Hodge–HelmholtzdecompositionsofweightedSobolevspaces ... · commutator of the Laplacian and a cut-off function , which vanishes near the origin and equals one near inﬁnity, on

MATHEMATICAL METHODS IN THE APPLIED SCIENCESMath. Meth. Appl. Sci. 2008; 31:1509–1543Published online 17 June 2008 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/mma.982MOS subject classification: 35Q 60; 58A 10; 58A 14; 78A 25; 78A 30

Hodge–Helmholtz decompositions of weighted Sobolev spacesin irregular exterior domains with inhomogeneous

and anisotropic media

Dirk Pauly∗,†

Fachbereich Mathematik, Universitat Duisburg-Essen, Campus Essen, Universitatsstr. 2, 45117 Essen, Germany

Communicated by W. Sproessig

SUMMARY

We study in detail Hodge–Helmholtz decompositions in nonsmooth exterior domains �⊂RN filled withinhomogeneous and anisotropic media. We show decompositions of alternating differential forms of rankq belonging to the weighted L2-space L2,q

s (�), s∈R, into irrotational and solenoidal q-forms. Thesedecompositions are essential tools, for example, in electro-magnetic theory for exterior domains. Tothe best of our knowledge, these decompositions in exterior domains with nonsmooth boundaries andinhomogeneous and anisotropic media are fully new results. In the Appendix, we translate our results tothe classical framework of vector analysis N =3 and q=1,2. Copyright q 2008 John Wiley & Sons, Ltd.

KEY WORDS: Hodge–Helmholtz decompositions; Maxwell equations; electro-magnetic theory; weightedSobolev spaces

1. INTRODUCTION

Hodge–Helmholtz decompositions of square integrable fields, i.e. decompositions in irrotationaland solenoidal fields, are important and strong tools for solving partial differential equations, forinstance, in electro-magnetic theory.

Since formally igrad and idiv resp. curl and curl are adjoint to each other and curlgrad=0 anddivcurl=0 hold as well, the ε-L2-orthogonal decompositions,

L2(�) =H(◦

curl0,�)⊕ε ε−1H(div0,�)⊕ε εH(�)

L2(�) =H(curl0,�)⊕ε ε−1H(◦div0,�)⊕ε εH(�)

(1)

∗Correspondence to: Dirk Pauly, Fachbereich Mathematik, Universitat Duisburg-Essen, Campus Essen, Universitatsstr.2, 45117 Essen, Germany.

†E-mail: [email protected]

Copyright q 2008 John Wiley & Sons, Ltd. Received 25 July 2006

1510 D. PAULY

where �⊂R3 is a domain, are easy consequences of the projection theorem in Hilbert space.Here ε :�→R3×3 is a real-valued, symmetric and uniformly bounded and positive-definite matrix,which models material properties, such as the dielectricity or the permeability of the medium, andεH(�) resp. εH(�) denotes the space of Dirichlet resp. Neumann fields. (See Appendix B.2 forthe exact definitions of all these spaces.)

This problem may be generalized if we formulate Maxwell’s equations in the framework ofalternating differential forms of order q , short q-forms, on some N -dimensional Riemannianmanifold �. Additionally to the generality and the easy and short notation, this approach providesalso a deeper insight into the structure of the underlying problems. It has become customaryfollowing Weyl [1] to denote the exterior derivative d by rot and the co-derivative � by div. Wewill use this notation throughout this paper and thus we have on q-forms

div=(−1)(q+1)N ∗ rot∗where ∗ is Hodge’s star operator. Since rot and div are formally skew adjoint to each other aswell as rot rot=0 and divdiv=0 hold, the corresponding Hodge–Helmholtz decompositions ofL2-forms

L2,q(�)=0◦Rq(�)⊕ε ε−1

0Dq(�)⊕ε εHq(�) (2)

again are easy consequences of the projection theorem. Here ε maps � to the real, linear, symmetricand uniformly bounded and positive-definite transformations on q-forms. Furthermore, we denoteby ⊕ε the orthogonal sum with respect to the 〈ε · , · 〉L2,q (�)-scalar product. (See Section 2 fordefinitions.) For N =3 and q=1 or q=2 we obtain the two classical decompositions (1).

