ u) = 0;

(4) α ∧ (β ∨ u) + β ∨ (α ∧ u) = 〈α, β〉u;

(5) 〈α ∧ u,w〉 = 〈u, α ∨ w〉;

(6) ∗(α ∧ u) = (−1)`α ∨ (∗u) and ∗(α ∨ u) = (−1)`−1α ∧ (∗u).

Moreover, if α is normalized such that 〈α, α〉 = 1, then also:

(7) u = α ∧ (α ∨ u) + α ∨ (α ∧ u);

(8) |α ∧ (α ∨ u)| = |α ∨ u| and |α ∨ (α ∧ u)| = |α ∧ u|.

Finally,

(9) d d = 0, δ δ = 0;

5

(10) δ = (−1)m(`+1)+1 ∗ d∗ and ∗δ = (−1)`d∗, δ∗ = (−1)`+1 ∗ d on `-forms.

The Hodge Laplacian on `-forms is defined as

∆` := −(dδ + δd). (2.5)

Note that ∆` =∑

j,k gjk∂xj∂xk +∑

j Pj(g)∂xj + Q(g), where Pj(g) and Q(g) are matrix-valuedfunctions with entries depending polynomially on ∂αgjk, ∂αgjk for, respectively, |α| ≤ 1 and |α| ≤ 2.Since we want this operator to have reasonably smooth coefficients, while at the same time tryingto limit the amount of smoothness allowed, we shall assume from now on that

the Riemannian metric tensor g has coefficients in H2,r, r > m, (2.6)

unless otherwise specified. Hereafter, Hs,p will stand for the (Lp-style) Sobolev scale of spaces. An`-form u is called harmonic if ∆`u = 0 in the sense of distributions.

A domain Ω ⊂ M is called Lipschitz provided its boundary is locally described by graphs ofLipschitz functions. See, e.g., [32] for more on this. Fix such a Lipschitz domain Ω and recall thatν ∈ T ∗M stands for the outward unit conormal to ∂Ω. A measurable section f : ∂Ω → Λ`TM iscalled tangential if ν ∨ f = 0 a.e. on ∂Ω, and normal if ν ∧ f = 0 on ∂Ω. We define the spaces

Lptan(∂Ω,Λ`TM) := v ∈ Lp(∂Ω, Λ`TM); ν ∨ v = 0, (2.7)

Lpnor(∂Ω, Λ`TM) := v ∈ Lp(∂Ω,Λ`TM); ν ∧ v = 0, (2.8)

which are dense, closed subspaces of Lp(∂Ω,Λ`TM).An `-form f ∈ Lp

tan(∂Ω, Λ`TM), 1 < p < ∞, is said to have its boundary (exterior) co-derivativein Lp if there exists an (`− 1)-form in Lp(∂Ω, Λ`−1TM), which we denote by δ∂f , so that

∫

∂Ω〈dψ, f〉 dσ =

∫

∂Ω〈ψ, δ∂f〉 dσ for any ψ ∈ C1(M,Λ`−1TM). (2.9)

We set

Lp,δtan(∂Ω,Λ`TM) :=

f ∈ Lptan(∂Ω,Λ`TM); δ∂f ∈ Lp(∂Ω,Λ`−1TM)

, (2.10)

and equip it with the natural norm

‖f‖Lp,δtan(∂Ω,Λ`TM) := ‖f‖Lp(∂Ω,Λ`TM) + ‖δ∂f‖Lp(∂Ω,Λ`−1TM). (2.11)

It is immediate that δ∂ is a local operator, i.e. supp (δ∂f) ⊆ supp f for any form f in Lp,δtan(∂Ω,Λ`TM).

Hence, in the smooth context, δ∂ is a first order differential operator. However, its coefficients wouldinvolve second order derivatives of the transition functions between local charts in the manifold ∂Ωand, hence, such a description is no longer possible if ∂Ω is merely Lipschitz. It is precisely forcircumventing this problem that we resort to the distributional definition above. In the Euclideansetting, this definition was first introduced in [29], [30] and subsequently utilized in [17], [23], [25].

It is not difficult to check that

δ∂δ∂f = 0 and ν ∨ δ∂f = 0 on ∂Ω for any f ∈ Lp,δtan(∂Ω,Λ`TM). (2.12)

For f ∈ Lpnor(∂Ω, Λ`TM), we define the distribution d∂f by requiring that

6

∫

∂Ω〈δψ, f〉 dσ =

∫

∂Ω〈ψ, d∂f〉 dσ for any ψ ∈ C1(M, Λ`+1TM). (2.13)

Set

Lp,dnor(∂Ω,Λ`TM) :=

f ∈ Lpnor(∂Ω, Λ`TM); d∂f ∈ Lp(∂Ω, Λ`+1TM)

, (2.14)

equipped with the natural norm

‖f‖Lp,dnor(∂Ω,Λ`TM) := ‖f‖Lp(∂Ω,Λ`TM) + ‖d∂f‖Lp(∂Ω,Λ`+1TM). (2.15)

Analogously to (2.12), we have that

d∂d∂f = 0 and ν ∧ d∂f = 0 on ∂Ω for any f ∈ Lp,dnor(∂Ω, Λ`TM). (2.16)

Let Bs,qp (∂Ω), 0 < |s| < 1, 1 < p, q < ∞, be the usual scale of Besov spaces on ∂Ω. As usual, we

shall simplify the notation a bit when p = q and in this case simply write Bs,p(∂Ω). In particular,the trace map

Tr : Hs,p(Ω) −→ Bs−1/p,p(∂Ω) (2.17)

is well-defined and bounded for 1 < p < ∞, 1/p < s < 1 + 1/p, and has a bounded right inverse(Gagliardo’s lemma). Also, Bs,q

p (Ω), s > 0, 1 < p < ∞, will denote the class of Besov spaces on Ω.More detailed accounts on these matters can be found in [19], [18], [34].

Next, fix some positive, not identically zero function V ∈ C∞(M). As in [28], under the currentassumptions, the operator

∆` − V : H1,2(M, Λ`TM) −→ H−1,2(M,Λ`TM) (2.18)

has an inverse, (∆`−V )−1, whose Schwartz kernel, Γ`(x, y), is a symmetric double form of bidegree(`, `). In local coordinates, in which the metric tensor is given by (2.1) we can set

E0` (x− y, y) := Cn

(∑

j,k

gjk(y)(xj − yj)(xk − yk))−(m−2)/2

γ`(x, y) (2.19)

for appropriate C = Cm, where γ` is the double form of bidegree (`, `) given by

γ`(x, y) :=

∑

|I|=`∑

|J |=` det ((gij(y))i∈I,j∈J) dxI ⊗ dyJ , if ` ≥ 1,

1, if ` = 0.(2.20)

Note that E0` (z, y) is smooth and homogeneous of degree −(m − 2) in z ∈ Rm \ 0 and C2+µ in

y, for some µ > 0. Then define the remainder E1` (x, y) so that

Γ`(x, y)√

g(y) = E0` (x− y, y) + E1

` (x, y). (2.21)

Below we collect some useful estimates for E1` (x, y) and its derivatives. They follow from Proposi-

tions 2.5 and 2.8 of [28].

Proposition 2.2 For each ` ∈ 0, 1, ..., m, the remainder E1` (x, y) satisfies

|∇jx∇k

yE1` (x, y)| ≤ C|x− y|−(m−3+j+k), (2.22)

for each j, k ∈ 0, 1.

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To be more exact, this estimate has been obtained in [28] with m − 3 + j + k + ε in place ofm − 3 + j + k where ε > 0 is as small as we want. However, one can get rid of ε by working inthe context of Morrey spaces and invoking the results in [42]. We owe this observation to MichaelTaylor.

In the sequel, we shall also need information about the “commutators” between d, δ on theone hand and the forms Γ`(x, y) on the other hand. This is made precise in the proposition below,which is taken from §6 of [28].

Proposition 2.3 There exists a double form R`(x, y) of bidegree (`, ` + 1) so that

R` ∈ C1+µloc ((M×M\ diag) ∪ (x, y); x /∈ supp dV ), some µ > 0,

|∇jx∇k

yR`(x, y)| ≤ C|x− y|−(m−4+j+k), 0 ≤ j, k ≤ 1, (2.23)

and so that

δx(Γ`+1(x, y)) = dy(Γ`(x, y)) + R`(x, y). (2.24)

Let us point out that R`(x, y) ≡ 0 if the potential V is constant.Analogously, we have

dx(Γ`(x, y)) = δy(Γ`+1(x, y)) + Q`(x, y) (2.25)

where Q`(x, y) is a double form of bidegree (` + 1, `) which exhibits a similar behavior; in factQ`(x, y) = −R`(y, x). Once again, Q`(x, y) ≡ 0 if the potential V is constant.

Next, denote by S` the single layer potential operator on ∂Ω with kernel Γ`(x, y), i.e.,

S`f(x) :=∫

∂Ω〈Γ`(x, y), f(y)〉 dσ(y), x ∈M \ ∂Ω, (2.26)

where f ∈ Lp(∂Ω, Λ`TM). Note that (∆` − V )S`f = 0 in M\ ∂Ω. Also, set S`f := S`f |∂Ω.Going further, let us introduce the principal value singular integral operators

M`f(x) := p.v.∫

∂Ω〈ν(x) ∨ dxΓ`(x, y), f(y)〉 dσ(y), x ∈ ∂Ω, (2.27)

and

N`f(x) := p.v.∫

∂Ω〈ν(x) ∧ δxΓ`(x, y), f(y)〉 dσ(y), x ∈ ∂Ω. (2.28)

Here, p.v.∫

∂Ω . . . is taken in the sense of removing geodesic balls (with respect to some smoothbackground metric); see [32] for more details.

These operators, which are the higher degree analogue of the so-called magnetostatic and elec-trostatic operators arising in scattering theory in R3 (cf., e.g., [4]), have been studied in detail in[28]. For further reference, some basic properties are collected in the theorems stated below. Forproofs, the reader is referred to [28], [34]. More related material is in [25], [26]. Before stating them,let us mention that all restrictions to the boundary of ∂Ω are taken in the pointwise nontangentialsense. That is,

u∣

∣

∣

∂Ω(x) := lim

y∈γ(x), y→xu(y), x ∈ ∂Ω, (2.29)

8

where γ(x) ⊆ Ω is an appropriate nontangential approach region. Finally, N is going to denote thenontangential maximal operator defined on some `-form u in Ω by

Nu(x) := sup |u(y)|; y ∈ γ(x), x ∈ ∂Ω. (2.30)

Theorem 2.4 Let Ω ⊂M be a Lipschitz domain. Then for each 1 < p < ∞ we have

‖N (S`f)‖Lp(∂Ω), ‖N (dS`f)‖Lp(∂Ω), ‖N (δS`f)‖Lp(∂Ω) ≤ C‖f‖Lp(∂Ω,Λ`TM), (2.31)

uniformly for f ∈ Lp(∂Ω, Λ`TM), and

‖N (S`f)‖Lp(∂Ω) ≤ C‖f‖H−1,p(∂Ω,Λ`TM), (2.32)

uniformly for f ∈ H−1,p(∂Ω, Λ`TM).Also, with boundary traces taken in the pointwise nontangential sense, the following jump-

relations are valid:

ν ∨ dS`f∣

∣

∣

∂Ω±= ∓1

2(ν ∨ (ν ∧ f)) + M`f, ν ∧ δS`f∣

∣

∣

∂Ω±= ±1

2(ν ∧ (ν ∨ f)) + N`f (2.33)

a.e. on ∂Ω for each f ∈ Lp(∂Ω,Λ`TM) and 1 < p < ∞.Moreover,

δS`f = S`−1(δ∂f) +∫

∂Ω〈R`−1(x, ·), f(y)〉 dσ(y), ∀ f ∈ Lp,δ

tan(∂Ω,Λ`TM), (2.34)

dS`g = S`+1(d∂g) +∫

∂Ω〈Q`(x, ·), g(y)〉 dσ(y), ∀ g ∈ Lp,d

nor(∂Ω, Λ`TM), (2.35)

and

M`∗ = − ∗Nm−` and ∗M` = −Nm−` ∗ on `-forms. (2.36)

Also, for any constant potential V , the adjoint of M` acting on Lptan(∂Ω, Λ`TM) is the operator

M t` acting on Lq

tan(∂Ω,Λ`TM), with 1/p + 1/q = 1, given by

M t` = ν ∨N`+1(ν ∧ ·). (2.37)

Finally, for 1 < p < ∞, 0 ≤ s ≤ 1 and 0 ≤ ` ≤ m, the operator

S` : H−s,p(∂Ω, Λ`TM) −→ B1−s+1/p,p∗p (Ω,Λ`TM) (2.38)

is well defined and bounded. Hereafter, we set p∗ := max p, 2.

Theorem 2.5 Let Ω ⊂ M be a Lipschitz domain. There exists ε = ε(Ω) > 0 so that for each` ∈ 0, 1, ...,m and each 2− ε < p < 2 + ε,

the operators ± 12I + M` are Fredholm with index zero

on the spaces Lp,δtan(∂Ω,Λ`TM) and Lp

tan(∂Ω,Λ`TM) (2.39)

and

9

the operators ± 12I + N` are Fredholm with index zero

on the spaces Lp,dnor(∂Ω,Λ`TM) and Lp

nor(∂Ω, Λ`TM). (2.40)

Moreover, there exists a real, discrete set of values U , with no finite accumulation point, such thatthe same operators above are in fact isomorphisms (on the respective spaces) if V (x) ≡ ik for somek ∈ C \ U .

We shall also need the following regularity result from [28], [34].

Theorem 2.6 For any Ω arbitrary Lipschitz domain in M there exists ε = ε(Ω) > 0 with thefollowing significance. Assume that 2−ε < p < 2+ε, 0 ≤ l ≤ m and that the l-differential form u ∈Lp(Ω,Λ`TM) has, in the sense of distributions, du ∈ Lp(Ω,Λ`+1TM) and δu ∈ Lp(Ω,Λ`−1TM).

Then the following are equivalent:

(i) ν∨u, initially considered as a distribution (cf. (3.4)) belongs, in fact, to the space Lp(∂Ω, Λ`−1TM)(and, hence, to Lp

tan(∂Ω, Λ`−1TM));

(ii) ν ∧ u, initially considered as a distribution, actually belongs to Lp(∂Ω,Λ`+1TM) (and, thus,to Lp

nor(∂Ω, Λ`−1TM)).

Moreover, if (i) or (ii) above is valid, then u ∈ B1/p,p∗p (Ω, Λ`TM) (recall that p∗ := max p, 2).

Also, naturally accompanying estimates are valid in each case.

We discuss one more result which is going to be of importance for us later on. To state it, definethe Newtonian (volume) potential

Π`u(x) :=∫ ∫

Ω〈Γ`(x, y), u(y)〉 dVol(y), x ∈ Ω, (2.41)

and recall the parameter r > m := dimM from (2.1).

Proposition 2.7 Assume that the metric tensor on M satisfies (2.1) and that Ω ⊆M is a Lips-chitz domain. Then, for V as before,

(i) Π` : Lp(Ω, Λ`TM) → H2,p(Ω, Λ`TM) is bounded for every 1 < p < r.

(ii) S` : B− 1p ,p(∂Ω, Λ`TM) → H1,p(Ω,Λ`TM) is bounded for every r

r−1 < p < ∞.

Proof. Part (i) is proved in [28], [34]. Part (ii) follows from (i) and duality. Specifically, from(2.21) and Proposition 2.2 it follows that ∂xiΓ`(x, y) = −∂yiΓ`(x, y) + Ri(x, y) where |Ri(x, y)| +|x − y||∇Ri(x, y)| = O(|x − y|−(m−3)). Thus, if u ∈ C∞

comp(Ω,Λ`TM) and f ∈ B− 1p ,p(∂Ω,Λ`TM)

have sufficiently small supports but are otherwise arbitrary then, ∀ 1 ≤ i ≤ m,

∣

∣

∣

∣

∫ ∫

Ω〈∂iS`f, u〉 dVol

∣

∣

∣

∣

=∣

∣

∣

∣

∫

∂Ω〈Tr [∂iΠ`u], f〉 dσ

∣

∣

∣

∣

+O(‖u‖Lq(Ω)‖f‖B−

1p ,p(∂Ω)

)

≤ ‖f‖B−

1p ,p(∂Ω)

‖Tr [∂iΠ`u]‖B1− 1

q ,q(∂Ω)+O(‖u‖Lq(Ω)‖f‖

B−1p ,p(∂Ω)

)

≤ C‖f‖B−

1p ,p(∂Ω)

‖u‖Lq(Ω), (2.42)

10

by (i) and Gagliardo’s trace lemma. 2

The reader should be aware that, in order to simplify the notation, we may occasionally dropthe dependence of the various norms on the exterior power bundle (as already done in (2.42)).

3 Distinguished spaces of boundary forms

As far as the operator d∂ is concerned, Lpnor(∂Ω, Λ`TM) and Lp,d

nor(∂Ω,Λ`TM) can be thought of asLp-Sobolev spaces of order zero and one, respectively. In this section, our aim is to introduce somerelated spaces which are well adapted to finite energy solutions of Maxwell’s equations in Lipschitzdomains. Homogeneity considerations dictate that such boundary spaces should be, in the previous(informal) description, Lp-Sobolev spaces of order −1/p for d∂ .

A direct attempt to do just that, i.e. define one such space as the collection of all f ∈B− 1

p ,p(∂Ω, Λ`TM) for which ν ∧ f = 0 and d∂f ∈ B− 1p ,p(∂Ω, Λ`+1TM) would only work in the

smooth setting. In the case we are interested in, i.e. when ∂Ω is merely Lipschitz, there are obviousproblems in making sense of the above conditions. It is precisely because of these difficulties thatwe adopt the more convoluted construction below.

As in §2 we let M be a smooth, compact, connected, boundaryless manifold of real dimensionm. Equip this with a Lipschitz metric tensor g and denote by d the exterior derivative operatorand by δ its formal transpose, i.e. δ = ∗d∗, where ∗ is the Hodge star operator associated with g.For Ω a Lipschitz domain of M we set Ω+ := Ω, Ω− := M\ Ω. If 1 < p < ∞, 0 ≤ ` ≤ m, define

Dp` (d, Ω) := u ∈ Lp(Ω, Λ`TM); du ∈ Lp(Ω, Λ`+1TM), (3.1)

Dp` (δ,Ω) := u ∈ Lp(Ω, Λ`TM); δu ∈ Lp(Ω, Λ`−1TM). (3.2)

The action of d and δ is taken in the sense of distributions. Friedrichs’s identity of weak and strongextensions of first order PDO’s ([12]) shows that

C∞(Ω,Λ`TM) → Dp` (d, Ω) densely, (3.3)

where the last space is considered equipped with the natural graph norm.Consider now 1 ≤ ` ≤ m and u ∈ Lp(Ω, Λ`TM) such that δu ∈ Lp(Ω, Λ`−1TM) for some

1 < p < ∞. Then we define the distribution ν ∨ u on ∂Ω by requiring that

〈ν ∨ u, ϕ〉 := −∫ ∫

Ω〈δu, v〉 dVol +

∫ ∫

Ω〈u, dv〉 dVol, (3.4)

for any v ∈ H1,q(Ω, Λ`−1TM), 1/p + 1/q = 1, with Tr v = ϕ. Thus, the right side of (3.4) is welldefined for ϕ ∈ B1/p,q(∂Ω,Λ`−1TM), independently of the choice of such v, so we have

ν ∨ u ∈ B−1/p,p(∂Ω, Λ`−1TM) (3.5)

with naturally accompanying estimates. If u is a zero-form then we set ν ∨ u := 0. Further, ifu ∈ Lp(Ω, Λ`TM) is such that du ∈ Lp(Ω, Λ`+1TM) then we can define the distribution ν ∧ u bya similar procedure inspired by (2.4). Once again,

ν ∧ u ∈ B−1/p,p(∂Ω,Λ`+1TM) (3.6)

plus natural estimates. It follows that

11

Dp` (d, Ω) 3 u 7→ ν ∧ u ∈ B− 1

p ,p(∂Ω,Λ`+1TM) (3.7)

and

Dp` (δ,Ω) 3 u 7→ ν ∨ u ∈ B− 1

p ,p(∂Ω, Λ`−1TM) (3.8)

are well-defined and bounded, when the spaces in the left side are equipped with the natural graphnorm. Also, set

Dp` (d∧, Ω) := u ∈ Dp

` (d, Ω); ν ∧ u = 0, (3.9)

Dp` (δ∨, Ω) := u ∈ Dp

` (δ,Ω); ν ∨ u = 0. (3.10)

For later use we also introduce

Np` (d, Ω) := u ∈ Dp

` (d, Ω); du = 0, (3.11)

Np` (d∧, Ω) := u ∈ Np

` (d,Ω); ν ∧ u = 0. (3.12)

Similar definitions are given to Np` (δ,Ω) and Np

` (δ∨,Ω). It is immediate that

∗Dp` (d, Ω) = Dp

m−`(δ,Ω), ∗Dp` (δ,Ω) = Dp

m−`(d, Ω), (3.13)

∗Dp` (d∧, Ω) = Dp

m−`(δ∨,Ω), ∗Dp` (δ∨,Ω) = Dp

m−`(d∧,Ω). (3.14)

Next we shall prove that Dp` (d,Ω) and Dp

` (δ,Ω) enjoy a very useful extension property. Specifically,we have:

Proposition 3.1 Given Ω, p, `, as before, there exists a linear map

Dp` (d,Ω) 3 u 7→ u ∈ Dp

` (d,M) (3.15)

so that u|Ω = u and

‖u‖Lp(M) + ‖du‖Lp(M) ≤ C(‖u‖Lp(Ω) + ‖du‖Lp(Ω)). (3.16)

A similar extension property is valid for Dp` (δ,Ω).