In the case of unbounded domains, it is often necessary and useful to work with weightedSobolev spaces. Especially in our efforts to completely determine the low-frequency asymptotics ofthe solutions of the time-harmonic Maxwell equations in exterior domains [2, 3] and a forthcomingthird paper [4], it has turned out that decompositions of weighted L2-spaces are necessary andessential tools.

Hence motivated by this paper, we wish to answer the question, in which way the weightedL2-space of q-forms

L2,qs (�) :={F ∈L2,q

loc (�) :�s F ∈L2,q(�)}, s∈R

where �⊂RN is an exterior domain, i.e. a domain with compact complement, and � :=(1+r2)1/2

with r(x) :=|x | for x ∈RN denotes a weight function, may be decomposed into irrotational andsolenoidal forms, i.e. q-forms with vanishing rotation rot resp. divergence div.

For the special case s=0 Picard has shown (2) in [5, 6] and (in the classical framework) in[7, 8]. Moreover, for domains � possessing the ‘Maxwell local compactness property’ (MLCP)(See Section 2.) Picard proved the representations

0◦Rq(�) = rot

◦R

q−1−1 (�)= rot(

◦R

q−1−1 (�)∩0D

q−1−1 (�))

0Dq(�) = divDq+1−1 (�)=div(D

q+1−1 (�)∩0

◦R

q+1−1 (�))

i.e. any form from 0◦Rq(�) may be represented as a rotation of a solenoidal form and any form

from 0Dq(�) may be represented as a divergence of a irrotational form.

Copyright q 2008 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2008; 31:1509–1543DOI: 10.1002/mma

HODGE–HELMHOLTZ DECOMPOSITIONS IN EXTERIOR DOMAINS 1511

Now one may expect for arbitrary s∈R the direct decomposition

L2,qs (�)=0

◦R

qs (�)�ε−1

0Dqs (�)�εH

qs (�)

However, as we will see, this holds only for s ‘near’ zero, since for small s we lose the directnessof the decomposition and for large s the right-hand side is too small. However, both negativeeffects are of finite dimensional nature.

For general s∈R\I introducing the countable discrete set of (bad) weights

I={N/2+n :n∈N0}∪{1−N/2−n :n∈N0}Weck and Witsch showed in [9] for the special case �=RN and ε= Id, where no Dirichlet formsand no boundary exist, the decompositions

L2,qs =

⎧⎪⎪⎨⎪⎪⎩0R

qs +0D

qs , s∈(−∞,−N/2)

0Rqs �0D

qs , s∈(−N/2,N/2)

0Rqs �0D

qs �S

qs , s∈(N/2,∞)

and the representations

0Rqs = rot R

q−1s−1 , 0D

qs =divD

q+1s−1

Thereby Sqs is a finite dimensional subspace of

◦C∞,q(RN \{0}) generated by the action of the

commutator of the Laplacian and a cut-off function �, which vanishes near the origin and equalsone near infinity, on the linear hull of some finitely many decaying potential forms in RN \{0}, i.e.generalized spherical harmonics multiplied by a negative power of r solving Laplace’s equation.(Here we omit the dependence on the domain RN and denote the direct sum by �.) We notethat Weck and Witsch in [9] even decomposed the Lebesgue–Banach spaces Lp,q

s with p∈(1,∞)

instead of p=2. The proof of their results uses heavily the corresponding results for the scalarLaplacian in RN developed by McOwen [10]. For the Hilbert space case p=2 these results havebeen generalized to smooth (at least C3) exterior domains �⊂RN by Bauer [11]. Unfortunatelyby their second-order approach, these techniques cannot be applied to handle inhomogeneities ε

and the smoothness of � is essential as well.Results in the classical case q=N−1 have been given by Specovius-Neugebauer in [12] for

ε= Id and a smooth (C2) exterior domain �⊂RN , N�3. She considered only this special case and

additionally only a weaker version of (2), which reads as L2,N−1(�)=0◦R N−1(�)⊕divD N

−1(�)

resp. in the classical language

L2(�)=gradH−1(grad,�)⊕H(◦div0,�)

She was able to show for s∈R\I

L2s (�)=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩gradHs−1(grad,�)+Hs(

◦div0,�), s∈(−∞,−N/2)

gradHs−1(grad,�)�Hs(◦div0,�), s∈(−N/2,N/2)

gradHs−1(grad,�)�Hs(◦div0,�)�Ss, s∈(N/2,∞)


1512 D. PAULY

where Ss corresponds to SN−1s . We note that she proved the corresponding decompositions even

for Banach spaces Lps (�) with 1<p<∞. Since she used heavily trace operators and convolution

techniques, her results cannot be generalized to nonsmooth boundaries or inhomogeneities ε.

Moreover, she showed no further decomposition of Hs(◦div0,�) into Neumann fields and images

of curl-terms (for N =3), which is highly important in electro-magnetic theory.Our main focus is to treat nonsmooth boundaries, i.e. Lipschitz boundaries or even weaker

assumptions, and most of all nonsmooth inhomogeneities corresponding to inhomogeneous andanisotropic media, which are only asymptotically homogeneous. To the best of our knowledge, itwas an open question, if those weighted L2-decompositions hold for inhomogeneous and anisotropicmedia or for nonsmooth boundaries. We will allow our transformations ε to be L∞-perturbationsof the identity, i.e. ε= Id+ ε, where ε does not need to be compactly supported but decays atinfinity. Moreover, ε is not assumed to be smooth. We require only ε∈C1 in the outside of anarbitrarily large ball. Omitting some details for this introductory remarks, we will show essentiallyfor small s∈(−∞,−N/2)\ I

L2,qs (�)=0

◦R

qs (�)+ε−1

0Dqs (�)

and for large s∈(−N/2,∞)\ I

L2,qs (�)∩εH

q(�)⊥ε =⎧⎨⎩0

◦R

qs (�)�ε−1

0Dqs (�), s<N/2

0◦R

qs (�)�ε−1

0Dqs (�)��ε�P

qs−2, s>N/2

Here �ε�Pqs−2 is a finite dimensional subspace of

◦H

1,qs (�)∩C1,q(�), whose elements have

supports in the outside of an arbitrarily large ball, and � is a cut-off function as before but nowvanishing near the boundary ��. The forms from P

qs−2 are potential forms, i.e. solve Laplace’s

equation in RN \{0}, and�ε = rotdiv+ε−1divrot

In the special case ε= Id since �= rotdiv+divrot (Here the Laplacian � is to be understoodcomponentwise in Euclidean coordinates.) we have like above

��Pqs−2=C�,�P

qs−2=S

qs ⊂◦

C∞,q(�)

where CA,B :=AB−BA denotes the commutator of two operators A and B. (For details seeTheorem 3.2.) Furthermore, L2,q

s (�) decomposes for large s into L2,qs (�)∩εH

q(�)⊥ε and thelinear hull of finitely many smooth forms, which have bounded supports. We note that for allt ∈[−N/2,N/2−1) the spaces of Dirichlet forms εH

qt (�) coincide. Moreover, for all s∈R\ I the

irrotational forms from 0◦R

qs (�) resp. the solenoidal forms from 0D

qs (�) can be represented as

rotations resp. divergences, i.e.