In order to prove this proposition, we shall first need a lemma.

Lemma 3.2 Let Ω ⊆ M be a Lipschitz domain and 0 ≤ ` ≤ m. If w ∈ Lp(M,Λ`TM) is suchthat w|Ω± ∈ Dp

` (d, Ω±), then

dw ∈ Lp(M,Λ`TM) ⇔ ν ∧ (w|Ω+) = ν ∧ (w|Ω−) in B− 1p ,p(∂Ω, Λ`+1TM). (3.17)

A similar property is valid for the operator δ.

12

Proof. Let v ∈ H1,q(M,Λ`+1TM), 1p + 1

q = 1, be arbitrary and denote the Sobolev trace of v on

∂Ω by Tr v ∈ B1− 1q ,q(∂Ω, Λ`+1TM). Then

〈ν ∧ (w|Ω+), Tr v〉 =∫ ∫

Ω+

〈dw, v〉 −∫ ∫

Ω+

〈w, δv〉, (3.18)

and

〈ν ∧ (w|Ω−), Tr v〉 = −∫ ∫

Ω−〈dw, v〉+

∫ ∫

Ω−〈w, δv〉. (3.19)

Subtracting the identities (3.18) and (3.19) yields∫ ∫

M〈ω, v〉 −

∫ ∫

M〈w, δv〉 = 〈ν ∧ (w|Ω+)− ν ∧ (w|Ω−), Tr v〉 (3.20)

where ω := d(w|Ω±) in Ω±. Then (3.17) follows easily from this. The last part in the statementfollows from what we have proved so far and (3.13). 2

Proof of Proposition 3.1. In order to show that Dp` (d, Ω) has the extension property described

in the statement of Proposition 3.1, observe that this property localizes via a partition of unityand is stable under pull-back (via bi-Lipschitz functions). Also, by (3.3) and a weak-∗ argument,it suffices to prove the extension property in the following context. Assume that

M≡ Rm, Ω± = Rm± , u ∈ C∞

comp(Rm, Λ`Rm). (3.21)

We are looking for u ∈ Dp` (d,Rm) with u|Rm

±= u and so that (3.16) is valid.

To this end, let Φ : Rm− → Rm

+ be the reflection Φ(x′, xm) := (x′,−xm) and define

u :=

u in Rm+ ,

Φ∗(u|Rm+

) in Rm− ,

(3.22)

where the superscript ∗ stands for pull-back. Clearly, u ∈ Lp(Rm, Λ`Rm), ‖u‖Lp(Rm) ≤ 2‖u‖Lp(Rm+ ),

and u|Rm+

= u. According to Lemma 3.2, in order to conclude that du ∈ Lp(Rm, Λ`+1Rm) we needto check that

dxm ∧ (u|Rm+

) = dxm ∧ (Φ∗(u|Rm+

)|Rm−

) (3.23)

in B− 1p ,p(∂Rm

+ ,Λ`+1TRm). In fact, this holds in the pointwise sense. Indeed, if

u =∑

m∈I

u′IdxI +(

∑

m/∈I

u′′IdxI)

∧ dxm (3.24)

then, since Φ|∂Rm = Id, the identity operator, we have

dxm ∧(

Φ∗(u|Rm+

)|Rm−

)

= dxm ∧[

(∑

m∈I

u′IdxI)− (∑

m/∈I

u′′IdxI) ∧ dxm

]

= dxm ∧ (u|Rm+

), (3.25)

as desired. Obviously, the assignment u 7→ u is linear. Finally,

‖du‖Lp(Rm) ≤ ‖du‖Lp(Rm+ ) + ‖dΦ∗(u|Rm

+)‖Lp(Rm

− ) ≤ 2‖du‖Lp(Rm+ ), (3.26)

so that (3.16) holds. The last part of the statement also follows on account of (3.13). 2

13

Next, for 1 < p < ∞, ` ∈ 0, 1, . . . , m we introduce some subspaces of B−1/p,p(∂Ω,Λ`TM).More concretely, we define

X p`,±(∂Ω) :=

f ∈ B− 1p ,p(∂Ω,Λ`TM); ∃u ∈ Dp

`−1(d, Ω±) so that ν ∧ u = f

, (3.27)

equipped with the natural norm

‖f‖X p`,±(∂Ω) := inf‖u‖Lp(Ω±) + ‖du‖Lp(Ω±); u ∈ Dp

`−1(d, Ω±), ν ∧ u = f (3.28)

and

Yp`,±(∂Ω) :=

g ∈ B− 1p ,p(∂Ω,Λ`TM); ∃ v ∈ Dp

`+1(δ,Ω±) so that ν ∨ v = g

, (3.29)

endowed with the norm

‖g‖Yp`,±(∂Ω) := inf‖v‖Lp(Ω±) + ‖dv‖Lp(Ω±); v ∈ Dp

`+1(δ,Ω±), ν ∨ v = g. (3.30)

We also define the (bounded) operators

d∂ : X p`,±(∂Ω) −→ X p

`+1,±(∂Ω), (3.31)

δ∂ : Yp`,±(∂Ω) −→ Yp

`−1,±(∂Ω), (3.32)

by setting

d∂(ν ∧ u) := −ν ∧ du, u ∈ Dp`−1(d, Ω±), (3.33)

δ∂(ν ∨ v) := −ν ∨ δv, v ∈ Dp`+1(δ,Ω±). (3.34)

Proposition 3.3 Let Ω be Lipschitz, 1 < p < ∞, ` ∈ 0, . . . , m. Then the following hold.

(i) X p`,+(∂Ω) ≡ X p

`,−(∂Ω), in the sense that the two spaces coincide as sets and the norms areequivalent. Thus, from now on, we may drop the subscripts ± and simply write X p

` (∂Ω).

(ii) X p` (∂Ω) is a reflexive Banach space.

(iii) X p` (∂Ω) → B− 1

p ,p(∂Ω,Λ`TM) continuously.

(iv) If the metric tensor satisfies (2.6), then Lp,dnor(∂Ω,Λ`TM) → X p

` (∂Ω) continuously and densely.Moreover d∂ defined in (3.31), (3.33) is compatible with d∂ defined on Lp,d

nor(∂Ω,Λ`TM) in§2.

(v) ∗ : X p`,±(∂Ω)→Yp

m−`,±(∂Ω) is an isomorphism. In particular, Yp`,+(∂Ω) ' Yp

`,−(∂Ω) and wemay drop the subscripts ± and simply write Yp

` (∂Ω). Also, the definition of δ∂ in (3.32),(3.34) agrees with that on the space Lp,δ

tan(∂Ω, Λ`TM) given in §2.

14

(vi) For each 1 < p, q < ∞ conjugate exponents, the mapping

ν ∧ · : Yq` (∂Ω) −→

(

X p`+1(∂Ω)

)∗(3.35)

defined by

〈ν ∧ f, g〉 :=∫ ∫

Ω〈U, dW 〉 dVol−

∫ ∫

Ω〈δU,W 〉 dVol (3.36)

for U ∈ Dq`+1(δ,Ω), with f = ν ∨U and W ∈ Dp

` (d,Ω) with g = ν ∧W is well-defined, in factan isomorphism, for each ` ∈ 0, 1, ..., m.Finally, a similar statement is valid for ν ∨ · : X p

`+1(∂Ω) −→(

Yq` (∂Ω)

)∗.

(vii) For each 1 < p < ∞, X p1 (∂Ω) = ν B1−1/p,p(∂Ω), while Yp

0 (∂Ω) = B−1/p,p(∂Ω).

Proof. For f ∈ X p`,+(∂Ω) arbitrary, fixed, let u ∈ Dp

`−1(d, Ω+) be so that

12

[

‖u‖Lp(Ω) + ‖du‖Lp(Ω)

]

≤ ‖f‖X p`,+(∂Ω) ≤ ‖u‖Lp(Ω) + ‖du‖Lp(Ω) (3.37)

and f = ν∧u in B− 1p ,p(∂Ω, Λ`TM). Let u ∈ Dp

`−1(d,M) be the extension of u as in Proposition 3.1and set g := ν ∧ (u|Ω−) ∈ X p

`,−(∂Ω). Then

‖g‖X p`,−

(∂Ω) ≤ C(‖u‖Lp(Ω) + ‖du‖Lp(Ω)) ≤ C‖f‖X p`,+(∂Ω)

and, by Lemma 3.2,

g = ν ∧ (u|Ω−) = ν ∧ (u|Ω+) = f in B− 1p ,p(∂Ω,Λ`TM).

This proves that X p`,+(∂Ω) → X p

`,−(∂Ω). Similarly, X p`,−(∂Ω) → X p

`,+(∂Ω) and (i) follows. Paren-thetically, let us note that we may arrive at the same conclusion by inspecting the diagram:

Dp`−1(d, Ω+)

Dp`−1(d∧, Ω+)

ν ∧ ·∼- X p

`,+(∂Ω) ⊂- B− 1p ,p(∂Ω,Λ`TM)

Dp`−1(d,Ω−)

Dp`−1(d∧, Ω−)

?ν ∧ ·∼- X p

`,−(∂Ω)?

...............⊂- B− 1

p ,p(∂Ω,Λ`TM)

Id?

(3.38)

where the first vertical arrow is the mapping

Dp`−1(d,Ω+) 3 u 7→ u|Ω− ∈ Dp

`−1(d, Ω−) (3.39)

at the level of quotient spaces. Since this, as well as ν∧·, are isomorphisms, it follows that the dottedarrow (which is naturally induced) is also an isomorphism. Note that (ii), except for reflexivitywhich follows from (vi), and (iii) can be seen directly from (the first row of) the commutativediagram (3.38).

The fact that the inclusion in (iv) is well-defined and continuous is seen as follows. If f belongsto the space Lp,d

nor(∂Ω, Λ`TM), set u± := δS`f in Ω±. Here S` is as in (2.26), correspondingto some fixed, constant potential V > 0. By Theorem 2.4, N (u±), N (du±) ∈ Lp(∂Ω), so that

15

u± ∈ Dp`−1(d, Ω±), and ν ∧ u+ − ν ∧ u− = f in the Lp sense on ∂Ω. This and (i) show that

f ∈ X p` (∂Ω) and

‖f‖X p` (∂Ω) ≤

∑(

‖u±‖Lp(Ω) + ‖du±‖Lp(Ω)

)

≤ C∑

(

‖N (u±)‖Lp(∂Ω) + ‖N (du±)‖Lp(∂Ω)

)

≤ C‖f‖Lp,dnor(∂Ω,Λ`TM). (3.40)

To see that inclusion (iv) is also dense, fix f ∈ X p` (∂Ω), f = ν ∧ u, u ∈ Dp

`−1(d,Ω). By (3.3),there exists a sequence uj ∈ C∞(Ω) so that uj → u and duj → du in Lp(Ω). Then ν ∧ uj ∈Lp,d

nor(∂Ω, Λ`TM) and, by the first row in (3.38), ν ∧ uj → ν ∧ u = f in X p` (∂Ω).

Next, (v) follows from (3.14), Lemma 2.1 and what we have proved so far.As for (vi), it is clear that the map (3.35) is well-defined, linear, bounded and one-to-one. There

remains to shows that it is also onto. To this end, let θ ∈(

X p`+1(∂Ω)

)∗be arbitrary and consider

θ : Dp` (d,Ω) → R defined by θ(u) := θ(ν ∧ u). Thinking of Dp

` (d, Ω) as a (closed) subspace ofLp(Ω, Λ`TM) ⊕ Lp(Ω, Λ`+1TM) via the identification u 7→ (u, du), the Hahn-Banach and Riesztheorems allow us to conclude that there exist v1 ∈ Lq(Ω,Λ`+1TM) and v2 ∈ Lq(Ω, Λ`TM) suchthat

θ(u) =∫ ∫

Ω〈v1, du〉 dVol−

∫ ∫

Ω〈v2, u〉 dVol, ∀u ∈ Dp

` (d,Ω). (3.41)

Note that choosing u ∈ C∞comp(Ω) yields δv1 = v2. In particular v1 ∈ Dq

` (δ,Ω). Utilizing this backin (3.41) gives that θ(ν ∧u) = 〈ν∧ (ν ∨v1), ν ∧u〉, for each u which, in turn, entails ν ∧ (ν∨v1) = θ.Hence, the map (3.35) is onto and this finishes the proof of (vi).

Finally, (vii) follows more or less directly from definitions; we omit the details. 2

Proposition 3.4 Let Ω ⊆M be a Lipschitz domain. Then, for each 0 ≤ ` ≤ m, the classes

X p` (∂Ω)1<p<∞, Yp

` (∂Ω)1<p<∞ (3.42)

are two complex interpolation scales (in the sense of (1.10)).

The proof of this proposition rests on several results of independent interest which we now state.

Lemma 3.5 Let Ω ⊆M be a Lipschitz domain. Then, for each 0 ≤ ` ≤ m, the family of Banachspaces (equipped with the graph norm) Dp

` (d,Ω)1<p<∞ is a complex interpolation scale.

Lemma 3.6 If the metric tensor is smooth and ∂Ω ∈ C∞ then, for each 1 < p < ∞ and ` ∈0, 1, . . . ,m,

X p` (∂Ω) ≡

f ∈ B− 1p ,p(∂Ω, Λ`TM); ν ∧ f = 0, d∂f ∈ B− 1

p ,p(∂Ω,Λ`+1TM)

(3.43)

in the sense that the two spaces coincide as sets and

‖f‖X p` (∂Ω) ≈ ‖f‖

B−1p ,p(∂Ω,Λ`TM)

+ ‖d∂f‖B−

1p ,p(∂Ω,Λ`+1TM)

. (3.44)

16

Here and elsewhere, we write A ≈ B for two positive quantities depending on some set of parametersif A/B and B/A are bounded by constants independent of the relevant parameters.

Lemma 3.7 Let Ω1, Ω2 ⊆ M be two Lipschitz domains which are homeomorphic to each otherunder a (global) C0,1-diffeomorphism of M. Also, for 1 < p < ∞, consider the spaces X p

` (∂Ω1)and X p

` (∂Ω2) corresponding, respectively, to two metric tensors g1, g2, with Lipschitz coefficientson M. Then for each ` ∈ 0, 1, . . . ,m, we have the isomorphism

X p` (∂Ω1) ∼= X p

` (∂Ω2), (3.45)

independently of 1 < p < ∞.

Modulo the proofs of Lemmas 3.5-3.7, we are now ready to present the

Proof of Proposition 3.4. Let 1 < p1, p2 < ∞, 0 < θ < 1, 1p0

= 1−θp1

+ θp2

. Our goal is to showthat

[X p1` (∂Ω),X p1

` (∂Ω)]θ ≡ X p0` (∂Ω). (3.46)

To this end, since

ν ∧ · : Dpj` (d,Ω) → X pj

` (∂Ω) → B− 1

pj,pj (∂Ω, Λ`TM), j = 1, 2, (3.47)

it follows, by (3.7) and Lemma 3.5, that

ν ∧ · : Dp0` (d, Ω) → X p0

` (∂Ω) → B− 1p0

,p0(∂Ω, Λ`TM) (3.48)

is compatible with

ν ∧ · : Dp0` (d, Ω) → [X p1

` (∂Ω),X p2` (∂Ω)]θ → B− 1

p0,p0(∂Ω, Λ`TM). (3.49)

Now, the first arrow in (3.48) is onto, so that

X p0` (∂Ω) → [X p1

` (∂Ω),X p2` (∂Ω)]θ, (3.50)

continuously.There remains to prove the opposite inclusion in (3.50). We do so but, first, under the additional

assumption that ∂Ω ∈ C∞. Recall first that

d∂ : X pj` (∂Ω) → B

− 1pj

,pj (∂Ω, Λ`+1TM), j = 1, 2, (3.51)

are well-defined and bounded. Thus, by interpolation,

d∂ : [X p1` (∂Ω),X p2

` (∂Ω)]θ → B− 1p0

,p0(∂Ω, Λ`+1TM) (3.52)

(so far, this is actually valid in the Lipschitz context). It is at this point that we utilize thesmoothness assumptions. Specifically, (3.52) and (3.43) in Lemma 3.6 readily imply that

[X p1` (∂Ω),X p2

` (∂Ω)]θ → X p0` (∂Ω), (3.53)

as desired.Summarizing, we have proved that (3.46) holds provided ∂Ω ∈ C∞ and the metric is smooth.

Thus, X p` (∂Ω)1<p<∞ is a complex interpolation scale in this case. Granted Lemma 3.7, the same

17

conclusion holds for ∂Ω and the metric only Lipschitz, since Ω can be always mapped via a bi-Lipschitz homeomorphism onto a smooth domain (and the metric can be replaced by a smoothone). This finishes the proof of Proposition 3.4 (modulo those of Lemmas 3.5–3.7). 2

Next we turn attention to the proof of Lemma 3.5. In doing so, we shall need the following wellknown result (whose proof is an exercise).

Lemma 3.8 Let Xj , j = 0, 1, be Banach spaces. Assume that there exists a common projection,i.e. bounded, linear operator P : Xj → Xj, j = 0, 1, such that P 2 = P . Then P ([X0, X1]θ) =[PX0, PX1]θ for each 0 < θ < 1.

We thank Nigel Kalton for pointing out to us the utility of this lemma in the present context.

Proof of Lemma 3.5. The first step is to show that, if 1 < p1, p2 < ∞, 0 < θ < 1, 1p0

= 1−θp1

+ θp2

,then

[Dp1` (d,Ω), Dp2

` (d, Ω)]θ → Dp0` (d, Ω). (3.54)

This follows easily by interpolation since

d : Dpj` (d,Ω) → Lpj (Ω, Λ`+1TM), (3.55)

is bounded for j = 1, 2. Second, it is not too hard to see that (3.54) holds with equality if a localversion of it holds with equality. Specifically, it is enough to show that for each x ∈ Ω there existsU 3 x, open, and so that

[Dp1` (d, U), Dp2

` (d, U)]θ = Dp0` (d, U). (3.56)

That this forces equality in (3.54) is seen by considering the mapping

Dpj` (d, U) 3 u 7→ u|Ω ∈ Dpj

` (d, Ω), j = 1, 2, (3.57)

where u ∈ Dpj` (d,M) is obtained via the extension operation (relative for U ⊆ M) described in

Proposition 3.1, and interpolation.Next, we tackle the right-to-left inclusion in (3.56); the opposite one is proved as in (3.54). To

this end, we can assume that 1 ≤ ` ≤ m− 1. Also, fix a small, open set U with a smooth boundaryand which is homeomorphic to an Euclidean ball, but otherwise arbitrary. Since pull-backing by asmooth diffeomorphism Φ induces an isomorphism between Dp

` (d, U) and Dp` (d,Φ(U)), there is no

loss of generality to assume that

the underlying manifold M is a homology sphere. (3.58)

Continuing the series of reductions, we note that it suffices to show that (3.56) is valid with Min place of U . Indeed, granted (3.56) with M in place of U , and since by Proposition 3.1 therestriction operator RΩu := u|Ω is bounded from Dpj

` (d,M) onto Dpj` (d, U), j = 1, 2, interpolation

further implies

Dp0` (d, U) = RΩ

(

Dp0` (d,M)

)

→ [Dp1` (d, U), Dp2

` (d, U)]θ, (3.59)

as desired.Thus, we are left with proving the right-to-left inclusion in (3.56) with M in place of U , i.e.

18

Dp0` (d,M) → [Dp1

` (d,M), Dp2` (d,M)]θ. (3.60)

To see this, we proceed in a sequence of steps. In Step One, we shall argue to the effect that

dDp` (d,M)1<p<∞ is a complex interpolation scale. (3.61)

This is clear for ` = 0 and ` = m. On the other hand, for 1 ≤ ` ≤ m− 1, let Π` be the Newtonianpotential on M, i.e. the integral operator whose kernel is the Schwartz kernel of ∆−1

` . Here ∆` :=−(dδ + δd) is the Hodge-Laplacian on `-forms and δ is taken with respect to a smooth backgroundmetric. Note that ∆−1

` exists because of (3.58). Then define P` : Lp(M, Λ`TM) → Lp(M, Λ`TM)by

P`u := −dδΠ`u, u ∈ Lp(M,Λ`TM). (3.62)

From Calderon-Zygmund theory, P` is bounded on Lp(M,Λ`TM). Also, since δ2 = 0 and δΠ` =Π`−1δ (in the sense of unbounded operators), we see that

P 2` u = dδΠ`dδΠ`u = −dδΠ`∆`Π`u = −dδΠ`u = P`u, (3.63)

i.e., P` is idempotent. Hence, (3.61) will follow from Lemma 3.8 as soon as we show that

P`+1Lp(M,Λ`+1TM) = dDp` (d,M). (3.64)

Turning our attention to (3.64), notice that the left-to-right inclusion is obvious. To see the oppositeone, if u ∈ Dp

` (d,M) is arbitrary then P`+1(du) = −dδΠ`+1du = d(∆`Π`u) = du. This completesthe proof of (3.64) and concludes Step One.