0◦R

qs (�)= rot

◦R

q−1s−1 (�), 0D

qs (�)=divD

q+1s−1 (�)

hold except of some special values of s or q . However, contrary to the case s=0 for large

s>1+N/2 we lose integrability properties, if we wish to represent forms in 0◦R

qs (�) resp. 0D

qs (�)



by rotations of solenoidal resp. divergences of irrotational forms. Looking at Theorem 3.5 weobtain

0◦R

qs (�) = rot((

◦R

q−1s−1 (�)��H

q−1s−1 )∩0D

q−1<N/2(�))

0Dqs (�) = div((D

q+1s−1 (�)��H

q+1s−1 )∩0

◦R

q+1<N/2(�))

i.e. the representing solenoidal resp. irrotational forms no longer belong to L2,q∓1s−1 (�) but to

L2,q∓1t (�) for all t<N/2. (For details see Theorems 3.4, 3.5 and 3.8.)If we project onto the orthogonal complement of εH

q−s(�), i.e. of more Dirichlet forms, we

finally obtain even for large s>N/2

L2,qs (�)∩εH

q−s(�)⊥ε =0

◦R

qs (�)⊕ε ε−1

0Dqs (�)

2. DEFINITIONS AND PRELIMINARIES

We consider an exterior domain �⊂RN , i.e. RN \� is compact, as a special smooth Riemannianmanifold of dimension 3�N ∈N, and fix a radius r0 and radii rn :=2nr0, n∈N, such that RN \�is a compact subset of Ur0 :={x ∈RN : |x |<r0}. Moreover, we choose a cut-off function �, suchthat [2, (2.1)–(2.3)] hold. We then have �=0 in Ur1 and �=1 in Ar2 :={x ∈RN : |x |>r2} and thussupp∇�⊂ Ar1 ∩Ur2 .

Throughout this paper we will use the notations from [2, 3]. Considering alternating differentialforms of rank q∈Z (short q-forms), we denote the exterior derivative d by rot and the co-derivative�=±∗ d∗ (∗: Hodge star operator) by div to remind of the electro-magnetic background. On◦C∞,q(�) (the vector space of all C∞-q-forms with compact support in �) we have a scalar product

〈�,�〉L2,q (�) :=∫

��∧∗� ∀�,�∈◦

C∞,q(�)

and an induced norm ||·||L2,q (�) :=〈 · , · 〉1/2L2,q (�)

. Thus we may define (taking the closure in the latternorm)

L2,q(�) := ◦C∞,q(�)

the Hilbert space of all square integrable q-forms on �. Moreover, due to Stokes’ theorem on◦C∞,q(�) the linear operators rot and div are formally skew adjoint to each other, i.e.

〈rot�,�〉L2,q+1(�) =−〈�,div�〉L2,q (�)

for all (�,�)∈◦C∞,q(�)× ◦

C∞,q+1(�), which gives rise to weak formulations of rot and div. Usingthese and the weight function �(r) :=(1+r2)1/2 for s∈R, we introduce the following weighted


1514 D. PAULY

Hilbert spaces (endowed with their natural norms) of q-forms:

L2,qs (�) := {E ∈L2,q

loc (�) :�s E ∈L2,q(�)}

Rqs (�) := {E ∈L2,q

s (�) : rotE ∈L2,q+1s+1 (�)}

Dqs (�) := {H ∈L2,q

s (�) :divH ∈L2,q−1s+1 (�)}

All these spaces equal zero if q /∈{0, . . . ,N }. Furthermore, taking the closure in Rqs (�) we introduce

the Hilbert space

◦R

qs (�) := ◦

C∞,q(�)

which generalizes the boundary condition of vanishing tangential component of a q-form at theboundary ��. More precisely this generalizes the boundary condition �∗E=0, which means thatthe pull-back of E on the boundary of � (considered as a (N−1)-dimensional Riemanniansubmanifold of �) vanishes. Here � :�� ↪→� denotes the natural embedding.

A lower left index 0 indicates vanishing rotation resp. divergence.For weighted Sobolev spaces Vs , s∈R, we define

V<t :=⋂s<t

Vs

We consider only exterior domains �, which possess the MLCP, i.e. for all q and all t<s theembeddings

◦R

qs (�)∩D

qs (�) ↪→L2,q

t (�)

are compact. (See [2, Definition 2.4, Remark 2.5] and the literature cited there.)We assume our real-valued transformations to be �-admissible resp. �-C1-admissible as defined

in [2, Definitions 2.1 and 2.2]. This means shortly that they generate scalar products on L2,q(�) andare asymptotically the identity mapping. The parameter � always denotes this rate of convergenceand the perturbations only have to be C1 in the outside of an arbitrarily large ball. Hence, we maychoose r0, such that the transformations are C1 in Ar0 .