In Step Two, we shall prove that for each 1 < p < ∞,

H1,p(M, Λ`TM) + dDp`−1(d,M) = Dp

` (d,M). (3.65)

Indeed, the left-to-right inclusion is obvious. In the opposite direction, if u ∈ Dp` (d,M), then

u = ∆`Π`u = δΠ`+1(du) + d(δΠ`u) ∈ H1,p(M, Λ`TM) + dDp`−1(d,M), (3.66)

as claimed. It is important to point out that the above decomposition is amenable to interpolationin the sense that

A : Dp` (d,M) → H1,p(M, Λ`TM), Au := δΠ`+1(du), (3.67)

B : Dp` (d,M) → dDp

`−1(M, Λ`TM), Bu := d(δΠ`u), (3.68)

are linear, bounded and independent of p.In the third (and last) step we tackle (3.60). First, from the obvious inclusions

H1,pj (M, Λ`TM) → Dpj` (d,M), dDpj

`−1(d,M) → Dpj` (d,M), j = 1, 2,

(3.61) and interpolation we get

H1,p0(M, Λ`TM) → [Dp1` (d,M), Dp2

` (d,M)]θ, (3.69)

dDp0`−1(d,M) → [Dp1

` (d,M), Dp2` (d,M)]θ. (3.70)

19

Adding up the two inclusions and invoking (3.65) finally yields (3.60). This finishes the proof ofLemma 3.5. 2

We now present the

Proof of Lemma 3.6. If f ∈ X p` (∂Ω) then there exists u ∈ Dp

`−1(d,Ω) so that ν ∧ u = f and‖u‖Lp(Ω) + ‖du‖Lp(Ω) ≈ ‖f‖X p

` (∂Ω). Let uj ∈ C1(Ω,Λ`−1TM) be so that uj → u and duj → du in

Lp(Ω). Then, for any ϕ ∈ B1p ,q(∂Ω,Λ`+1TM),

〈ν ∧ f, ϕ〉 = 〈f, ν ∨ ϕ〉 = limj〈ν ∧ uj , ν ∨ ϕ〉 = lim

j〈ν ∧ (ν ∧ uj), ϕ〉 = 0, (3.71)

i.e. ν ∧ f = 0. Also

d∂f = d∂(ν ∧ u) = −ν ∧ du ∈ B− 1p ,p(∂Ω, Λ`+1TM), (3.72)

since du ∈ Dp`+1(d,Ω). Moreover, it is implicit in the above calculation that

‖f‖B−

1p ,p(∂Ω)

+ ‖d∂f‖B−

1p ,p(∂Ω)

≤ C‖f‖X p` (∂Ω), (3.73)

uniformly for f ∈ X p` (∂Ω). This proves the left-to-right inclusion in (3.43). Parenthetically, let us

observe that the estimate (3.73) is actually valid in arbitrary Lipschitz domains.Proving the opposite inclusion makes essential use of the smoothness of ∂Ω and requires deeper

results. To begin with, let us temporarily denote by B− 1

p ,pd (∂Ω, Λ`TM) the space in the right side

of (3.43), i.e.

B− 1

p ,pd (∂Ω, Λ`TM) :=

f ∈ B− 1p ,p(∂Ω, Λ`TM); ν ∧ f = 0 and d∂f ∈ B− 1

p ,p(∂Ω,Λ`+1TM)

.

Furthermore, equip it with the norm given by the right side of (3.44). Next, fix a constant potentialV > 0 and recall the integral operators introduced in this context as in the second part of §2. Theincisive claim is that, for each ` ∈ 0, 1, . . . , m and 1 < p < ∞,

the operator 12I + N` is invertible on B

− 1p ,p

d (∂Ω, Λ`TM). (3.74)

Accepting this for the time being, we shall now indicate how to finish the proof of Lemma 3.6. To

this effect, for f ∈ B− 1

p ,pd (∂Ω,Λ`TM) set

u := δS`

(

(12I + N`)

−1f)

in Ω. (3.75)

Since, by Proposition 2.7 (with r = ∞), the operator S` maps B− 1p ,p(∂Ω,Λ`TM) boundedly into

H1,p(Ω, Λ`TM), we have that u ∈ Lp(Ω, Λ`−1TM). In order to continue, we need one more identitywhich is actually valid for Lipschitz domains and metrics satisfying (2.6). Specifically, in analogywith (2.34)–(2.35), we claim that

dS`f = S`+1(d∂f), ∀ f ∈ X p` (∂Ω) (3.76)

δS`g = S`−1(δ∂g), ∀ g ∈ Yp` (∂Ω). (3.77)

In order to prove (3.76), let f ∈ X p` (∂Ω) and u ∈ Dp

`−1(d,Ω) so that ν ∧ u = f . Then, using thedefinition of S`, (2.25), integrations by parts and (3.33), we obtain that

20

dS`(f) =∫

∂Ω〈dxΓ`, ν ∧ u〉 dσ =

∫

∂Ω〈δyΓ`+1, ν ∧ u〉 dσ (3.78)

=∫ ∫

Ω〈δΓ`+1, du〉 = −

∫

∂Ω〈Γ`+1, ν ∧ du〉 dσ = S`+1(d∂f),

as desired. The proof of (3.77) is similar.

In fact, if ∂Ω ∈ C∞, (3.76) is valid for f ∈ B− 1

p ,pd (∂Ω,Λ`TM). This is seen via a limiting

argument based on the fact that the set f ∈ C∞(∂Ω,Λ`TM); ν ∧ f = 0 is densely embedded inB− 1

p ,p(∂Ω, Λ`TM). In turn, the latter assertion can be proved by localizing and mollifying. Hence,making use of (3.76) we also have that

du = dδS`

(

(

12I + N`

)−1f)

= −δdS`

(

(

12I + N`

)−1f)

−∆`S`

(

(

12I + N`

)−1f)

= −δS`+1

(

d∂

(

12I + N`

)−1f)

(3.79)

−V S`

(

(

12I + N`

)−1f)

∈ Lp(Ω,Λ`TM) + H1,p(Ω, Λ`TM).

Thus, u ∈ Dp`−1(d, Ω) and

‖u‖Lp(Ω) + ‖du‖Lp(Ω) ≤ C(‖f‖B−

1p ,p(∂Ω)

+ ‖d∂f‖B−

1p ,p(∂Ω)

). (3.80)

Also,

ν ∧ u = ν ∧(

δS`

(

(

12I + N`

)−1f))

=(

12I + N`

) (

12I + N`

)−1f = f (3.81)

in B− 1p ,p(∂Ω, Λ`TM); the above formal calculation can be easily justified via a density argument.

The bottom line is that f ∈ X p` (∂Ω) and

‖f‖X p` (∂Ω) ≤ C(‖f‖

B−1p ,p(∂Ω)

+ ‖d∂f‖B−

1p ,p(∂Ω)

). (3.82)

This proves the right-to-left inclusion in (3.43) and finishes the proof of Lemma 3.6, modulo theproof of (3.74), which we now tackle.

The first order of priorities is to extend the action of the operator 12I + N` so that it maps

B− 1

p ,pd (∂Ω, Λ`TM) boundedly into itself. Given the jump formula (2.33), it is natural to set

(

12I + N`

)

f := ν ∧ (δS`f) for any f ∈ B− 1

p ,pd (∂Ω, Λ`TM). (3.83)

Clearly, this definition agrees with the one given for 12I + N` on the space Lp,d

nor(∂Ω,Λ`TM). Go-

ing further, we need to verify that(

12I + N`

)

f ∈ B− 1p ,p(∂Ω, Λ`TM), ν ∧

(

12I + N`

)

f = 0 and

d∂

(

12I + N`

)

f ∈ B− 1p ,p(∂Ω, Λ`+1TM). The first two conditions are immediate consequences of

(3.83) plus the fact that S` maps B− 1p ,p(∂Ω, Λ`TM) into H1,p(Ω,Λ`TM). Furthermore,

21

d∂

(

12I + N`

)

f = d∂(ν ∧ δS`f) = −ν ∧ dδS`f

= ν ∧ δdS`f + ν ∧∆`S`f (3.84)

= ν ∧ δS`+1(d∂f) + V ν ∧ S`f ∈ B− 1p ,p(∂Ω, Λ`+1TM).

The proof for the fact that 12I + N` is well defined on B

− 1p ,p

d (∂Ω, Λ`TM), 1 < p < ∞, is thereforecompleted.

Turning to the actual job of proving the Fredholmness of this operator on the space B− 1

p ,pd (∂Ω, Λ`TM),

1 < p < ∞, we first note that the kernel of N` acting on normal forms can also be written in theform ν(y)∧ [ν(y)∨ (ν(x)∧ δxΓ`(x, y))]. Consequently, a basic ingredient in accomplishing our goalis the fact that

ν(y) ∨ (ν(x) ∧ δxΓ`(x, y)) = O(

|x− y|−(m−2))

as |x− y| → 0, x, y ∈ ∂Ω. (3.85)

Granted this, an inspection of (3.84) and (3.83) shows that N` is a smoothing operator of order−1, indeed N` and d∂N` belong to OPS−1(∂Ω); see [41], Vol. II, Proposition 11.2 for similar cir-cumstances. This, further, will imply several things. First, so we claim, N` turns out to be a

compact operator on B− 1

p ,pd (∂Ω, Λ`TM). Indeed, N` and d∂N` map H−1,p(∂Ω) boundedly into

Lp(∂Ω) so that, by real interpolation and classical embedding results, they both map the Besovspace B− 1

p ,p(∂Ω, Λ`TM) compactly into itself. From this, the claim follows. In particular,

12I + N` is Fredholm with index zero on B

− 1p ,p

d (∂Ω, Λ`TM) for 1 < p < ∞ (3.86)

and

Ker(

12I + N`; B

− 1p ,p

d (∂Ω,Λ`TM))

= Ker(

12I + N`; L2(∂Ω, Λ`TM)

)

. (3.87)

From Theorem 2.5 we know that the space in the right side of (3.87) is trivial which, together with(3.86), gives (3.74).

Hence, at this point we are left with proving (3.85). To this effect, we first recall from §2 that

δx(Γ`(x, y)) = dy(Γ`−1(x, y)), (3.88)

(the absence of any residue in (3.88) is due to the fact that V is constant), so that it suffices totreat ν(y) ∨ (ν(x) ∧ dyΓ`−1(x, y)). We also know that, locally,

Γ`−1(x, y) = Cm1

√

g(y)

∑

j,k

gjk(y)(xj − yj)(xk − yk)

−(m−2)/2

×∑

|I|=`−1

∑

|J |=`−1

det[

(gij(y)) i∈Ij∈J

]

dxI ⊗ dyJ

+O(|x− y|−(m−3)). (3.89)

In order to continue, we need one more piece of notation. For an arbitrary matrix A = (aij)1≤i,j≤m

and 0 ≤ ` ≤ m we set T`A := (aIJ) |I|=`|J|=`

, where aIJ := det[

(aij) i∈Ij∈J

]

. The operator A 7→ T`A is

22

known to commute with powers and transposition (see II, §2.4 in [11] where this is referred to as the`-th compound of A; cf. also [16]). Also, set G := (gij)1≤i,j≤m, G1/2 =: (bij)ij and ωj :=

∑

k bjkdxk.Then ωj1≤j≤m is an orthonormal basis for T ∗M and

∀ |I| = `− 1, dxI =∑

|K|=`−1

(T`−1G−1/2)IK ωK , (3.90)

where ωK := ωk1 ∧ ... ∧ ωk`−1 if K = (k1, ..., k`−1). Thus,

∑

|I|=`−1

∑

|J |=`−1

(T`−1G)IJ(y) dxI ⊗ dyJ

=∑

I,J,K,L

(T`−1G)IJ(y) (T`−1G−1/2)IK(x) (T`−1G−1/2)JL(y) ωKx ⊗ ωL

y

=∑

K,L

(

(T`−1G−1/2)(x) (T`−1G)(y) (T`−1G−1/2)(y))

KLωK

x ⊗ ωLy

=∑

K,L

(

(T`−1G)−1/2(x) (T`−1G)1/2(y))

KLωK

x ⊗ ωLy

=∑

K,L

(

(T`−1G)−1/2(x)[(T`−1G)1/2(y)− (T`−1G)1/2(x)])

KLωK

x ⊗ ωLy

+∑

K

ωKx ⊗ ωK

y =: I + II. (3.91)

Clearly, I = O(|x− y|) so its contribution is residual. Thus, in subsequent analysis, we shall onlyconsider II. In other words, at this stage it suffices to treat

ν(y) ∨ (ν(x) ∧ dy[E0` (x, y)γ(x, y)]) (3.92)

where, recall from §2 that E0` (x, y) :=

[

∑

j,k gjk(y)(xj − yj)(xk − yk)]−(m−2

2 )and we set γ(x, y) :=

∑

K ωKx ⊗ ωK

y . In fact, the main source of singularities in dy(E0` γ) = dyE0

` ∧ γ + E0` dyγ is dyE0

` ∧ γso we shall focus on it. Based on Lemma 2.1, we write

νy ∨ (νx ∧ (dyE0` ∧ γ)) = −νy ∨ (dyE0

` ∧ (νx ∧ γ))

= dyE0` ∧ (νy ∨ (νx ∧ γ))− 〈νy, dyE0

` 〉νx ∧ γ

=: III + IV. (3.93)

Inspection of III reveals that, with ν =∑

j νjωj ,

III =∑

j,k

∑

J,M

∑

I

εjIJ εkM

I νj(x)νk(y) ωJx ⊗ ωM

y . (3.94)

Here, for any two ordered arrays J , K, the generalized Kronecker symbol εJK is given by

εJK :=

det ((δj,k)j∈J,k∈K) , if |J | = |K|,0, otherwise,

(3.95)

23

where δj,k := 1 if j = k, and zero if j 6= k.Now, the only non-zero terms in (3.94) occur for j 6= k, j /∈ I, k ∈ I, in which case we have

εjIJ εkM

I = εjIJ εjkM

jI = εjkMJ . Thus, from (3.94),

III =( m− 2

`

)∑

J,M

∑

j 6=k

εjkMJ νj(x)νk(y)

ωJx ⊗ ωM

y . (3.96)

Now, for each fixed J,M , due to the antisymmetry of εjkMJ in j and k, we have

∑

j 6=k

εjkMj νj(x)νk(y) =

∑

j<k

εjkMJ [νj(x)νk(y)− νj(y)νk(x)] = O(|x− y|), (3.97)

since ν is Lipschitz by assumption. Hence, the contribution from III is of the right order.Going further, if n = (nj)j denotes the Euclidean unit normal to ∂Ω (looked at in local coordi-

nates) then ν = (∑

j,k gjknjnk)−1

2 (∑

l nldxl) so that

〈νy, dyE0` (x, y)〉 =

(∑

j,k

gjk(y)nj(y)nk(y))−1

2∑

l,s

∂ylE0` (x, y)ns(y)gls(y).

In particular, the main singularity in IV is contained in

∑

l,s

ns(y)gls(y)∂

∂yl

(∑

j,k

gjk(y)(xj − yj)(xk − yk))−(m−2

2 )−1

= −2∑

l,s

ns(y)gls(y)glr(y)(xr − yr)(∑

j,k

gjk(y)(xj − yj)(xk − yk))−m/2

= −2〈n(y), x− y〉(∑

j,k

gjk(y)(xj − yj)(xk − yk))−m/2

, (3.98)

where, this time, the inner product in the last term is taken with respect to the ordinary Euclideanmetric. Now, due to the smoothness of the boundary, 〈n(y), x− y〉 = O(|x− y|2) so that the entireexpression in (3.98) is O(|x− y|−(m−2)) as desired. This concludes the proof of (3.85) and, with it,the proof of Lemma 3.6. 2

Proof of Lemma 3.7. Let Φ : M→M be a bi-Lipschitz homeomorphism so that Φ(Ω1) = Ω2and set Ψ := Φ−1. For 1 < p < ∞, 0 ≤ ` ≤ m, define

A : X p` (∂Ω1) → X p

` (∂Ω2), B : X p` (∂Ω2) → X p

` (∂Ω1) (3.99)

by

A(ν1 ∧ u) := ν2 ∧Ψ∗u, u ∈ Dp` (d, Ω1),

B(ν2 ∧ v) := ν1 ∧ Φ∗v, v ∈ Dp` (d,Ω2), (3.100)

where the superscript star denotes pull-back, and νj stands for the unit conormal to ∂Ωj withrespect to the metric gj , j = 1, 2. Note that Ψ∗ : Dp

` (d,Ω1) → Dp` (d,Ω2) is an isomorphism with

inverse Φ∗. To see that A,B are well-defined we need to show that, e.g., Ψ∗ maps Dp` (d∧,Ω1) into

Dp` (d∧, Ω2), i.e. prove that

24

u ∈ Dp` (d, Ω1), ν1 ∧ u = 0 ⇒ ν2 ∧Ψ∗u = 0. (3.101)

To this end, we first remark that for j = 1, 2, given u ∈ Dp` (d, Ωj),

u ∈ Dp` (d∧, Ωj) ⇔ u ∈ Dp

` (d,M), (3.102)

where tilde is extension by zero in M\ Ωj .

This is easily seen using the distributional definition of d and integrating by parts. Based onthis, the fact that pull-backing by Ψ commutes with tilde and preserves Dp

` (d,M), (3.101) follows.Finally, observe that A B = Id, B A = Id, hence the conclusion in the lemma.

One final note is that the isomorphisms A,B are independent of p and, hence, are amenable tointerpolation. 2

Next, define

X p,0` (∂Ω) := f ∈ X p

` (∂Ω); d∂f = 0 (3.103)

so that, clearly, the image of d∂ on X p` (∂Ω) is a subset of X p,0

`+1(∂Ω). As the next proposition shows,we can be more precise about this inclusion.

Proposition 3.9 For any 1 < p < ∞, the operator d∂ : X p` (∂Ω) → X p,0

`+1(∂Ω) has closed range andits cokernel is isomorphic to H`

sing(∂Ω;R), the `-th singular homology group of ∂Ω over the reals.In particular,

d∂ :X p

` (∂Ω)

X p,0` (∂Ω)

−→ X p,0`+1(∂Ω) (3.104)

is a Fredholm operator with index b`(∂Ω), the `-th Betti number of ∂Ω.Finally, similar results hold for δ∂ acting on the scale Yp

` (∂Ω).

Proof. Our proof is modeled upon an argument in [28] where a related statement (at the Lp-level)has been proved. First, we localize the definition of d∂ .

More specifically, if 1 < p < ∞, for U an arbitrary, fixed open subset of ∂Ω, we define X p` (U)

as the subspace of B−1/p,ploc (U,Λ`TM) consisting of distributions f on ∂Ω enjoying the following

property: for each x ∈ U , there exist D, open neighborhood of x in M with D ∩ ∂Ω ⊂ U andu ∈ Dp

`−1(d,Ω) such that f |D∩∂Ω = (ν ∧ u)|D∩∂Ω. Also, introduce

d∂ : X p` (U) −→ X p

`+1(U) (3.105)

by setting d∂f := −ν ∧ du near x, if f is locally given by ν ∧ u near x. Clearly, each X p` (U) is

an additive Abelian group and also a module over the algebra Lip(∂Ω). It follows that the familyX p

` := (X p` (U))U , indexed by open subsets in ∂Ω, is a fine sheaf on the topological space ∂Ω.

Going further, we observe that the operator d∂ induces a natural sequence of sheaf morphisms

0−−−→ LCFι

−−−→ X p1

d∂−−−→ X p

2d∂

−−−→ X p3

d∂−−−→ · · · (3.106)

where LCF stands for the sheaf of germs of locally constant functions on ∂Ω and the embeddingworks according to f 7→ fν. Since d∂ d∂ = 0, the above is a complex. In fact, so we claim, (3.106)provides a fine resolution of the sheaf LCF. The essential ingredient in the proof of this claim is the

25

acyclicity of the complex (3.106). Granted this, the so-called abstract de Rham theorem applies toour context and gives that

X p`+1(∂Ω)

d∂(

X p` (∂Ω)

)∼= H`

sing(∂Ω;R) (3.107)

for ` = 0, 1, .., m and 1 < p < ∞. With this at hand, all the claims in the proposition follow. Thereader is referred to §9 in [28] for somewhat similar circumstances.

Next, we aim to prove the acyclicity of the sheaf (3.106). It is not hard to see that this isequivalent to the following claim:

∀x ∈ ∂Ω and u ∈ Dp` (d, Ω) with ν ∧ du = 0 near x, (3.108)

∃ v ∈ Dp`−1(d, Ω) such that ν ∧ u = ν ∧ dv near x.

Below we outline the main steps in the proof of this claim. First, much as in the proof of Lemma 3.7,the statement can be pull-backed to Rm. Thus, without loss of generality, we can assume thatΩ = D ∩Rm

+ , where D is an open, bounded, starlike (with respect to the origin) Lipschitz domainin Rm, equipped with the standard Euclidean metric, and such that x = 0 ∈ D. Set D± := D∩Rm

± .In order to continue, we need a “localized” Lp-Poincare type lemma to the effect that

∀w ∈ Np` (d,D) such that w ≡ 0 in D−, (3.109)

∃ω ∈ Dp`−1(d,D) with dω = w and ω|D− ≡ 0.

In turn, this is proved much as in the classical case. For example, if w is smooth, one could constructω via familiar path integrals along rays emerging from 0. In this scenario, the support restrictionon ω is an automatic consequence of this construction and our hypothesis on suppw. Finally, inthe case when w is only p-th power integrable, mollifying w and using a priori estimates plus aweak-limit argument readily reduce matters to the smooth situation. We leave the standard detailsto the interested reader.

Let us now indicate how (3.109) can be used to conclude the proof of (3.108). To this end,assume that u is as in (3.108) and Ω = D−, with D as before. This implies that the form w,defined as du in D+ and 0 in D−, belongs to Dp

`+1(d, U) for some open neighborhood U of 0 ∈ Rm.Furthermore, dw = 0 and w|U∩Rm

−≡ 0. The localized Poincare lemma (3.109) then guarantees the

existence of ω ∈ Dp` (d, U) with dω = w in U and ω|U∩Rm

−≡ 0. In particular,

d(u− ω) = 0 in U ∩Rm+ and ν ∧ ω = 0 near 0. (3.110)

A (weaker) version of (3.109) then gives that there exists ω ∈ Dp`−1(d, U∩Rm

+ ) such that u−ω = dω.Thus, near the origin, ν ∧u− ν ∧dω = ν ∧ω = 0. Hence, ν ∧u = ν ∧dω near the origin, as desired.

This finishes the proof of the proposition. 2

4 Finite energy solutions for Maxwell’s equations

For each 0 ≤ ` ≤ m, ∆` : H1,2(M, Λ`TM) → H−1,2(M, Λ`TM) is a bounded, negative, formallyself-adjoint operator. Since (∆` − λ)−1 gives rise to a negative, self-adjoint compact operator onL2(M, Λ`TM) for λ ∈ R with |λ| large, it follows that there exists Spec (∆`) ⊆ (−∞, 0], a discreteset (which accumulates only at −∞) so that

26

z /∈ Spec (∆`) ⇒ (∆` − z) : H1,2(M, Λ`TM) → H−1,2(M,Λ`TM) is invertible. (4.1)

Select kj ∈ [0,∞) so that

−(kj)2j =⋃

0≤`≤m

Spec (∆`). (4.2)

For k /∈ ±kjj , let Γk,` be the Schwartz kernel of ∆` + k2 on `-forms.Note that, in the notation of §2, this formally corresponds to choosing V := −k2. To emphasize

the dependence on the parameter k we should append it as a subindex to the various objectsconstructed in connection with Γk,`. Once a Lipschitz domain Ω ⊂ M has been fixed, we cantherefore talk about the (family of) Newtonian potentials Πk,`, boundary integral operators Sk,`,Mk,`, Nk,`, etc. As explained in §2,

δx(Γk,`+1(x, y)) = dy(Γk,`(x, y)), dx(Γk,`(x, y)) = δy(Γ`+1(x, y)), (4.3)

since the potential V is constant.For k /∈ ±kjj , we aim at extending the action of the operators Mk,` and Nk,` to the spaces

Yp` (∂Ω) and X p

` (∂Ω), respectively. As a preliminary step, we first analyze the action of Sk,` onthese spaces.

Lemma 4.1 Let Ω be a Lipschitz domain in M and fix an arbitrary complex number k /∈ ±kjj.Then, for each 1 < p < ∞ and 0 ≤ ` ≤ m, the operators

Sk,` : X p` (∂Ω) −→ H1,p(Ω, Λ`TM), (4.4)

Sk,` : Yp` (∂Ω) −→ H1,p(Ω, Λ`TM), (4.5)

are well-defined and bounded. In particular, Sk,` := Tr Sk,` maps X p` (∂Ω) and Yp

` (∂Ω) boundedlyinto B1−1/p,p(∂Ω, Λ`TM).

Proof. Consider the case of (4.4); (4.5) is handled similarly. If r/(r−1) < p < ∞, then the desiredconclusion follows from (iii) in Proposition 3.3 and Proposition 2.7.

Assume next that 1 < p < r and select an arbitrary u ∈ Dp`−1(d, Ω). Thanks to (4.3) and an

integration by parts we may write

Sk,`(ν ∧ u)(x) = 〈Γk,`(x, ·), ν ∧ u〉

=∫ ∫

Ω〈Γk,`(x, ·), du〉 dVol−

∫ ∫

Ω〈δyΓk,`(x, ·), u〉 dVol

=∫ ∫

Ω〈Γk,`(x, ·), du〉 dVol−

∫ ∫

Ω〈dxΓk,`−1(x, ·), u〉 dVol

= Πk,`(du)(x)− dΠk,`−1u(x). (4.6)

From this and Proposition 2.7 it is clear that Sk,`(ν∧u) ∈ H1,p(Ω, Λ`TM) and ‖Sk,`(ν∧u)‖H1,p(Ω) ≤C(‖u‖Lp(Ω) + ‖du‖Lp(Ω)). With this at hand, it is trivial to finish the proof. 2

We next turn our attention to the operators Mk,`, Nk,`.

27

Proposition 4.2 Let Ω be a Lipschitz domain and let k /∈ ±kjj. Then for any 1 < p < ∞ and0 ≤ ` ≤ m, the operators ±1

2I +Nk,`, originally defined on Lp(∂Ω,Λ`TM) have bounded extensionsto X p

` (∂Ω), i.e.

±12I + Nk,` : X p

` (∂Ω) −→ X p` (∂Ω). (4.7)

Moreover, similar results are valid for the operators

±12I + Mk,` : Yp

` (∂Ω) −→ Yp` (∂Ω) (4.8)

and, if 1/p + 1/q = 1, the diagram

Yq` (∂Ω)

ν ∧ ·∼

-(

X p`+1(∂Ω)

)∗

Yq` (∂Ω)

±12I + Mk,`

? ν ∧ ·∼

-(

X p`+1(∂Ω)

)∗

(±12I + Nk,`+1)t

?

(4.9)

is commutative.

Proof. With an eye on (4.7), let f ∈ X p` (∂Ω) and set v := δSk,`f in Ω±. The immediate goal is to

show that v ∈ Dp`−1(d, Ω±) and that

‖v‖Lp(Ω±) + ‖dv‖Lp(Ω±) ≤ C‖f‖X p` (∂Ω), (4.10)

holds uniformly for f ∈ X p` (∂Ω). To this end, observe first that the part (ii) of Proposition 2.7

gives v ∈ Lp(Ω±, Λ`−1TM). Moreover,

dv = dδSk,`f = −δdSk,`f + k2Sk,`f = −δSk,`(d∂f) + k2Sk,`f, (4.11)

where, in the last equality, we have used (3.76). Now, (4.11) and Lemma 4.1 prove that dv ∈Lp(Ω±, Λ`TM) hence, v ∈ Dp

`−1(d,Ω±). It is also implicit in the above calculations that (4.10)holds.

Next, in analogy with (3.83) we define(

±12I + Nk,`

)

f := ν ∧ (v|Ω±) ∈ X p` (∂Ω). (4.12)

It is then a simple consequence of (4.10) that, for each 1 < p < ∞, the operators (4.7) are well-defined and bounded.

Our next goal is to show that the above definition of Nk,` is indeed compatible with the oldergiven in §2. A moment’s reflection shows that this comes down to checking the following. Letv ∈ Dp

`−1(d, Ω) be such that v, dv are continuous in Ω, N (v), N (dv) ∈ Lp(∂Ω) and v|∂Ω exists inthe pointwise nontangential sense; cf. (2.29). Then ν ∧ v, considered in the sense of (3.6), coincideswith ν ∧ (v|∂Ω). In order to see this, consider a sequence of smooth domains Ωj Ω as in [28]. Foreach `-form φ with Lipschitz continuous coefficients in Ω we may write

〈ν ∧ v, φ〉 =∫ ∫

Ω〈ν ∧ dv, φ〉 dVol−

∫ ∫

Ω〈ν ∧ v, δφ〉 dVol

28

= limj

∫ ∫

Ωj

〈ν ∧ dv, φ〉 dVol− limj

∫ ∫

Ωj

〈ν ∧ v, δφ〉 dVol

= limj

∫

∂Ωj

〈νj ∧ (v|∂Ωj ), φ〉 dσj =∫

∂Ω〈ν ∧ (v|∂Ω), φ〉 dσ, (4.13)

where the last equality follows from Lebesgue’s Dominated Convergence Theorem. This shows thatν ∧ v, initially considered in B−1/p,p(∂Ω,Λ`TM), coincides with ν ∧ (v|∂Ω) ∈ Lp(∂Ω, Λ`TM). Atthis stage, all claims pertaining to the operators (4.7) have been proved.

The treatment of (4.8) is very similar and we only outline the main steps. This time, the ideais to set

(

∓12I + Mk,`

)

g := ν ∨ (w|Ω±), (4.14)

where g ∈ Yp` (∂Ω) is arbitrary and w := dSk,`g in Ω±. That these are meaningful, bounded

assignments can be seen by paralleling the corresponding proof for (4.12) or, alternatively, usingwhat we have proved so far and Hodge duality. In this latter scenario, the relevant ingredients are:the point (6) in Lemma 2.1, (2.36) and the point (v) in Proposition 3.3.

Finally, the commutativity of the diagram (4.9) comes down to checking the identity

〈ν ∧ (±12I + Mk,`)f, g〉 = 〈(±1

2I + Nk,`+1)t(ν ∧ f), g〉 (4.15)

for any f ∈ Yq` (∂Ω) and g ∈ X p

`+1(∂Ω). By density, it suffices to prove (4.15) when f = ν ∨ u andg = ν ∧ w, for some u,w ∈ C1(Ω). In this latter case, (4.15) is a straightforward consequence ofthe definition (3.36) and the property (2.37). 2

The long term goal is to show that the operators (4.7)–(4.8) are actually isomorphisms for allp’s in some open interval containing 2 (further restrictions on the complex parameter k are alsoneeded). In order to proceed, introduce

H`(Ω) :=

(E, H) ∈ L2(Ω,Λ`TM)⊕ L2(Ω, Λ`+1TM);

δE = 0, dH = 0, ν ∧H = 0

. (4.16)

It follows that H`(Ω) is a closed subspace of L2(Ω, Λ`TM)⊕ L2(Ω, Λ`+1TM), hence Hilbert, withthe inherited inner product. Consider next the unbounded operator

M` :=

(

0 −iδid 0

)

, M` : H`(Ω) → H`(Ω), (4.17)

with domain

D(M`) :=

(E, H) ∈ H`(Ω); dE ∈ L2(Ω, Λ`+1TM),

ν ∧ E = 0, δH ∈ L2(Ω,Λ`TM)

(4.18)

and natural action.

Proposition 4.3 Let Ω ⊆ M be a Lipschitz domain and 0 ≤ ` ≤ m. Then the operator M` :D(M`) ⊆ H`(Ω) → H`(Ω) is self-adjoint and

29

D(M`) ⊆ H12 ,2(Ω, Λ`TM)⊕H

12 ,2(Ω,Λ`+1TM). (4.19)

In particular, M` has a compact resolvent so that its spectrum consists of only (real) eigenvaluesaccumulating at ±∞.

Proof. The inclusion M` ⊆ M∗` , i.e. the fact that M` is symmetric, is straightforward and we omit

the details. Conversely, let (U, V ) ∈ H`(Ω) belong to D(M∗` ) and set (U , V ) := M∗

` (U, V ) ∈ H`(Ω).Then, for any (E,H) ∈ D(M`),

−i〈δH, U〉+ i〈dE, V 〉 = 〈M`(E, H), (U, V )〉 = 〈(E, H), (U , V )〉 = 〈E, U〉+ 〈H, V 〉, (4.20)

where, in the present context, 〈·, ·〉 stands for the Hermitian L2-pairing in Ω. Taking H ≡ 0 andE ∈ N2

` (δ,Ω) ∩D2` (d∧,Ω) arbitrary forces

δV = iU ∈ L2(Ω, Λ`+1TM). (4.21)

Going further, we specialize (4.20) to the case when E ≡ 0 and H is selected as follows. For anarbitrary W ∈ D2

` (δ,Ω), we use the L2-Hodge decomposition in Ω (cf. Proposition 11.3 in [28]; seealso [39], [43] for related matters) to write W = dα + δβ + γ, where

α ∈ D2`−1(d∧, Ω), β ∈ D2

`+1(δ,Ω), γ ∈ N2` (d∧, Ω) ∩N2

` (δ,Ω). (4.22)

Then we take H := W − δβ = dα + γ ∈ N2`+1(d∧,Ω) ∩D2

`+1(δ,Ω). This forces

−i〈δW,U〉 = 〈W, V 〉, ∀W ∈ D2` (δ, Ω) (4.23)

which, in turn, entails

dU = −iV ∈ L2(Ω,Λ`+1TM) and ν ∧ U = 0. (4.24)

Hence, (U, V ) ∈ D(M`) and M`(U, V ) = M∗` (U, V ). This proves the self-adjointness of M`.

As for (4.19), this follows directly from definitions and Theorem 2.6. With this at hand, therest of the proposition follows from Rellich’s selection lemma and simple functional analysis. 2

In the sequel we shall denote by Spec(M`) the spectrum of the operator M` introduced in(4.17)-(4.18), i.e. the collection of all z ∈ C so that zI −M` does not have a bounded inverse onH`(Ω). As pointed out in the statement of the above proposition, Spec(M`) is a countable set ofreal eigenvalues (of finite multiplicity) of M` which only accumulate at ±∞.

Proposition 4.4 Let Ω ⊆ M be a Lipschitz domain and let 0 ≤ ` ≤ m. Then, for any k /∈Spec(M`), the L2-Poisson problem for the Maxwell system (with homogeneous boundary conditions)

(BV P1)k,`

E ∈ L2(Ω, Λ`TM), H ∈ L2(Ω, Λ`+1TM),

dE − ikH = K ∈ L2(Ω,Λ`+1TM),

δH + ikE = J ∈ L2(Ω, Λ`TM),

ν ∧ E = 0,

has a unique solution. This solution satisfies

‖E‖L2(Ω) + ‖H‖L2(Ω) ≤ C(‖K‖L2(Ω) + ‖J‖L2(Ω)). (4.25)

Moreover, if k 6= 0 is purely imaginary, we may take C = 2/|k| in the above estimate.

30

Proof. Invoking the L2-Hodge decomposition (used in the proof of Proposition 4.3) we can de-compose K = dα⊕ δβ ⊕ γ with

α ∈ D2` (d∧, Ω), β ∈ D2

`+2(δ,Ω), γ ∈ N2`+1(d∧, Ω) ∩N2

`+1(δ,Ω), (4.26)

and J = da⊕ δb⊕ c, where

a ∈ D2` (d∧, Ω), b ∈ D2

`+2(δ,Ω), c ∈ N2` (d∧, Ω) ∩N2

` (δ,Ω). (4.27)

In particular, (−i(c + δb), i(γ + dα)) ∈ H`(Ω). Since k /∈ Spec(M`) we can solve

(kI −M`)(E′,H ′) = (−i(c + δb), i(γ + dα)) (4.28)

in H`(Ω), which is the same as solving

E′ ∈ L2(Ω, Λ`TM), H ′ ∈ L2(Ω, Λ`+1TM),

dE′ − ikH ′ = γ + dα,

δH ′ + ikE′ = c + δb,

ν ∧ E′ = 0.

If we now take E := E′ + 1ikda, H := H ′ − 1

ikδβ, then (E,H) is a solution of (BV P1)k,` whichsatisfies (4.25). Uniqueness follows from the condition k /∈ Spec(M`) and Proposition 4.3.

The last part in the statement is seen from ‖(kI − M`)−1‖ ≤ |Im k|−1 which, in turn, is anelementary consequence of the self-adjoitness of M`. 2

Proposition 4.5 Let Ω ⊆ M be a Lipschitz domain and let 0 ≤ ` ≤ m. Also, fix a non-zerocomplex number k /∈ Spec(M`). Then the Maxwell boundary problem

(BV P2)k,`

E ∈ L2(Ω,Λ`TM), H ∈ L2(Ω,Λ`+1TM),

dE − ikH = 0 in Ω,

δH + ikE = 0 in Ω,

ν ∧ E = f ∈ B−12 ,2(∂Ω, Λ`+1TM),

has a solution if and only if f ∈ X 2`+1(∂Ω). The solution is unique and satisfies

‖E‖L2(Ω) + ‖H‖L2(Ω) ≈ ‖f‖X 2`+1(∂Ω). (4.29)

If k = i then the constants in (4.29) are universal.

Proof. If (BV P2)k,` is solvable then f = ν ∧ E ∈ X 2`+1(∂Ω), since E ∈ D2

` (d, Ω). Conversely, letf ∈ X 2

`+1(∂Ω) be arbitrary. Then there exists U ∈ D2` (d,Ω) so that f = ν ∧ U and ‖U‖L2(Ω) +

‖dU‖L2(Ω) ≈ ‖f‖X 2`+1(∂Ω). Let E′ ∈ L2(Ω, Λ`TM), H ′ ∈ L2(Ω,Λ`+1TM) solve

(BV P3)k,`

dE′ − ikH ′ = dU ∈ L2(Ω, Λ`+1TM),

δH ′ + ikE′ = ikU ∈ L2(Ω, Λ`TM),

ν ∧ E′ = 0.

That this is possible, is ensured by our hypothesis on k and Proposition 4.4. Finally, set E := U−E′

and H := −H ′. Then, (E, H) solve (BV P2)k,` and

31

‖E‖L2(Ω) + ‖H‖L2(Ω) ≤ C(‖U‖L2(Ω) + ‖dU‖L2(Ω)) ≤ C‖f‖X 2`+1(∂Ω). (4.30)

This proves one inequality in (4.29). The opposite inequality follows from

‖f‖X 2`+1(∂Ω) = ‖ν ∧ E‖X 2

` (∂Ω) ≤ ‖E‖L2(Ω) + ‖dE‖L2(Ω) ≤ C(‖E‖L2(Ω) + ‖H‖L2(Ω)). (4.31)

Uniqueness is a consequence of the fact that k /∈ Spec (M`). This finishes the proof. 2

Proposition 4.6 Let Ω ⊆M be a Lipschitz domain and let ` ∈ 0, 1, ..., m. Also, fix a non-zerocomplex number k so that

k /∈ Spec (M`) ∪ Spec (Mm−`) (4.32)

and let E ∈ L2(Ω, Λ`TM), H ∈ L2(Ω,Λ`+1TM) solve the homogeneous Maxwell equations (withwave number k) in Ω, i.e.

(Maxwell)

dE − ikH = 0 in Ω,

δH + ikE = 0 in Ω.(4.33)

Then

‖ν ∧ E‖X 2`+1(∂Ω) ≈ ‖ν ∨H‖Y2

` (∂Ω), (4.34)

uniformly in (E,H).

Proof. For E, H as above, let us define E′ := (−1)` ∗ H and H ′ := ∗E. It follows thatE′ ∈ L2(Ω, Λm−`−1TM), H ′ ∈ L2(Ω,Λm−`TM) and (E′,H ′) solves (4.33). Hence, by repeatedapplications of Proposition 4.5 and the point (v) in Proposition 3.3,

‖ν ∧ E‖X 2`+1(∂Ω) ≈ ‖E‖L2(Ω) + ‖H‖L2(Ω) = ‖E′‖L2(Ω) + ‖H ′‖L2(Ω)

≈ ‖ν ∧ E′‖X 2m−`(∂Ω) = ‖ ∗ (ν ∨H)‖X 2

m−`(∂Ω)

≈ ‖ν ∨H‖Y2` (∂Ω). (4.35)

The proof is complete. 2

Finally, we are ready to tackle the issue raised in the first part of this section. Recall the discreteset U ⊆ R from Theorem 2.5.

Theorem 4.7 For each Lipschitz domain in Ω ⊆M there exists ε > 0 with the following property.If 2− ε < p < 2 + ε, 0 ≤ ` ≤ m and k /∈ ±kjj ∪ U also satisfies

k /∈ Spec (M`−1) ∪ Spec (Mm−`+1), (4.36)

then the operators in (4.7) are in fact isomorphisms.Furthermore, similar results are valid for the operators (4.8).

32

Proof. Consider the issue of inverting ±12I + Nk,` on X p

` (∂Ω) with p near 2. Let us assume fora moment that this holds for p := 2 and k := k0, some fixed, purely imaginary complex number.Then the extension to the more general situation described in the statement of the theorem isaccomplished as follows. To begin with, since by Proposition 3.4 the family X p

` (∂Ω)1<p<∞ is acomplex interpolation scale, we see there exists ε = ε(Ω) > 0 so that the operators in (4.7) areisomorphisms if 2− ε < p < 2 + ε and k = k0. This is a consequence of known stability results (cf.,e.g., [20] for a discussion). Extending further the aforementioned invertibility result to arbitrarywave numbers k /∈ ±kjj ∪ U and so that (4.36) holds, requires two other ingredients. First, dueto the fact that the main singularity in Γk,` is actually independent of k (cf. (2.21), (2.19), (2.22))and the commutator identities (2.24), (2.25), it follows that Nk,` −Nk0,` is a compact operator onX p

` (∂Ω), for any 1 < p < ∞, 0 ≤ ` ≤ m, and k /∈ ±kjj . In particular, if 2 − ε < p < 2 + ε,the operators in (4.7) are Fredholm with index zero for any k /∈ ±kjj . The second ingredientis a general stability result (cf. [20]) to the effect that if a linear operator T , mapping a complexinterpolation scale X p boundedly into itself, is Fredholm with index zero on X p for 2−ε < p < 2+εand is invertible on X 2, then T is actually invertible on X p for each 2− ε < p < 2 + ε.

Summarizing, at this stage we are left with proving that the operators in (4.7) are isomorphismsif p = 2 and k /∈ ±kjj ∩ U satisfies (4.36). To this end, for f ∈ X 2

` (∂Ω) arbitrary, we set

E := δSk,`f in Ω±, (4.37)

and

H :=1ik

dE = −ikSk,`f −1ik

δSk,`+1(d∂f) in Ω±. (4.38)

It follows that (E, H) solves the Maxwell system (4.33) both in Ω+ and in Ω−. Consequently, byProposition 4.6 and our assumptions on k,

‖ν ∧ (E|Ω±)‖X 2` (∂Ω) ≈ ‖ν ∨ (H|Ω±)‖Y2

`−1(∂Ω). (4.39)

The claim we make at this stage is that

ν ∨ (H|Ω+) = ν ∨ (H|Ω−) in Y2`−1(∂Ω). (4.40)

In order to see this, we note that the applications

X 2` (∂Ω) 3 f 7→ ν ∨ (H|Ω±) ∈ Y2

`−1(∂Ω) (4.41)

are continuous. Hence, by (iv) in Proposition 3.3 (with p = 2), it suffices to prove (4.40) whenf ∈ L2,d

nor(∂Ω, Λ`TM). In this case, based on the results of §2 we see that, even in the sense ofnontangential convergence to the boundary,

ν ∨H|∂Ω± = −ikν ∨ Sk,`f −1ik

ν ∨[

±12ν ∨ (d∂f) + δSk,`+1(d∂f)

]

= −ikν ∨ Sk,`f −1ik

ν ∨ δSk,`+1(d∂f). (4.42)

From this, (4.40) follows.Next, notice that (4.12) gives

ν ∧ (E|Ω±) =(

±12I + Nk,`

)

f. (4.43)

33

Armed with (4.40) and (4.43) we are finally ready to tackle the issue of invertibility of the operators(4.7) when p = 2. We write:

‖f‖X 2` (∂Ω) ≤ ‖(1

2I + Nk,`)f‖X 2` (∂Ω) + ‖(−1

2I + Nk,`)f‖X 2` (∂Ω)

= ‖ν ∧ (E|Ω−)‖X 2` (∂Ω) + ‖ν ∧ (E|Ω+)‖X 2

` (∂Ω)

≤ C‖ν ∨ (H|Ω−)‖Y2`−1(∂Ω) + ‖ν ∧ (E|Ω+)‖X 2

` (∂Ω)

= C‖ν ∨ (H|Ω+)‖Y2`−1(∂Ω) + ‖ν ∧ (E|Ω+)‖X 2

` (∂Ω)

≤ C‖ν ∧ (E|Ω+)‖X 2` (∂Ω)

= C‖(12I + Nk,`)f‖X 2

` (∂Ω). (4.44)

A similar calculation shows that 12 in the last term above can be replaced by −1

2 . That is,

‖f‖X 2` (∂Ω) ≤ C‖(±1

2I + Nk,`)f‖X 2` (∂Ω), (4.45)

uniformly for f ∈ X 2` (∂Ω). In particular, ±1

2I + Nk,` : X 2` (∂Ω) → X 2

` (∂Ω) are one-to-one andwith closed range. Furthermore, Theorem 2.5 in concert with (iv) in Proposition 3.3 give that theoperators under discussion have also dense ranges. Thus, they are isomorphisms of X 2

` (∂Ω), asdesired.

That similar statements are valid in the case of the operators (4.8) is seen from (v) in Proposi-tion 3.3 and the fact that ∗Nk,` = − ∗Mk,m−` (cf. Theorem 2.4). 2

Next, for 1 < p < ∞, introduce

Zpnor(∂Ω, Λ`TM) := Lp

nor(∂Ω,Λ`TM) ∩ X p` (∂Ω) (4.46)

equipped with the norm

‖f‖Zpnor(∂Ω,Λ`TM) := ‖f‖Lp(∂Ω,Λ`TM) + ‖f‖X p

` (∂Ω). (4.47)

Clearly, this makes Zpnor(∂Ω,Λ`TM) a Banach space. Analogously, we introduce

Zptan(∂Ω, Λ`TM) := Lp

tan(∂Ω, Λ`TM) ∩ Yp` (∂Ω) (4.48)

and equip it with the natural norm. Finally, we set

Wpnor(∂Ω,Λ`TM) :=

f ∈ X p` (∂Ω); d∂f ∈ Lp(∂Ω, Λ`+1TM)

, (4.49)

endowed with

‖f‖Wpnor(∂Ω,Λ`TM) := ‖f‖X p

` (∂Ω) + ‖d∂f‖Lp(∂Ω,Λ`+1TM), (4.50)

and Wptan(∂Ω,Λ`TM) := ∗Wp

nor(∂Ω, Λm−`TM).

Theorem 4.8 Let Ω be a Lipschitz domain in M, ` ∈ 0, 1, ...,m and k /∈ ±kjj ∪ U . Then theoperators

34

±12I + Nk,` : Zp

nor(∂Ω, Λ`TM) → Zpnor(∂Ω, Λ`TM), (4.51)

±12I + Nk,` : Wp

nor(∂Ω,Λ`TM) →Wpnor(∂Ω,Λ`TM), (4.52)

are well defined and bounded for each 1 < p < ∞. Furthermore, there exists ε = ε(Ω) > 0 so that,if 2 − ε < p < 2 + ε and k also satisfies (4.32), then the operators in (4.51)–(4.52) are in factisomorphisms.

Finally, similar conclusions are valid for the operators

±12I + Mk,` : Zp

tan(∂Ω,Λ`TM) → Zptan(∂Ω,Λ`TM), (4.53)

±12I + Mk,` : Wp

tan(∂Ω,Λ`TM) →Wptan(∂Ω, Λ`TM). (4.54)

Proof. That the actions of Nk,` on Lpnor(∂Ω, Λ`TM) and on X p

` (∂Ω) are compatible on the inter-section, has been demonstrated before (in Proposition 4.2). Also, the boundedness of 1

2I + Nk,` onZp

nor(∂Ω, Λ`TM) follows from that of 12I + Nk,` on Lp

nor(∂Ω, Λ`TM) (cf. §2) and on X p` (∂Ω) (cf.

Proposition 4.2), separately.Next we consider the issue of the invertibility of the operators (4.51) when p is close to 2. First,

injectivity on X p` (∂Ω) clearly entails injectivity on Zp

nor(∂Ω, Λ`TM). To show ontoness, in the lightof Theorem 4.7, it suffices to prove the implication

f ∈ X p` (∂Ω)

(

12I + Nk,`

)

f ∈ Lpnor(∂Ω, Λ`TM)

⇒ f ∈ Lpnor(∂Ω, Λ`TM). (4.55)

To this end, let f be as in the left side of (4.55) and set u := δSk,`f in Ω±. Thus,

u ∈ Lp(Ω±,Λ`−1TM),

du ∈ Lp(Ω±, Λ`TM),

δu = 0 in Ω±,

ν ∧ (u|Ω+) =(

12I + Nk,`

)

f ∈ Lpnor(∂Ω, Λ`TM).

(4.56)

Theorem 2.6 applied to u in Ω+ then implies that

ν ∨ (u|Ω+) ∈ Lptan(∂Ω, Λ`−2TM). (4.57)

Now, as in (4.40),

ν ∨ (u|Ω+) = ν ∨ (u|Ω−) (4.58)

so that, by (4.57),

ν ∨ (u|Ω−) ∈ Lptan(∂Ω, Λ`−2TM). (4.59)

In turn, (4.59) and (4.56) together with the same regularity result (i.e. Theorem 2.6), applied thistime to u in Ω−, yield

ν ∧ (u|Ω−) ∈ Lpnor(∂Ω, Λ`TM). (4.60)

With this at hand and using the fact that, by definition,

35

ν ∧ (u|Ω±) =(

±12I + Nk,`

)

f (4.61)

we arrive at the conclusion that

f =(

12I + Nk,`

)

f −(

−12I + Nk,`

)

f

= ν ∧ (u|Ω+)− ν ∧ (u|Ω−) ∈ Lpnor(∂Ω,Λ`TM). (4.62)

This proves (4.55) and concludes the proof of the part in Theorem 4.8 which refers to the operators(4.51).

Turning attention to the operators (4.52), the first order of business is to show that they arewell-defined and bounded. However, if f ∈ Wp

nor(∂Ω,Λ`TM) is arbitrary and u := δSk,`f , then

d∂

(

12I + Nk,`

)

f = d∂(ν ∧ u) = −ν ∧ du

= ν∧(

δSk,`+1(d∂f))

−k2ν ∧ Sk,`f

=(

12I + Nk,`+1

)

(d∂f)− k2ν ∧ Sk,`f. (4.63)

In particular, by the last part in Lemma 4.1 plus the fact that Nk,` is a bounded mapping of X p` (∂Ω)

and of Lp(∂Ω, Λ`TM) for each 0 ≤ ` ≤ m, 1 < p < ∞, we see that the operators (4.52) are indeedwell-defined and bounded.

As for their invertibility for p near 2 (and the extra assumptions on k), much as before, we onlyneed to prove the implication

f ∈ X p` (∂Ω) and

d∂

(

12I + Nk,`

)

f ∈ Lpnor(∂Ω,Λ`+1TM)

⇒ d∂f ∈ Lpnor(∂Ω, Λ`+1TM). (4.64)

Now the left side of (4.64) together with (4.63) and the last part in Lemma 4.1 imply that d∂f ∈X p

`+1(∂Ω) has the property that(

12I + Nk,`+1

)

(d∂f) ∈ Lpnor(∂Ω, Λ`+1TM). Thus, by our assump-

tions on k and (4.55) (adapted to the level of `+1 forms), it follows that d∂f ∈ Lpnor(∂Ω, Λ`+1TM),

as wanted. This finishes the proof of the claim made in the statement of the theorem for the oper-ators (4.52).

Finally, the last part of the statement of the theorem follows from what we have proved up tothis point and applications of the Hodge star isomorphism. 2

Theorem 4.9 Let Ω be a Lipschitz domain in M. Then there exists ε = ε(Ω) > 0 so that if2− ε < p < 2 + ε and the non-zero complex number k satisfies

k /∈ ±kjj ∪ U , k /∈⋃

0≤`≤m

Spec (M`), (4.65)

the Maxwell boundary value problem

(BV P4)k,`

E ∈ Lp(Ω, Λ`TM), H ∈ Lp(Ω, Λ`+1TM),

dE − ikH = 0 in Ω,

δH + ikE = 0 in Ω,

ν ∧ E = f ∈ X p`+1(∂Ω),

36

is uniquely solvable for each 0 ≤ ` ≤ m. The solution (E, H) of (BV P4)k,` also satisfies

‖E‖Lp(Ω,Λ`TM) + ‖H‖Lp(Ω,Λ`+1TM) ≤ C‖f‖X p`+1(∂Ω). (4.66)

Moreover, the following regularity statements are true:

(i) N (E) ∈ Lp(∂Ω) ⇔ f ∈ Zpnor(∂Ω,Λ`+1TM). Moreover, if one (and, hence, both) of these

conditions is true then, actually, E ∈ B1/p,p∗p (Ω, Λ`TM);

(ii) N (H) ∈ Lp(∂Ω) ⇔ f ∈ Wpnor(∂Ω, Λ`TM). Also, if one (and, hence, both) of these conditions

is valid then, in fact, H ∈ B1/p,p∗p (Ω, Λ`+1TM);

(iii) N (E), N (H) ∈ Lp(∂Ω) ⇔ f ∈ Lp,dnor(∂Ω, Λ`+1TM). In addition, if one (and, hence, both) of

these conditions holds then

‖E‖B1/p,p∗

p (Ω,Λ`TM)+ ‖H‖

B1/p,p∗p (Ω,Λ`+1TM)

≤ C‖f‖Lp,dnor(∂Ω,Λ`+1TM). (4.67)

Proof. Let ε > 0 be so that the conclusions of Theorem 4.7 and Theorem 4.8 are valid for each` ∈ 0, 1, ..., m if 2− ε < p < 2 + ε. Granted this and (4.65), a solution to (BV P4)k,` in Ω can beexpressed in the form

E := δSk,`

[(

12I + Nk,`

)−1f]

, (4.68)

H :=1ik

dE =1ik

dδSk,`

[

(

12I + Nk,`

)−1f]

= − 1ik

δdSk,`

(

(

12I + Nk,`

)−1f)

− 1ik

∆`Sk,`

(

(

12I + Nk,`

)−1f)

= − 1ik

δSk,`+1

(

d∂

(

12I + Nk,`

)−1f)

− ik Sk,`

(

(

12I + Nk,`

)−1f)

. (4.69)

From this (and the mapping properties of the operators involved), it is clear that E, H satisfy thedesired Lp estimates.

Turning our attention to the uniqueness part, we first observe that, via repeated integrationsby parts, the Green type representation formula

E = δSk,`+1(ν ∧ E)− dSk,`−1(ν ∨ E)− Sk,`(ν ∨ dE) (4.70)

is seen to hold for any E ∈ Lp(Ω,Λ`TM) with dE ∈ Lp(Ω, Λ`+1TM), δE = 0 and such that(∆` + k2)E = 0 in Ω. If, in addition, ν ∧ E = 0 (as is the case for the electric component of anull-solution of (BV P4)k,`), then (4.70) reduces to

E = −dSk,`−1(ν ∨ E)− Sk,`(ν ∨ dE). (4.71)

Applying d to the above identity then taking ν ∨ · of both sides yields (12I + Mk,`)−1(ν ∨ dE) = 0.

Now, since ν ∨ dE ∈ Yp` (∂Ω) with p ∈ (2 − ε, 2 + ε), Theorem 4.7 applies and gives ν ∨ dE = 0.

Utilizing this back in (4.71) gives

E = −dSk,`−1(ν ∨ E). (4.72)

37

By taking ν ∨ · of both sides we arrive at (12I + Mk,`−1)−1(ν ∨ E) = 0 and, much as before, via

another application of Theorem 4.7, to ν ∨E = 0. Plugging this back in (4.72) finally allows us toconclude that E = 0 for any null-solution (E, H) of (BV P4)k,`. Since this also entails H = 0, theproof of uniqueness is finished.

Next, we consider the regularity statements. Clearly, the fact that f ∈ Zpnor(∂Ω,Λ`TM) entails

N (E) ∈ Lp(∂Ω). The converse implication follows from the fact that N (E) ∈ Lp(∂Ω) ⇒ ∃E|∂Ω ∈Lp(∂Ω) in the sense of the nontangential convergence; cf. [28]. Note that, granted the membershipof f to Zp

nor(∂Ω), (4.68) entails the fact that E belongs to B1/p,p∗p (Ω). This proves (i).

The left-to-right implication in (ii) is seen from the identity

ν ∧H = − 1ik

d∂(ν ∧ E) = − 1ik

d∂f, (4.73)

since (∆`+1 + k2)H = 0 and N (H) ∈ Lp(∂Ω) entail the existence of H|∂Ω in Lp(∂Ω,Λ`+1TM).The opposite implication is a consequence of (4.69), the fact that the operators (4.52), (4.7) areisomorphisms (cf. Theorems 4.8, 4.7), and the estimates (2.31)–(2.32).

That any of the two sides of the equivalence in (ii) implies the membership of H to B1/p,p∗p (Ω)

is a consequence of (4.69), Theorem 2.4 and Proposition 2.7.Finally, (iii) is a direct consequence of (i) and (ii). 2

Remark. The boundary problem (BV P4)k,` becomes, after eliminating H,

(BV P5)k,`

E ∈ Lp(Ω, Λ`TM), dE ∈ Lp(Ω, Λ`+1TM),

(∆` + k2)E = 0 in Ω,

δE = 0 in Ω,

ν ∧ E = f ∈ X p`+1(∂Ω).

Consequently, the conclusions (i)− (iii) in the Theorem 4.9 (with H := 1ikdE) apply to (BV P5)k,`

also.

Theorem 4.10 Let Ω be a Lipschitz domain and let k ∈ C, ε = ε(Ω) > 0 be as in Theorem 4.9.Then, for 2− ε < p < 2 + ε, and 0 ≤ ` ≤ m consider the mapping

Ck,` : X p`+1(∂Ω) −→ Yp

` (∂Ω) (4.74)

given by

Ck,`(f) := ν ∨H, (4.75)

where (E, H) is the solution of the boundary problem (BV P4)k,` with boundary datum f . Then:

(i) Ck,` is an isomorphism of X p`+1(∂Ω) onto Yp

` (∂Ω);

(ii) Ck,` maps Zpnor(∂Ω, Λ`+1TM) isomorphically onto Zp

tan(∂Ω,Λ`TM);

(iii) Ck,` maps Wpnor(∂Ω, Λ`+1TM) isomorphically onto Wp

tan(∂Ω,Λ`TM);

(iv) Ck,` maps Lp,dnor(∂Ω,Λ`+1TM) isomorphically onto Lp,δ

tan(∂Ω, Λ`TM);

(v) C−1k,` = (−1)m(`+1)+1 ∗ Ck,m−`−1∗.

38

Proof. Clearly, if (E, H) solves the boundary problem (BV P4)k,` with boundary datum f , thenthe pair (∗H , (−1)`(∗E)) solves (BV P4)k,m−`−1 with boundary datum ν∧ (∗H) = (−1)` ∗ (ν∨H).Thus,

(−1)m(`+1)+1(∗Ck,m−`−1∗)(ν ∨H) = (−1)(`+1)(m+1) ∗ [ν ∨ (−1)`(∗E)] = ν ∧ E = f. (4.76)

From this and Theorem 4.9, (i)− (v) in Theorem 4.10 follow easily. We omit the details. 2

Let us point out that Ck,` is sometimes refereed to as the Calderon operator (or, capacityoperator, current-to-voltage operator).

5 Green operators for the Hodge-Laplacian and the Hodge Diracoperator

Here, again, we retain our usual hypotheses on M and its metric; cf. §2. In this section, our aimis to discuss several basic boundary value problems for the Hodge-Laplacian and the Hodge Diracoperator in a Lipschitz subdomain Ω of M. For related problems in smoother domains the readeris also referred to [35], [36], [37], [38], [8], [9], [5]. Our treatment is based on integral methods. Inparticular, this allows for a constructive approach to the existence (and properties) of the associatedGreen operators, i.e. the solution operators for the problems under discussion.

To begin with, for 1 < p < ∞ and 0 ≤ ` ≤ m, set

Hp,`∨ (Ω) := Np

` (δ∨,Ω) ∩Np` (d, Ω)

= u ∈ Lp(Ω, Λ`TM); du = 0, δu = 0, ν ∨ u = 0, (5.1)

and

Hp,`∧ (Ω) := Np

` (d∧, Ω) ∩Np` (δ,Ω)

= v ∈ Lp(Ω, Λ`TM); dv = 0, δv = 0, ν ∧ v = 0. (5.2)

The spaces Hp,`∨ (Ω), Hp,`

∧ (Ω) will be referred to as the Dirichlet and Neumann, respectively, har-monic fields of Ω. It has been proved in [28] that for any Lipschitz domain Ω there existsε = ε(Ω) > 0 so that Hp,`

∨ (Ω), Hp,`∧ (Ω) are independent of p ∈ (2− ε, 2 + ε) and

dimHp,`∧ (Ω) = bm−`(Ω), dimHp,`

∨ (Ω) = b`(Ω), (5.3)

where bj(Ω) is the jth Betti number of Ω. In the light of these results, we may occasionally suppressthe dependence of p whenever |p−2| < ε and simply write H`

∧(Ω), H`∨(Ω). From [28], we also know

that N (u) ∈ Lp(∂Ω) for any u ∈ Hp,`∧ (Ω)∪Hp,`

∨ (Ω), provided |p− 2| < ε. In particular, u|∂Ω makessense in this case.

Our first theorem in this section is as follows.

Theorem 5.1 For any Ω Lipschitz domain there exists ε = ε(Ω) > 0 so that, for any 2− ε < p <2+ ε and 0 ≤ ` ≤ m, the Lp-Poisson boundary problem for the Hodge-Laplacian (with homogeneous

39

relative boundary conditions):

(BV P6)`

∆`u = η ∈ Lp(Ω, Λ`TM),

u, dδu, δdu ∈ Lp(Ω, Λ`TM),

du ∈ Lp(Ω, Λ`+1TM), δu ∈ Lp(Ω,Λ`−1TM),

ν ∨ u = 0 in B− 1p ,p(∂Ω, Λ`−1TM),

ν ∨ du = 0 in B− 1p ,p(∂Ω, Λ`TM),

has a solution if and only if the datum η satisfies the compatibility condition

η ∈ [H`∨(Ω)], (5.4)

where [. . .] refers to the annihilator of the set [. . .] in Lp(Ω, Λ`TM).The space of null-solutions for (BV P6)` is precisely H`

∨(Ω); in particular, its dimension isb`(Ω) < ∞.

Furthermore, if (5.4) is satisfied, the following are true:

(i) The forms du, δu are uniquely determined and

‖du‖Lp(Ω) + ‖δu‖Lp(Ω) + ‖δdu‖Lp(Ω) + ‖dδu‖Lp(Ω) ≤ C‖η‖Lp(Ω). (5.5)

Also,

du = 0 ⇔ dη = 0, δu = 0 ⇔ η ∈ Np` (δ∨, Ω). (5.6)

(ii) The form u itself is uniquely determined if one also imposes the normality condition

u ∈ [H`∨(Ω)], (5.7)

and an integral representation formula for u is available. Also, in this case,

‖u‖Lp(Ω) ≤ C‖η‖Lp(Ω). (5.8)

(iii) The forms u, du, δu belong to B1/p,p∗p (Ω). In particular, if p = 2, then u, du, δu ∈ H

12 ,2(Ω)

and the exponent 12 is sharp in the class of Lipschitz domains.

Finally, a similar set of conclusions holds true for the Hodge-adjoint problem, i.e. the Poissonproblem with homogeneous absolute boundary conditions:

(BV P7)`

∆`v = ξ ∈ Lp(Ω,Λ`TM),

v, dδv, δdv ∈ Lp(Ω, Λ`TM),

dv ∈ Lp(Ω,Λ`+1TM), δv ∈ Lp(Ω, Λ`−1TM),

ν ∧ v = 0 in B− 1p ,p(∂Ω, Λ`+1TM),

ν ∧ δv = 0 in B− 1p ,p(∂Ω,Λ`TM).

This time, the compatibility condition becomes

40

ξ ∈ [H`∧(Ω)] (5.9)

while the normality condition becomes

v ∈ [H`∧(Ω)]. (5.10)

Also, the space of null-solutions for (BV P7)` is H`∧(Ω). In particular, its dimension is bm−`(Ω) <

∞.

In the proof of this theorem, the following lemma is going to play a crucial role. To state it, letΓV (x, y) be the Schwartz kernel of (∆` − V )−1, for some V ∈ C∞(M) with V ≥ 0 and so that∅ 6= suppV ⊆M\ Ω.

Lemma 5.2 For any Lipschitz domain Ω there exists ε = ε(Ω) > 0 with the following significance.For each ` ∈ 0, 1, . . . , m and p ∈ (2− ε, 2 + ε), the boundary value problem

(BV P8)`

w ∈ C0(Ω, Λ`TM),

∆`w = 0 in Ω,

N (w),N (dw) ∈ Lp(∂Ω),

ν ∨ w = f ∈ Lptan(∂Ω, Λ`−1TM),

ν ∨ dw = g ∈ Lptan(∂Ω,Λ`TM),

is solvable if and only if

g ∈ h|∂Ω; h ∈ H`∨(Ω), (5.11)

where . . . refers to the annihilator of . . . in Lptan(∂Ω, Λ`TM). If (5.11) is true, then any

solution can be expressed in the form “single layer + double layer”, i.e.

w(x) =∫

∂Ω〈ΓV (x, y), k1(y)〉 dσ(y) +

∫

∂Ω〈δyΓV (x, y), k2(y)〉 dσ(y) x ∈ Ω, (5.12)

for suitable k1 ∈ Lptan(∂Ω, Λ`TM), k2 ∈ Lp

tan(∂Ω, Λ`−1TM). Furthermore, matters can be arrangedso that

‖k1‖Lp(∂Ω) + ‖k2‖Lp(∂Ω) ≤ C(‖f‖Lp(∂Ω) + ‖g‖Lp(∂Ω)). (5.13)

for some C > 0 independent of w.The space of null solutions for (BV P8)` is H`

∨(Ω); in particular, dw and δw are uniquelydetermined. Moreover, whenever (5.11) is satisfied,

δw ∈ Lp(Ω, Λ`−1TM) ⇔ f ∈ Zptan(∂Ω, Λ`−1TM) (5.14)

and

dδw , δdw ∈ Lp(Ω, Λ`TM) ⇔ g ∈ Zptan(∂Ω, Λ`TM). (5.15)

41

Proof. Results of this type have been established in [28]. In fact, everything except for (5.14) and(5.15) is contained in §5 of [28]. In dealing with the two remaining equivalences, the crux of thematter is proving that whenever w ∈ C0(Ω, Λ`TM) satisfies

∆` w ∈ Lp(Ω,Λ`TM), N (w), N (dw) ∈ Lp(∂Ω),

∃w|∂Ω ∈ Lp(∂Ω,Λ`TM), ∃ (dw)|∂Ω ∈ Lp(∂Ω, Λ`+1TM) (5.16)

and ν ∨ (w|∂Ω) ∈ Zptan(∂Ω,Λ`−1TM),

then also

δ w ∈ Lp(Ω, Λ`−1TM), (5.17)

plus a natural estimate.To see this, fix a purely imaginary complex number k (so that k /∈

⋃

0≤`≤m Spec (M`); cf.the discussion in §4) and then set up the corresponding integral operators corresponding to theSchwartz kernel of (∆` + k2)−1. We shall make use of an integral representation formula valid forw satisfying (5.16) to the effect that

w = Πk,`((∆` + k2)w)

−δ Sk,`+1

[

(−12I + Nk,`+1)

−1(ν ∧ Tr (Πk,`(∆` + k2)w))]

−Sk,`

[

(

−12I + Nk,`

)−1(ν ∧ Tr (δ Πk,`(∆` + k2)w))

]

+δ Sk,`+1

[

(

−12I + Nk,`+1

)−1[

ν ∧ Sk,`

(

(

−12I + Nk,`

)−1(ν ∧ Tr (δ Πk,`(∆` + k2)w))

)]]

−Sk,`(ν ∨ (dw)|∂Ω)

+δ Sk,`+1

[

(

−12I + Nk,`+1

)−1(ν ∧ Sk,`(ν ∨ (dw)|∂Ω))

]

+Sk,`

[

(

−12I + Nk,`

)−1(ν ∧ δ Sk,`(ν ∨ (dw)|∂Ω))

]

−δ Sk,`+1

[

(

−12I + Nk,`+1

)−1[

ν ∧ Sk,`

(

(

−12I + Nk,`

)−1(ν ∧ δ Sk,`(ν ∨ (dw)|∂Ω))

)]]

−dSk,`−1(ν ∨ (w|∂Ω))

+δ Sk,`+1

[

(

−12I + Nk,`+1

)−1(ν ∧ dSk,`−1(ν ∨ (w|∂Ω)))

]

−k2Sk,`

[

(

−12I + Nk,`

)−1(ν ∧ Sk,`−1(ν ∨ (w|∂Ω)))

]

+k2δSk,`+1

[

(

−12I + Nk,`+1

)−1(

ν ∧ Sk,`

(

−12I + Nk,`

)−1(ν ∧ Sk,`−1(ν ∨ (w|∂Ω)))

)]

−Sk,`

[

(

−12I + Nk,`

)−1(ν ∧ dSk,`−2(δ∂(ν ∨ (w|∂Ω))))

]

+δ Sk,`+1

[

(

−12I + Nk,`+1

)−1[

ν ∧ Sk,`

(

−12I + Nk,`

)−1(ν ∧ dSk,`−2(δ∂(ν ∨ (w|∂Ω))))

]]

=: I1 + I2 + . . . + I13 + I14, (5.18)

42

in Lp(Ω, Λ`TM). The identity (5.18) is proved by integrating w against the Green function of theproblem (BV P8)` and successive integrations by parts. We refer the reader to Lemma 13.1 in [28]where a formula in the same spirit is proved.

Observe that δ annihilates I2, I4, I6, I8, I10, I12 and I14. Also, δI1, δI3, δI5, δI7, δI11, clearlybelong to Lp(Ω, Λ`−1TM). There remains to analyze δI9 and δI13. We write

δI9 = δdSk,`−1(ν ∨ w|∂Ω)

= −dδ Sk,`−1(ν ∨ w|∂Ω) + k2Sk,`−1(ν ∨ w|∂Ω)

= −dSk,`−2(δ∂(ν ∨ w|∂Ω)) + k2Sk,`−1(ν ∨ w|∂Ω), (5.19)

where the last equality utilizes (3.77). In this latter form Lemma 4.1 applies, since ν ∨ w ∈Zp

tan(∂ΩΛ`−1TM) entails δ∂(ν ∨ w|∂Ω) ∈ Yp`−2(∂Ω). Thus, we arrive at the conclusion that δI9 ∈

Lp(Ω, Λ`−1TM).As for δI13, we have

U := dSk,`−2(δ∂(ν ∨ (w|∂Ω))) ∈ Dp`−1(d, Ω) ⇒ ν ∧ U ∈ X p

` (∂Ω)

⇒(

−12I + Nk,`

)−1(ν ∧ U) ∈ X p

` (∂Ω)

⇒ Sk,`

[

(

−12I + Nk,`

)−1(ν ∧ U)

]

∈ H1,p(Ω,Λ`TM), (5.20)

thanks to Lemma 4.1. This gives δI13 ∈ Lp(Ω, Λ`−1TM), as desired.At this point, we have proved that (5.16) implies (5.17) which takes care of the right-to-left

implication in (5.14). The opposite one is obvious from the definition of Zptan(∂Ω, Λ`TM) and this

completes the proof of the equivalence (5.14).Turning attention to (5.15), it is easy to see that this follows by applying (5.14) to the form dw

plus the fact that ∆`w = 0. 2

Now, we are ready to present the

Proof of Theorem 5.1. Consider V ∈ C∞(M) a scalar, positive function so that ∅ 6= supp V ⊆M \ Ω and denote by ΓV (x, y) the Schwartz kernel of (∆ − V )−1. Also, let ΠV be the integraloperator with kernel ΓV (x, y). Finally, let ε > 0 be such that the previous Lp-results work for2− ε < p < 2 + ε. We look for a solution for the boundary value problem (BV P6)` in the form

u := ΠV η + w, (5.21)

where w is yet to be determined. Specifically, we take w to be a solution of

(BV P9)`

∆`w = 0 in Ω,

N (w), N (dw) ∈ Lp(∂Ω),

ν ∨ w = −ν ∨ Tr [ΠV η] ∈ Lptan(∂Ω, Λ`−1TM),

ν ∨ dw = −ν ∨ Tr [d ΠV η] ∈ Lptan(∂Ω, Λ`TM).

For this strategy to work, we need that (BV P9)` is solvable which, by Lemma 5.2, is the samething as

43

ν ∨ Tr [d ΠV η] ∈ h|∂Ω; h ∈ H`∨(Ω). (5.22)

Let us verify (5.22). Fix h ∈ H`∨(Ω) arbitrary and, using the support condition for V , write

∫

∂Ω〈ν ∨ Tr [dΠV η], h〉 dσ = −

∫ ∫

Ω〈δd ΠV η, h〉 dVol

=∫ ∫

Ω〈dδ ΠV η + η, h〉 dVol

=∫ ∫

Ω〈η, h〉 dVol. (5.23)

Thus, (BV P9)` is solvable if and only if (5.4) holds. Assuming that (5.4) is valid, we now proceedto verify that u is indeed a solution of (BV P6)`. In fact, we only need to check that

δu ∈ Lp(Ω,Λ`−1TM) and δdu, dδu ∈ Lp(Ω, Λ`TM). (5.24)

Granted (5.21) this comes down to

δw ∈ Lp(Ω,Λ`−1TM) and dδw, δdw ∈ Lp(Ω, Λ`TM), (5.25)

and further, by (5.14)–(5.15) in Lemma 5.2, to

ν ∨ Tr [ΠV η] ∈ Zptan(∂Ω, Λ`−1TM), (5.26)

and

ν ∨ Tr [d ΠV η] ∈ Zptan(∂Ω, Λ`TM). (5.27)

In turn, these are immediate from the mapping properties of ΠV and the definition of the spaceZp

tan(∂Ω, Λ`TM).Summarizing, we have proved that (BV P6)` is solvable if (5.4) is satisfied. Furthermore, our

solution is constructive in the sense that u can be represented as a “volume potential plus a singlelayer plus a double layer,” i.e.

u(x) =∫ ∫

Ω〈ΓV (x, y), η(y)〉 dVol(y) +

∫

∂Ω〈ΓV (x, y), k1(y)〉 dσ(y)

+∫

∂Ω〈δyΓV (x, y), k2(y)〉 dσ(y), (5.28)

where k1 ∈ Lptan(∂Ω,Λ`TM) and k2 ∈ Lp

tan(∂Ω, Λ`−1TM) are appropriately chosen. This followsfrom (5.21) and (5.12). The fact that the solvability of (BV P6)` implies (5.4) is a consequence ofstraightforward integrations by parts.

Turning attention to the space of null-solutions, we first note that if u solves the homogeneousversion of (BV P6)` for some p ∈ (2− ε, 2+ ε) then u is “regular” enough so that the identity (5.18)can be applied with w := u. This yields

44

u = k2δ Sk,`+1

[(

−12I + Nk,`+1

)−1(ν ∧ Tr (Πk,`u))

]

+k2Sk,`

[(

−12I + Nk,`

)−1(ν ∧ Tr (δΠk,`u))

]

(5.29)

−k2δ Sk,`+1

[(

−12I + Nk,`+1

)−1(ν ∧ Sk,`((−1

2I + Nk,`)−1(ν ∧ Tr (δΠk,`u))))]

so that, further,

N (u) ∈ Lp(∂Ω), N (δu) ∈ Lp(∂Ω). (5.30)

In fact, a similar conclusion also holds for du, but the calculation is more involved. More specifically,since the form du enjoys similar properties to u, the identity (5.18) can also be written for w := du,that is

du = k2δ Sk,`+2

[(

−12I + Nk,`+2

)−1(ν ∧ Tr (Πk,`+1(du)))

]

+k2Sk,`+1

[(

−12I + Nk,`+1

)−1(ν ∧ Tr (δΠk,`+1(du)))

]

(5.31)

−k2δ Sk,`+2

[(

−12I + Nk,`+2

)−1(ν ∧ Sk,`+1((−1

2I + Nk,`+1)−1(ν ∧ Tr (δΠk,`+1(du)))))]

.

Given the discussion in §2, an inspection of (5.31) reveals that

N (du) ∈ Lp(∂Ω). (5.32)

Thus, (5.30), (5.32) and the uniqueness part in Lemma 5.2 allow us to conclude that u ∈ H`∨(Ω),

as desired.Parenthetically, let us point out that we could have reached the same conclusion by proceeding

as follows. Let u be a null-solution of (BV P6)` and fix an arbitrary form η ∈ [H`∨(Ω)], where the

annihilator is taken in Lq(Ω, Λ`TM), 1/p + 1/q = 1. Using the existence part, we can producew solution of the Lq version of (BV P6)` with datum η. Then, starting with

∫∫

Ω〈u, ∆`w〉, we canarrive, via successive integrations by parts, to the conclusion that

∫∫

Ω〈u, η〉 = 0. Thus, since η wasarbitrary, we get u ∈ H`

∨(Ω) as desired.Conversely, any element from H`

∨(Ω) is a null-solution for (BV P6)`, and this proves takes careof the uniqueness part in the theorem. Now, the fact that du and δu are uniquely determinedfollows directly from this. The estimate (5.5) is a consequence of (5.28) and (5.13).

The left-to-right implication in the first equivalence in (5.6) is a direct consequences of the factthat ∆` “commutes” with d. As for the opposite one, if dη = 0 then, clearly, du is a null-solutionfor (BV P6)`+1. Thus du ∈ H`+1

∨ (Ω) which, in turn, implies∫∫

Ω〈du, du〉 = 0 via an integration byparts. Hence, du = 0 as desired.

Essentially the same type of argument works for the second equivalence in (5.6). Specifically,since ∆` commutes with δ, δu = 0 forces δη = 0. Moreover, ν ∨ η = −ν ∨ (δdu) = δ∂(ν ∨ du) = 0.This proves the right-to-left implication in the second equivalence in (5.6). Next, if δη = 0, thenw := δu ∈ Lp(Ω,Λ`−1TM) satisfies ν ∨ w = −δ∂(ν ∨ u) = 0 and

ν ∨ dw = ν ∨ dδu = −ν ∨ δdu− ν ∨ η = δ∂(ν ∨ du) = 0,

45

by assumptions. Thus, w = δu is a null-solution of (BV P6)`−1, i.e. δu ∈ H`−1∨ (Ω). Now, starting

with∫∫

Ω〈δu, δu〉 and utilizing dδu = 0, ν∨u = 0, an integration by parts gives δu = 0. This finishesthe proof of (5.6).

Obviously, granted the independence of Hp,`∨ (Ω) on p ∈ (2 − ε, 2 + ε), the condition (5.7)

determines u uniquely. The point (iii) is seen from Theorem 2.6 and the assumptions on u.Finally, the fact that similar conclusions hold for the Poisson problem for the Hodge Laplacian

with homogeneous absolute boundary conditions follows from what we have proved up to this pointand the properties of the Hodge star-operator. This completes the proof of Theorem 5.1. 2

In the sequel, fix a Lipschitz domain Ω and let ε > 0 be so that all our previous Lp-results arevalid for 2− ε < p < 2 + ε. Going further, for each ` ∈ 0, 1, ..., m, denote by

h`1, h

`2, . . . , h

`bm−`(Ω), (5.33)

a fixed basis for H`∨(Ω) and set

A` :=[

(

〈h`j , h

`k〉

)

j,k

]−1, A` a bm−`(Ω)× bm−`(Ω) matrix. (5.34)

Also, for each p ∈ (2− ε, 2 + ε), introduce the (projection) operator

P `∨ : Lp(Ω, Λ`TM) −→ Lp(Ω, Λ`TM),

P `∨u :=

∑

j

αjh`j , where (αj)j := A`(〈u, h`

j〉j). (5.35)

It is clear that the definition of P `∨ is independent of the basis (5.33). The Green operator for the

Hodge-Laplacian with relative boundary conditions is then defined for each p ∈ (2− ε, 2 + ε) by

G`∨ : Lp(Ω, Λ`TM) −→ Lp(Ω,Λ`TM)

G`∨ω := the unique solution of (BV P6)`, (5.4) and (5.7) (5.36)

with datum P `∨ω − ω.

Similarly, for p ∈ (2−ε, 2+ ε), the Green operator for the Hodge-Laplacian with absolute boundaryconditions is defined by

G`∧ : Lp(Ω,Λ`TM) −→ Lp(Ω, Λ`TM)

G`∧ω := the unique solution of (BV P7)`, (5.9) and (5.10) (5.37)

with datum P `∧ω − ω.

Here, P `∧ := ∗Pm−`

∨ is the projection operator of Lp(Ω, Λ`TM) onto the finite dimensional spaceH`∧(Ω), 2− ε < p < 2 + ε.

Several properties which are simple consequences of the results proved so far are as follows.First, the operator dual to G`

∧ acting on Lp(Ω,Λ`TM) is G`∧ acting on Lq(Ω, Λ`TM), where

p, q ∈ (2− ε, 2 + ε), 1p + 1

q = 1. Also,

46

dG`∨ = G`+1

∨ d on Dp` (d,Ω) and dG`

∧ = G`+1∧ d on Dp

` (d∧, Ω), (5.38)

δG`∧ = G`−1

∧ δ on Dp` (δ,Ω) and δG`

∨ = G`−1∨ δ on Dp

` (δ∨, Ω), (5.39)

∗G`∧ = Gm−`

∨ ∗ on Dp` (δ,Ω) and ∗G`

∨ = Gm−`∧ ∗ on Dp

` (d,Ω). (5.40)

One important aspect of our approach is the constructive proof of the existence of the Greenoperators G`

∨, G`∧. As explained in the course of the Proof of Theorem 5.1, there are explicit integral

representations of G`∨, G`

∧ in terms of potentials. Theorem 5.1 also yields mapping properties forG`∧ and G`

∨. More specifically, for each 2 − ε < p < 2 + ε, ` ∈ 0, 1, ..., m, the point (iii) inTheorem 5.1 gives

G`∧, G`

∨ : Lp(Ω, Λ`TM) −→ B1/p,p∗p (Ω, Λ`TM),

dG`∧, dG`

∨ : Lp(Ω,Λ`TM) −→ B1/p,p∗p (Ω, Λ`+1TM), (5.41)

δG`∧, δG`

∨ : Lp(Ω,Λ`TM) −→ B1/p,p∗p (Ω, Λ`−1TM)

are well-defined and bounded operators. In particular, when p = 2, the above operators map L2(Ω)

boundedly into H12 ,2(Ω) and (as simple counterexample in sectors of the complex plane show), the

exponent 12 is optimal in the class of Lipschitz domains. This stands in sharp contrast with the

case of smooth domains for which H12 ,2 can be replaced by H1,2.

It should be pointed out that under additional (suitable) convexity assumptions, it has beenproved in [31] that

G`∧, G`

∨ : L2(Ω,Λ`TM) −→ H12 ,2(Ω, Λ`TM),

dG`∧ : L2(Ω, Λ`TM) −→ H

12 ,2(Ω, Λ`+1TM), (5.42)

δG`∨ : L2(Ω, Λ`TM) −→ H

12 ,2(Ω, Λ`−1TM)

are, nonetheless, bounded operators. We shall not dwell on this here and refer the interested readerto [31] instead.

Certain spectral properties of independent interest for these Green operators are summarizedbelow.

Proposition 5.3 Let Ω be a Lipschitz domain in M. Then there exists ε = ε(Ω) > 0 with thefollowing significance. For each 2 − ε < p < 2 + ε and 0 ≤ ` ≤ m, the operator G`

∧ is compacton Lp(Ω, Λ`TM). Its spectrum, σ(G`

∧; Lp(Ω, Λ`TM)), is a bounded, countable set of nonnegativenumbers, accumulating at most at zero and which is independent of p.

Furthermore, λ > 0 has the property that λ−1 ∈ σ(G`∧; Lp(Ω, Λ`TM)) if and only if there exists

a nonzero `-form v so that

−∆`v = λv,

v, dδv, δdv ∈ Lp(Ω, Λ`TM),

dv ∈ Lp(Ω, Λ`+1TM), δv ∈ Lp(Ω,Λ`−1TM),

ν ∧ v = 0 in B− 1p ,p(∂Ω, Λ`+1TM),

ν ∧ δv = 0 in B− 1p ,p(∂Ω, Λ`TM).

(5.43)

47

If Ep`,λ(Ω) stands for the linear space of solutions of (5.43), then this is finite dimensional, indepen-

dent of p and satisfies (a generalized McKean-Singer telescopic formula)

m∑

`=0

(−1)`dim(

Ep`,λ(Ω)

)

= 0, ∀λ. (5.44)

Moreover, the Kunneth type formula

Ep`,λ(Ω1 × Ω2) =

∑

r1+r2=`

∑

µ1+µ2=λ

Epr1,µ1

(Ω1)⊗ Epr2,µ2

(Ω2), (5.45)

is valid for any two Lipschitz domains Ωj ⊂Mj, j = 1, 2.Finally, similar properties are valid for the operator G`

∨.

Proof. The fact that G`∧ is compact on Lp follows from (5.41). In particular, its spectrum consists

of eigenvalues which accumulate at most at zero. It is also clear that Ep`,λ(Ω) is precisely the

eigenspace corresponding to λ−1 ∈ σ(G`∧; Lp). Since, by (5.41) and classical embedding results,

this latter space is independent of p ∈ (2− ε, 2+ ε), it follows that σ(G`∧;Lp) is in fact independent

of p. That any eigenvalue is nonnegative is a consequence of fact that G`∧ ≥ 0, as an operator on

L2(Ω).We now turn our attention to (5.44). To this end, let us introduce the finite dimensional spaces

Ap`,λ(Ω) := v ∈ Ep

`,λ(Ω); dv = 0, Bp`,λ(Ω) := v ∈ Ep

`,λ(Ω); δv = 0. (5.46)

Then, so we claim,

Ep`,λ(Ω) = Ap

`,λ(Ω)⊕ Bp`,λ(Ω). (5.47)

Indeed, this is a simple consequence of the identity u = d(δG`∧u)⊕δ(dG`

∧u) valid for any u ∈ Ep`,λ(Ω).

Next, it is not difficult to check that d : Bp`−1,λ(Ω) → Ap

`,λ(Ω) is an isomorphism. Consequently,dimBp

`−1,λ(Ω) = dimAp`,λ(Ω). With this at hand, (5.47) gives

m∑

`=0

(−1)`dim(

Ep`,λ(Ω)

)

= dim(

Ap0,λ(Ω)

)

+ (−1)mdim(

Bpm,λ(Ω)

)

= 0, (5.48)

where the last equality follows from simple degree considerations.As for (5.45), which answers a question raised by B. Colbois, the right-to-left inclusion is simple

so we concentrate on the opposite one. Note that there is no loss of generality in assuming thatp = 2. Fix some u ∈ E2

`,λ(Ω1 × Ω2) and, for each r1 + r2 = `, consider the orthonormal basesin L2(Ωj , ΛrjTMj), j = 1, 2, consisting of forms urj

µj ,n ∈ E2rj ,µj

(Ωj), indexed by n and µj . Heren = 1, 2, ..., and µj is either 0 or µ−1

j belongs to the spectrum of the Green operator (5.37) onL2(Ωj , ΛrjTMj).

Writing u =∑

c(r1, r2, µ1, µ2, n′, n′′) ur1µ1,n′ ⊗ ur2

µ2,n′′ and the fact that λu = −∆M1×M2u, it isimmediate that µ1 + µ2 = λ. This, in turn, readily entails (5.45). The proof is finished. 2

We are now ready to state and prove the “full” Lp-Poisson problem for the Hodge-Laplacianwith relative and absolute boundary conditions.

48

Theorem 5.4 Let Ω be a Lipschitz domain in M. Then there exists ε = ε(Ω) > 0 so that, if2−ε < p < 2+ε and 0 ≤ ` ≤ m, then the Lp-Poisson problem for the Hodge-Laplacian with relativeboundary conditions

(BV P10)`

∆`u = η ∈ Lp(Ω, Λ`TM),

u, dδu, δdu ∈ Lp(Ω,Λ`TM),

du ∈ Lp(Ω, Λ`+1TM), δu ∈ Lp(Ω, Λ`−1TM),

u ∈ [H`∨(Ω)],

ν ∨ u = f ∈ Yp`−1(∂Ω),

ν ∨ du = g ∈ Yp` (∂Ω),

has a solution if and only if the data satisfy the compatibility condition∫ ∫

Ω〈η, ω〉 dVol =

∫

∂Ω〈g, ω〉 dσ, ∀ω ∈ H`

∨(Ω). (5.49)

The pairing in the right side of (5.49) should be understood as∫∫

Ω〈δW,ω〉 dVol if g = ν ∨ W ∈Yp

` (∂Ω) for some W ∈ Dp`+1(δ,Ω).

Granted (5.49), the solution is unique and satisfies

‖u‖Lp(Ω) + ‖du‖Lp(Ω) + ‖δu‖Lp(Ω) + ‖dδu‖Lp(Ω) + ‖δdu‖Lp(Ω)

≤ C(‖η‖Lp(Ω) + ‖f‖Yp`−1(∂Ω) + ‖g‖Yp

` (∂Ω)). (5.50)

Furthermore,

du = 0 ⇔ dη = 0 and g = 0. (5.51)

A similar set of results is valid for the Lp-Poisson problem with absolute boundary conditions,i.e.

(BV P11)`

∆v = ξ ∈ Lp(Ω,Λ`TM),

v, dδv, δdv ∈ Lp(Ω, Λ`TM),

dv ∈ Lp(Ω, Λ`+1TM), δv ∈ Lp(Ω, Λ`−1TM),

v ∈ [H`∧(Ω)],

ν ∧ v = h ∈ X p`+1(∂Ω),

ν ∧ δv = k ∈ X p` (∂Ω),

subject to the (necessary) compatibility condition∫ ∫

Ω〈ξ, ω〉 dVol = −

∫

∂Ω〈k, ω〉 dσ, ∀ω ∈ H`

∧(Ω). (5.52)

Proof. We shall only deal with (BV P10)` since (BV P11)` is its Hodge dual version. The strategyis to reduce (BV P10)` to the case when f = 0, g = 0, i.e. to the problem (BV P6)`. This can be

49

done as follows. Let k ∈ C be a fixed, purely imaginary number and consider a form w so that

(BV P12)`

(∆` + k2)w = 0 in Ω,

w, dδw, δdw ∈ Lp(Ω,Λ`TM),

dw ∈ Lp(Ω,Λ`+1TM), δw ∈ Lp(Ω, Λ`−1TM),

ν ∨ w = f ∈ Yp`−1(∂Ω),

ν ∨ dw = g ∈ Yp` (∂Ω).

Also, let u solve (BV P6)` for the datum η := η − ∆`w in Ω. Note that since P `∨(η) = 0, this is

equivalent to taking

u := −G`∨(η). (5.53)

Then, so we claim,

u := u + w − P `∨(u + w) (5.54)

solves (BV P10)` and satisfies (5.50). For this program to work, we need that w and u exist andsatisfy natural estimates. Now, the existence of w is assured by taking

w := Sk,`

[

(

−12I + Mk,`

)−1g]

(5.55)

+dSk,`−1

[

f − ν ∨ Sk,`

[

(

−12I + Mk,`

)−1g]]

where Sk,`, Mk,`, are as in §2-3. Note that, from (5.55) and the mapping properties of the operatorsinvolved, w constructed above also satisfies

‖w‖Lp(Ω) + ‖dw‖Lp(Ω) + ‖δw‖Lp(Ω) + ‖dδw‖Lp(Ω) + ‖δdw‖Lp(Ω)

≤ C(‖f‖Yp`−1(∂Ω) + ‖g‖Yp

` (∂Ω)). (5.56)

Second, the fact that u exists (plus accompanying estimates) is a consequence of the fact that(BV P6)` is solvable for the datum η, i.e. the compatibility condition η ∈ [H`

∨(Ω)] is satisfied. Inturn, this can be readily justified from (BV P12)` and integrations by parts.

Finally, (5.51) is a consequence of the uniqueness part in the statement of the theorem appliedto du. Also, the corresponding set of results for the Lp-Poisson problem with absolute boundaryconditions follows from what we have proved so far by applying the Hodge star operator. 2

Next, we introduce the space

H`,p(Ω) := u ∈ Lp(Ω, Λ`TM); du = 0, δu = 0 in Ω. (5.57)

Lemma 5.5 Assume that Ω is a Lipschitz domain in M. For 2 − ε < p < 2 + ε, ε = ε(Ω) > 0,and ` ∈ 0, 1, . . . ,m consider the application

π` : Lp(Ω, Λ`TM) −→ Lp(Ω, Λ`TM), π`(u) := u− δdG`∨u− dδG`

∧u. (5.58)

Then:

50

(i) π` is linear and bounded;

(ii) π`(u) = u ⇔ u ∈ H`,p(Ω);

(iii) π2` = π`;

(iv) The adjoint of π` acting on Lp(Ω, Λ`TM) is π` acting on Lq(Ω, Λ`TM), for 1p + 1

q = 1.

In particular, for p = 2, π` is the orthogonal projection of L2(Ω, Λ`TM) onto the closed subspaceH`,2(Ω).

Proof. (i) is clear from the properties of G`∧, G`

∨. The right-to-left implication in (ii) is a directconsequence of (5.39). To see the opposite one, fix some u ∈ Lp(Ω,Λ`TM) so that π`(u) = uand let uj ∈ Dp

` (d, Ω) ∩ Dp` (δ,Ω) be so that uj → u in Lp(Ω,Λ`TM). Then, d π`(uj) = duj +

∆`+1G`+1∨ (duj) = 0, δπ`(uj) = δuj + ∆`−1G`−1

∧ (δuj) = 0, i.e. π`(uj) ∈ H`,p(Ω) and π`(uj) →π`(u) = u. It follows that u ∈ H`,p(Ω) as desired. The rest of lemma follows if we show that, forp = 2, π` is the orthogonal projection of L2(Ω, Λ`TM) onto H2,`(Ω). Indeed, the only thing left tocheck is that

〈u− π`(u), h〉 = 0, ∀h ∈ H`,2(Ω). (5.59)

However, this is immediate from the definition of π` and repeated integrations by parts. 2

Our next result is a mixed Lp-Poisson problem for the Hodge-Laplacian with homogeneousboundary conditions.

Theorem 5.6 For any Lipschitz domain Ω there exists ε = ε(Ω) > 0 with the following significance.For any 2 − ε < p < 2 + ε and any ` ∈ 0, 1, . . . , m, the mixed Poisson boundary problem (withhomogeneous boundary conditions) for the Hodge-Laplacian

(BV P13)`

∆`u = µ ∈ Lp(Ω, Λ`TM),

u, dδu, δdu ∈ Lp(Ω,Λ`TM),

du ∈ Lp(Ω, Λ`+1TM), δu ∈ Lp(Ω, Λ`−1TM),

ν ∨ du = 0 in B− 1p ,p(∂Ω, Λ`TM),

ν ∧ δu = 0 in B− 1p ,p(∂Ω, Λ`TM),

is solvable if and only if

µ ∈ [H`,q(Ω)],1p

+1q

= 1; (5.60)

here [. . .] refers to the annihilator of [. . .] in Lp(Ω, Λ`TM).

(i) The space of null-solutions is precisely the infinite-dimensional space H`,p(Ω). In particular,du, δu are uniquely determined and

‖du‖Lp(Ω) + ‖δu‖Lp(Ω) ≤ C‖µ‖Lp(Ω). (5.61)

51

(ii) Granted (5.60), the solution is uniquely determined by the specification

u ∈ [H`,q(Ω)]. (5.62)

and, in this case,

‖u‖Lp(Ω) ≤ C‖µ‖Lp(Ω). (5.63)

(iii) Assuming that (5.60) is satisfied, any solution u has the property that

du ∈ B1/p,p∗p (Ω, Λ`+1TM), δu ∈ B1/p,p∗

p (Ω,Λ`−1TM). (5.64)

In particular, when p = 2, du and δu have H12 ,2(Ω)-coefficients.

(iv) For any solution u, there hold du = 0 ⇔ µ ∈ Np` (d∧, Ω) and δu = 0 ⇔ µ ∈ Np

` (δ∨,Ω).

Proof. Retain the notation introduced at the beginning of the proof of Theorem 5.1. We look fora solution to (BV P13)` in the form

u := ΠV µ + w, (5.65)

where w is a solution of

(BV P14)`

∆`w = 0 in Ω,

N (w), N (dw), N (δw) ∈ Lp(∂Ω),

ν ∨ dw = −ν ∨ Tr [dΠV µ] ∈ Lptan(∂Ω, Λ`TM),

ν ∧ δw = −ν ∧ Tr [δ ΠV µ] ∈ Lpnor(∂Ω,Λ`TM).

From Theorem 5.2 in [28] it is known that the boundary problem (BV P14)` is solvable if and onlyif

ν ∨ Tr [d ΠV µ]− ν ∧ Tr [δ ΠV µ]

belongs to the annihilator (taken in Lp(∂Ω,Λ`TM), 1p + 1

q = 1) of the set

h|∂Ω; h ∈ C0(Ω,Λ`TM), N (h) ∈ Lq(∂Ω), dh = 0, δh = 0

. (5.66)

Let us verify that (5.60) implies the aforementioned compatibility condition. Indeed, with h as in(5.66), granted (5.60) and the fact that suppV ∩ Ω = ∅ we have that

∫

∂Ω〈ν ∨ Tr [d ΠV µ], h〉 dσ = −

∫ ∫

Ω〈δd ΠV µ, h〉 dVol

=∫ ∫

Ω〈dδ ΠV µ, h〉 dVol

=∫

∂Ω〈ν ∧ Tr [δ ΠV µ], h〉 dσ. (5.67)

Hence, we can find a solution w of (BV P14)`, which also satisfies natural estimates. Moreover,from [28], w is constructively described as a single layer potential plus a derivative of a single

52

layer potential, both of which act on Lp boundary forms. In particular, from the results in §2,w ∈ B1/p,p∗

p (Ω, Λ`TM). A similar conclusion applies to dw and δw.Going further, with w at hand, we can construct u as in (5.65). For u to solve (BV P13)`,

we only need to check that dδu, δdu ∈ Lp(Ω, Λ`TM) or, since ∆`w = 0, δdw ∈ Lp(Ω,Λ`TM).However, this follows from Lemma 5.2 given that the boundary datum satisfies

ν ∨ Tr [dΠV µ] ∈ Zptan(∂Ω, Λ`TM). (5.68)

At this stage, we have shown that, given (5.60), the boundary problem (BV P14)` is solvable. Theconverse implication follows by simple integration by parts which we omit.

Consider next a null-solution u of (BV P13)`. It follows that du is a null-solution of (BV P6)`+1and, hence, du ∈ H`+1

∨ (Ω). This forces du = 0 as simple integrations by parts show. Similarlyδu = 0, so that u ∈ H`,p(Ω). Consequently, the space of null-solution for (BV P13)` is preciselyH`,p(Ω). Now, the right-to-left implications in (iv) are readily implied by this and integrationsby parts. As for the first left-to-right implication in (iv) observe that, granted the membershipof µ to Np

` (d∧, Ω), the form w := du ∈ Lp(Ω, Λ`+1TM) is a null-solution of (BV P13)`+1. Hence,du ∈ H`+1,p(Ω) and, further, du ∈ H`+1,p

∨ (Ω) by assumptions. As before, this ultimately yieldsdu = 0, as desired. The second left-to-right implication in (iv) is handled similarly. The oppositeimplications are simpler and follow from commutation relations. This finishes the proof of (iv).

Next, (5.61) and (iii) are consequences of the constructive way in which the solutions aredescribed, plus the mapping properties of the operators involved. As for (ii), we first note thatthe normalization (5.62) is possible by replacing u, constructed before, by u := u − π`(u). Now,π`(u) = 0 so that, if h ∈ H`,q(Ω),

〈u, h〉 = 〈u, π`(h)〉 = 〈π`(u), h〉 = 0. (5.69)

Regarding the uniqueness of a solution satisfying (5.62), observe that, if u is a null-solution for(BV P13)` so that (5.62) holds, then u ∈ H`,p(Ω) and, ∀ v ∈ Lq(Ω, Λ`TM), 1

p + 1q = 1,

〈u, v〉 = 〈π`(u), v〉 = 〈u, π`(v)〉 = 0 (5.70)

since π`(v) ∈ H`,q(Ω). Thus, necessarily, u = 0. This completes the proof of Theorem 5.6. 2

At this stage we can define the Green operator for the boundary problem (BV P13)` by setting

G`0 : Lp(Ω,Λ`TM) −→ Lp(Ω,Λ`TM)

G`0ω := the unique solution of (BV P13)`, (5.60) and (5.62) (5.71)

with datum π`(ω)− ω.

Note that, once again, the description of this operator is constructive, in that the solution of(BV P13)` is obtained via integral operators. Also, Theorem 5.6 can be rephrased in terms ofmapping properties for G`

0.We are now in a position to formulate and solve the “full” mixed Lp-Poisson problem for the

Hodge-Laplacian.

53

Theorem 5.7 Let Ω be a Lipschitz domain in M. Then there exists ε = ε(Ω) > 0 so that for each2− ε < p < 2 + ε and 0 ≤ ` ≤ m, the mixed Lp-Poisson problem

(BV P15)`

∆`u = µ ∈ Lp(Ω,Λ`TM),

u, dδu, δdu ∈ Lp(Ω, Λ`TM),

du ∈ Lp(Ω, Λ`+1TM), δu ∈ Lp(Ω, Λ`−1TM),

ν ∨ du = f ∈ Yp` (∂Ω),

ν ∧ δu = g ∈ X p` (∂Ω),

u ∈ [H`,q(Ω)], 1p + 1

q = 1,

has solution if and only if the compatibility condition∫ ∫

Ω〈µ, ω〉 dVol =

∫

∂Ω〈f − g, ω〉 dσ, ∀ω ∈ H`,q(Ω), (5.72)

is satisfied. Granted (5.72), the solution is unique and satisfies

‖u‖Lp(Ω) + ‖du‖Lp(Ω) + ‖δu‖Lp(Ω) + ‖dδu‖Lp(Ω) + ‖δdu‖Lp(Ω)

≤ C(‖µ‖Lp(Ω) + ‖f‖Yp` (∂Ω) + ‖g‖X p

` (∂Ω)). (5.73)

Also,

du = 0 ⇔ dµ = 0 and f = 0, δu = 0 ⇔ δµ = 0 and g = 0. (5.74)

Proof. We look for u in the form

u := u + w − π`(u + w), (5.75)

where w solves

(BV P16)`

(∆` + k2)w = 0 in Ω,

w, dδw, δdw ∈ Lp(Ω, Λ`TM),

dw ∈ Lp(Ω, Λ`+1TM), δw ∈ Lp(Ω, Λ`−1TM),

ν ∨ dw = f ∈ Yp` (∂Ω),

ν ∧ δw = g ∈ X p` (∂Ω),

and u solves (BV P13)` for the datum µ := µ−∆`w. Since π`(µ) = 0 (see below), this comes downto taking u := −G`

0(µ).Thus, all we need to check is that w and u exist and satisfy natural estimates. To this effect,

consider

w := k−2δdSk,`

[

(

−12I + Mk,`

)−1f]

+ k−2dδ Sk,`

[

(

12I + Nk,`

)−1g]

, (5.76)

for some k ∈ C, Im k > 0. Note that the membership of w to Lp(Ω, Λ`TM) follows from (5.76)much as in (3.79). Proceeding in a similar fashion, it is not difficult to check that w solves (BV P16)`and obeys natural estimates.

54

As for u, we only need to verify that µ ∈ [H`,q(Ω)], a compatibility condition which ensures thesolvability of (BV P13)` for the datum µ. However, this is a consequence of (5.72) and simple inte-grations by parts. We omit the straightforward details. Finally, (5.74) follows from the uniquenesspart in Theorem 5.4. 2

Next, we study the Green operators associate with boundary value problems for the Diracoperator D := d + δ on the Grassmann algebra

G := ⊕m`=0Λ

`TM. (5.77)

Specifically, for a Lipschitz domain Ω in M set

H∨(Ω) := ⊕m`=0H`

∨(Ω) and H∧(Ω) := ⊕m`=0H`

∧(Ω).

Also, for 1 < p < ∞, consider

(BV P17)

u, du, δu ∈ Lp(Ω,G),

(d + δ)u = ζ ∈ Lp(Ω,G),

ν ∨ u = 0 in B− 1p ,p(∂Ω,G).

The main result regarding the solvability of (BV P17) is as follows.

Theorem 5.8 Assume that Ω is a Lipschitz domain in M. There exists ε = ε(Ω) > 0 so that theboundary problem (BV P17) is solvable for 2− ε < p < 2 + ε if and only if

ζ ∈ [H∨(Ω)]. (5.78)

Also:

(i) The space of null-solutions is H∨(Ω), whose dimension is∑m

`=0 b`(Ω) < ∞. In particular, duand δu are uniquely determined and

‖du‖Lp(Ω) + ‖δu‖Lp(Ω) ≈ ‖ζ‖Lp(Ω). (5.79)

(ii) Granted (5.78), the extra condition

u ∈ [H∨(Ω)] (5.80)

determines a unique solution u which also satisfies

‖u‖Lp(Ω) ≤ C‖ζ‖Lp(Ω). (5.81)

(iii) Any solution u satisfies

dδu ∈ Lp(Ω,G) ⇔ dζ ∈ Lp(Ω,G), dδu = 0 ⇔ dζ = 0 (5.82)

and

δdu ∈ Lp(Ω,G) ⇔ δζ ∈ Lp(Ω,G), δdu = 0 ⇔ δζ = 0. (5.83)

55

(iv) Any solution u belongs to B1/p,p∗p (Ω,G). In particular, if p = 2, then u ∈ H

12 ,2(Ω,G). The

exponent 1/2 is sharp in the class of Lipschitz domains.

Similar conclusions are valid for the Hodge-dual boundary problem, i.e.

(BV P18)

v, dv, δv ∈ Lp(Ω,G),

(d + δ)v = k ∈ Lp(Ω,G),

ν ∧ v = 0 in B− 1p ,p(∂Ω,G).

In this case, the counterpart of (5.80) reads

v ∈ [H∧(Ω)], (5.84)

while the counterpart of (5.78) is

k ∈ [H∧(Ω)]. (5.85)

Finally, an analogous statement holds for the Dirac operator Dα := d + αδ, where α ∈ R \ 0.

Proof. Retain the notation introduced at the beginning of the proof of Theorem 5.1. We look fora solution expressed as

u := (d + δ)ΠV ζ + (d + δ)v (5.86)

where v is assumed to solve the boundary problem

(BV P19)

∆v = 0 in Ω,

N (v), N (dv) ∈ Lp(∂Ω,G),

ν ∨ v = 0 on ∂Ω,

ν ∨ dv = −ν ∨ Tr [(d + δ)ΠV ζ] ∈ Lptan(∂Ω,G).

One important aspect is that (5.80) decouples according to the natural splitting of v ∈ ⊕`Λ`TM informs of homogeneous degrees. Consequently, the theory in [28] applies for each degree separatelyand gives that (BV P19) is solvable if and only if

ν ∨ Tr [(d + δ)ΠV ζ] ∈ h|∂Ω; h ∈ H∨(Ω), (5.87)

which we now proceed to verify. Indeed, if h ∈ H∨(Ω), then

∫

∂Ω〈ν ∨ Tr [(d + δ)ΠV ζ], h〉 dσ =

∫ ∫

Ω〈δd ΠV ζ, h〉 dVol

=∫ ∫

Ω〈dδ ΠV ζ, h〉 dVol +

∫ ∫

Ω〈ζ, h〉 dVol

=∫ ∫

Ω〈ζ, h〉 dVol

= 0, (5.88)

where the last equality follows from (5.78). The fact that (5.78) is necessary for the solvabilityof (BV P17) follows from integrations by parts. To check that (5.86) is actually a solution for(BV P17) we only need to verify that

56

δv, dδv, δdv ∈ Lp(Ω,G). (5.89)

However, this is a consequence of the last part of Lemma 5.2. The first part in the statement ofTheorem 5.8 follows.

To describe the space of null-solutions, observe that if u solves the homogeneous version of(BV P17) then δu ∈ H∨(Ω)(=: ⊕m

`=0H`∨(Ω)) and, further, du ∈ H∨(Ω). Based on this and simple

integrations by parts, we may then conclude that du = 0. This entails δu = 0 so that, finally,u ∈ H∨(Ω). From this (i) is easily seen.

Following the same pattern as before, the rest of the theorem is more or less a direct consequenceof the results proved so far; we omit the details. 2

Theorem 5.8 allows us to introduce two Green operators for D = d + δ by setting

GD∨ω := the unique solution of (BV P17), (5.78) and (5.80)

for the datum ω − P∨ω, (5.90)

where ω ∈ Lp(Ω,G), 2− ε < p < 2 + ε, and P∨ := ⊕m`=0P

`∨, as well as

GD∧ ω := the unique solution of (BV P18), (5.84) and (5.85)

for the datum ω − P∧ω, (5.91)

where ω ∈ Lp(Ω,G), 2− ε < p < 2 + ε, and P∧ := ⊕m`=0P

`∧.

A direct calculation shows that ∗GD∧ = GD

∨ ∗, the adjoint of GD∧ on Lp(Ω,G) is GD

∧ on Lq(Ω,G),1p + 1

q = 1, and

GD∧ GD

∧ = ⊕m`=0G

`∧, GD

∨ GD∨ = ⊕m

`=0G`∨. (5.92)

Here, G`∨, G`

∨ are the Green operators for the Hodge-Laplacian, introduced in (5.36)–(5.37).Once again, Theorem 5.8 can be rephrased in terms of the mapping properties of GD

∧ , GD∧ .

Among other things,

GD∧ , GD

∨ : Lp(Ω,G) −→ B1/p,p∗p (Ω,G) (5.93)

are well defined and bounded. In particular, when p = 2, then GD∧ and GD

∨ map L2(Ω,G) into

H12 ,2(Ω,G) and the exponent 1/2 is sharp in the class of Lipschitz domains.

Finally, we are now ready to present the “full” Lp-Poisson problem for the Hodge Dirac operatorD = d + δ.

Theorem 5.9 Let Ω be a Lipschitz domain in M. Then there exists ε = ε(Ω) > 0 so that if2− ε < p < 2 + ε and ` ∈ 0, 1, . . . , m then the Lp-Poisson problem for the Hodge Dirac operator

(BV P20)

u, du, δu ∈ Lp(Ω,G),

(d + δ)u = ζ ∈ Lp(Ω,G),

ν ∨ u = f ∈ Yp(∂Ω) := ⊕m`=0Y

p` (∂Ω),

u ∈ [H∨(Ω)],

57

has a solution if and only if the compatibility condition∫ ∫

Ω〈ζ, ω〉 dVol = −

∫

∂Ω〈f, ω〉 dσ, ∀ω ∈ H∨(Ω) (5.94)

is satisfied (here the pairing in the right side should be interpreted in the natural sense). Granted(5.94), the solution is unique and there holds

‖u‖Lp(Ω) + ‖du‖Lp(Ω) + ‖δu‖Lp(Ω) ≤ C(‖ζ‖Lp(Ω) + ‖f‖Yp(∂Ω)). (5.95)

Moreover, there exists ` ∈ 0, 1, ..., m so that a solution u of (BV P20) can be found in Lp(Ω, Λ`TM),i.e. u is a homogeneous form of degree `, if and only if the following conditions are satisfied:

ζ = a + b, a ∈ Np`+1(d, Ω) ∩ [H`+1

∨ (Ω)], b ∈ Np`−1(δ,Ω), (5.96)

f ∈ Yp`−1(∂Ω), δ∂f = −ν ∨ b, (5.97)

∫ ∫

Ω〈b, ω〉 dVol = −

∫

∂Ω〈f, ω〉 dσ, ∀ω ∈ H`−1

∨ (Ω). (5.98)

In this latter case, (BV P20) becomes

(BV P20)`

u ∈ Lp(Ω,Λ`TM),

du = a ∈ Np`+1(d,Ω) ∩ [H`+1

∨ (Ω)],

δu = b ∈ Np`−1(δ,Ω),

ν ∨ u = f ∈ Yp`−1(∂Ω),

u ∈ [H`∨(Ω)].

Finally, similar conclusions are valid for the Hodge-dual problem, i.e. for

(BV P21)

u, du, δu ∈ Lp(Ω,G),

(d + δ)v = ξ ∈ Lp(Ω,G),

ν ∧ u = g ∈ X p(∂Ω) := ⊕m`=0X

p` (∂Ω),

u ∈ [H∧(Ω)].

In this case, the corresponding compatibility condition reads∫ ∫

Ω〈ξ, w〉 dVol =

∫

∂Ω〈g, w〉 dσ, ∀w ∈ H∧(Ω). (5.99)

Proof. The solution is sought in the form

u := u + w − P∨(u + w), (5.100)

where, for some k ∈ C, purely imaginary, w is given by

w :=∑

0≤`≤m

dSk,`

[(

−12I + Mk,`

)−1f`

]

if f =∑

0≤`≤m

f` ∈ ⊕m`=0Y

p` (∂Ω), (5.101)

and u solves (BV P17) for the datum ζ := ζ − (d + δ)w ∈ Lp(Ω,G), i.e. u = GD∨ (ζ), since one

can check that ζ ∈ [H∨(Ω)]. The estimate (5.95) is a direct consequence of this construction.

58

Uniqueness follows by the same token as before, while the last part in the statement of the theorem(regarding (BV P21)), is seen via an application of the Hodge star isomorphism.

There remains for us to show that the existence of a solution in Lp(Ω, Λ`TM) is equivalent to(5.96)–(5.98). The left-to-right implication is clear. To see the opposite one, note that (5.98) andTheorem 5.4 ensure the existence of w ∈ Lp(Ω, Λ`−1TM) with dw, δw, dδw ∈ Lp(Ω) and so that

∆`w = −b, ν ∨ w = 0, ν ∨ dw = f. (5.102)

By subtracting dw matters are reduced to finding u ∈ Lp(Ω, Λ`TM) so that

du = a, δu = b′ := b− δdw, ν ∨ u = 0. (5.103)

Since δb′ = δb − δ2dw = 0 and ν ∨ b′ = ν ∨ b + δ∂(ν ∨ dw) = 0 by (5.97) and (5.102), we see thatb′ ∈ Np

`−1(δ∨, Ω). Also, thanks to (5.98), we have∫∫

Ω〈b′, ω〉 = 0 for any ω ∈ H`−1∨ (Ω). At this

stage, we need some simple consequences of the Hodge decompositions which we shall prove in thenext section (and which, in turn, are independent of the present theorem). Specifically, we have

Np`−1(δ∨, Ω) = δDp

` (δ∨,Ω)⊕H`−1∨ (Ω), Np

`+1(d,Ω) ∩ [H`+1∨ (Ω)] = dDp

` (d, Ω),

for the range of p’s we are currently considering. Thus, based on the previous observations, we mayconclude that, in fact, b′ ∈ δDp

` (δ∨, Ω) and a ∈ dDp` (d,Ω).

Going further, we claim that the membership of a, b′ to, respectively, dDp` (d, Ω) and δDp

` (δ∨,Ω)is equivalent to the existence of α ∈ Dp

`+1(δ∨,Ω) and β ∈ Dp`−1(d, Ω) with δα ∈ Dp

` (d,Ω), dβ ∈Dp

` (δ∨, Ω) and so that a = dδα, b′ = δdβ. Again, this follows easily from the Hodge decompositionsproved in §6. Then, u := δα + dβ is an `-form which solves (5.103). 2

6 Constructive Lp-Hodge decompositions on Lipschitz domains

The conventional wisdom is that the most complete form of the Hodge decompositions can beachieved through the use of the corresponding Green operators. Having done that in a constructivefashion and at the Lp level, 2− ε < p < 2 + ε, on arbitrary Lipschitz domains, we can now proceedto develop a simple and unified approach to decompositions of forms which retains the same basicfeatures.

Theorem 6.1 (Hodge decompositions) Let Ω be an arbitrary Lipschitz subdomain of M. Thenthere exists ε = ε(Ω) > 0 so that, for any 2− ε < p < 2 + ε, and any ` ∈ 0, 1, . . . ,m,

Lp(Ω, Λ`TM) = dDp`−1(d∧, Ω)⊕ δDp

`+1(δ,Ω)⊕Hp,`∧ (Ω), (6.1)

Lp(Ω,Λ`TM) = dDp`−1(d, Ω)⊕ δDp

`+1(δ∨, Ω)⊕Hp,`∨ (Ω), (6.2)

where the direct sums are topological. Furthermore, the decompositions (6.1)–(6.2) are obtained ina constructive fashion (in the sense explained below).

Proof. This is more or less an immediate consequence of the results in the previous section.Specifically, if ε > 0 is so that the results in §5 hold for 2− ε < p < 2 + ε then, for this range of p’s,any u ∈ Lp(Ω,Λ`TM) can be written in the form

u = dα + δβ + γ, α ∈ Dp`−1(d∧, Ω), β ∈ Dp

`+1(δ,Ω), γ ∈ H`∧(Ω), (6.3)

59

where

α := δG`∧u, β := dG`

∧u, γ := P `∧u. (6.4)

Note that, by Theorem 5.1,

‖dα‖Lp(Ω) + ‖δβ‖Lp(Ω) + ‖γ‖Lp(Ω) ≤ C(Ω, p)‖u‖Lp(Ω) (6.5)

and that, given the constructive character of the Green operator G`∧, integral representation for-

mulas are valid for α, β, γ.As far as (6.1) is concerned, there remains to establish the uniqueness of the decomposition

(6.3). To this end, we assume that u = 0 and seek to prove that dα = 0, δβ = 0 and γ = 0. Let qbe the conjugate exponent of p. It follows that γ ∈ Lq(Ω, Λ`TM) and, via integrations by parts,

0 =∫ ∫

Ω〈dα + δβ + γ, γ〉 =

∫ ∫

Ω〈dα, γ〉+

∫ ∫

Ω〈δβ, γ〉+

∫ ∫

Ω〈γ, γ〉

=∫ ∫

Ω〈γ, γ〉. (6.6)

Thus, necessarily, γ ≡ 0 which also entails dα + δβ = 0. Going further, consider an arbitrary formv ∈ Lq(Ω,Λ`TM) which, thanks to the existence part in Lq, can be decomposed as

v = da + δb + c, a ∈ Dq`−1(d∧, Ω), b ∈ Dq

`+1(δ,Ω), c ∈ H`∧(Ω). (6.7)

Now, using dα = −δβ and integrating by parts we obtain∫ ∫

Ω〈dα, v〉 = −

∫ ∫

Ω〈δβ, da〉+

∫ ∫

Ω〈dα, δb〉+

∫ ∫

Ω〈dα, c〉 = 0. (6.8)

Since v was arbitrary this implies that dα = 0 and, further, δβ = 0. This shows that the sums in(6.1) are also direct.

That analogous conclusions apply to (6.2) is seen from what we have proved up to this pointplus an application of the Hodge star isomorphism (alternatively, we may base our discussion onG`∨). 2

Remark. The counterexamples in [10] show that there are necessary restrictions on the range of p’sfor which (6.1)–(6.2) are valid in the class of Lipschitz domains. For example, both decompositionsmay fail (even if the metric tensor is constant and ` = 1) if p /∈ [32 , 3]. The optimal range of p’s forwhich (6.1)–(6.2) hold in arbitrary Lipschitz domains remains an open problem at the moment butwe conjecture that Theorem 6.1 is sharp in its present form.

As a corollary of the (proof of) Theorem 6.1 we have the following.

Theorem 6.2 (Helmholtz-Weyl decompositions) Under the same hypothesis as in the Theo-rem 6.1,

Lp(Ω, Λ`TM) = dDp`−1(d, Ω)⊕Np

` (δ∨, Ω), (6.9)

Lp(Ω, Λ`TM) = Np` (d∧, Ω)⊕ δDp

`+1(δ,Ω), (6.10)

Lp(Ω, Λ`TM) = dDp`−1(d∧, Ω)⊕Np

` (δ,Ω), (6.11)

Lp(Ω,Λ`TM) = Np` (d, Ω)⊕ δDp

`+1(δ∨, Ω), (6.12)

60

where the sums are direct and topological. Once again all decompositions are obtained in a con-structive fashion.

Remark. Following work in [10] in the flat Euclidean context, it has been proved in [33] that (6.9)and (6.11) are valid for 3/2 − ε < p < 3 + ε if ` = 1. Furthermore, as far as these decompositionsare concerned, this range is sharp in the class of Lipschitz domains. Note that, by Hodge duality,the same results are valid for (6.10) and (6.12) if ` = m− 1.

Proof of Theorem 6.2. Let us prove, for example, (6.10). Granted (6.1), it only suffices to showthat

dDp`−1(d∧,Ω)⊕H`

∧(Ω) = Np` (d∧, Ω). (6.13)

The left-to-right inclusions is obvious. If u ∈ Np` (d∧, Ω) then (the proof of) Theorem 6.1 gives

u = dδ G`∧u + δdG`

∧u + P `∧u

= dδ G`∧u + δG`+1

∧ (du) + P `∧u

= d(δG`∧u) + P `

∧u ∈ dDp`−1(d∧,Ω)⊕Hp,`

∧ (Ω), (6.14)

where the second equality utilizes (the second identity in) (5.38). This proves the opposite inclusionin (6.13) and, hence, finishes the proof of (6.10). All the other decompositions are proved in a similarfashion. 2

Theorem 6.3 (Hodge-Morrey decompositions) Under the same hypothesis as before,

Lp(Ω,Λ`TM) = dDp`−1(d∧,Ω)⊕ δDp

`+1(δ∨,Ω)⊕H`,p(Ω), (6.15)

where the sums are direct and topological. Furthermore, the decomposition (6.15) is obtained in aconstructive way.

Proof. We only indicate the main step in the proof. The idea is to use the Green operator G`0

from §5 and write

u = d(δG`0u) + δ(dGd

0u) + π`u (6.16)

for arbitrary u ∈ Lp(Ω, Λ`TM). With this at hand, (6.15) follows. 2

Theorem 6.4 (Friedrichs decompositions) With the same hypotheses as in Theorem 6.1,

H`,p(Ω) = H`,p∧ (Ω)⊕ [H`,p(Ω) ∩ dDp

`−1(d, Ω)], (6.17)

H`,p(Ω) = H`,p∨ (Ω)⊕ [H`,p(Ω) ∩ δDp

`+1(δ,Ω)], (6.18)

where the sums are direct and topological. Moreover, both decompositions can be done in a con-structive fashion.

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Proof. Let us deal with, e.g., (6.17). The right-to-left inclusion is obvious. Now, if u ∈ H`,p(Ω)then, as in (6.14),

u = dδG`∧u + δdG`

∧u + P `∧u = P `

∧u + dδG`∧u. (6.19)

Here we have used the fact that dG`∧u = G`+1

∧ du = 0. Since, clearly, dδG`∧u ∈ H`,p(Ω)∩dDp

`−1(d,Ω),it follows that u ∈ H`,p

∧ (Ω)⊕ [H`,p(Ω) ∩ dDp`−1(d,Ω)] so that the left-to-right inclusion in (6.17) is

valid as well. 2

Theorem 6.5 (Hodge-Dirac decompositions) Under the same hypothesis as in Theorem 6.1,

Lp(Ω,⊕m`=0Λ

`TM) = (d + δ)[(⊕m`=0D

p` (d, Ω)) ∩ (⊕m

`=0Dp` (δ∨, Ω))]⊕ (⊕m

`=0H`∨(Ω))

= (d + δ)[(⊕m`=0D

p` (d∨, Ω)) ∩ (⊕m

`=0Dp` (δ,Ω))]⊕ (⊕m

`=0H`∧(Ω)), (6.20)

where the sums are direct and topological. Once again, these decompositions can be done in aconstructive way.

Proof. This follows from the existence of the Green operators GD∧ , GD

∨ from §5, by the same tokenas before. 2

We return to the Maxwell system one more time in order to discuss the “full” Lp-Poissonproblem.

Theorem 6.6 Let Ω be a Lipschitz domain. If k, ε are as in Theorem 4.9 and 2− ε < p < 2 + ε,then the Lp-Poisson problem for the Maxwell system

(BV P22)k,`

E ∈ Lp(Ω, Λ`TM), H ∈ Lp(Ω, Λ`+1TM),

dE − ikH = K ∈ Lp(Ω, Λ`+1TM),

δH + ikE = J ∈ Lp(Ω,Λ`TM),

ν ∧ E = f ∈ X p`+1(∂Ω),

has a unique solution. The solution satisfies

‖E‖Lp(Ω) + ‖H‖Lp(Ω) ≤ C(‖K‖Lp(Ω) + ‖J‖Lp(Ω) + ‖f‖X p`+1(∂Ω)). (6.21)

Proof. Matters can be reduced to the case K = 0, J = 0 by reasoning as in the proof ofProposition 4.4. Note that, in this step, use is made of the Lp-Hodge decompositions (6.1)–(6.2).At this point, Theorem 4.9 can be used to conclude the proof. 2

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Dorina MitreaDepartment of MathematicsUniversity of Missouri-ColumbiaMathematical Sciences BuildingColumbia, MO 65211, USAe-mail: dorina@math.missouri.edu

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Marius MitreaDepartment of MathematicsUniversity of Missouri-ColumbiaMathematical Sciences BuildingColumbia, MO 65211, USAe-mail: marius@math.missouri.edu

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