Let ε be a �-C1-admissible transformation on q-forms with some �>0. We need the finitedimensional vector space of Dirichlet forms

εHqt (�) :=0

◦R

qt (�)∩ε−1

0Dqt (�), t ∈R

(Here we neglect the indices ε or t in the cases ε= Id or t=0.) Citing [3, Lemma 3.8] we have

εHq−N/2(�)= εH

q(�)= εHq<N/2−1(�)

and even εHq(�)= εH

q<N/2(�) if q /∈{1,N−1}. Thus, εH

q(�)⊂L2,q−s (�) for s>1−N/2 and even

for s>−N/2 if q /∈{1,N−1}.Furthermore, for R�s>1−N/2 we introduce the Hilbert spaces

0Dqs (�)=0D

qs (�)∩εH

q(�)⊥, 0◦R

qs (�)=0

◦R

qs (�)∩εH

q(�)⊥ε (3)



where we denote by ⊥ε the orthogonality with respect to the 〈ε · , · 〉L2,q (�)-scalar product, i.e. the

duality between L2,qt (�) and L2,q

−t (�). If ε= Id we simply express ⊥:=⊥Id. The restrictions on

the weights s guarantee εHq(�)⊂L2,q

−s (�). However, for ranks q /∈{1,N−1} also the inclusion

εHq(�)⊂L2,q

<N/2(�) holds and these definitions extend to s>−N/2. Since there are no Dirichletforms εH

q(�) for q∈{0,N } in these special cases the definitions (3) may be extended to all s∈R

and we have

0◦R0

s (�)=0◦R0

s (�)={0}, 0◦R N

s (�)=0◦R N

s (�)=L2,Ns (�)

0D0s (�)=0D

0s (�)=L2,0

s (�), 0DNs (�)=0D

Ns (�)=

{{0}, s�−N/2

Lin{∗1}, s<−N/2

Moreover, there are some other characterizations of these spaces. We remind of the finitely many

special smooth forms◦Bq(�)⊂0

◦Rq(�) and Bq(�)⊂0Dq(�) presented in [3, Section 4], which

have compact resp. bounded supports in � and the properties

εHq(�)∩ ◦

Bq(�)⊥ε = εHq(�)∩Bq(�)⊥ ={0} (4)

We note in passing

dim εHq(�)=dimHq(�)=#

◦Bq(�)=#Bq(�)=:dq ∈N0

Using [3, Corollary 4.4] we observe in fact that

0Dqs (�) = 0D

qs (�)∩Hq(�)⊥ =0D

qs (�)∩ ◦

Bq(�)⊥

0◦R

qs (�) = 0

◦R

qs (�)∩Hq(�)⊥ =0

◦R

qs (�)∩Bq(�)⊥

do not depend on the transformation ε. Since Bq(�) is only defined for q �=1, the last character-

ization in the second equation holds only for q �=1. Now the definitions of 0Dqs (�) and 0

◦R

qs (�)

extend to arbitrary weights s∈R, because the forms◦Bq(�),Bq(�) have bounded supports. We

say that � possesses the ‘static Maxwell property’ (SMP), if and only if � has the MLCP and

the forms◦Bq(�) and Bq(�) exist. For instance, the SMP is guaranteed for Lipschitz domains �.

(See [3, Section 4] and the literature cited there.) We may choose r0, such that suppb⊂Ur0 for

all b∈◦Bq(�)∪Bq(�) and all q .

Finally, for s>1−N/2 or s>−N/2 and q /∈{1,N−1} we put

εL2,qs (�) :=L2,q

s (�)∩εHq(�)⊥ε

We also need the negative ‘tower forms’ −Dq,��,m , −Rq,�

�,m for the values �=0,1,2 and �∈N0,m∈{1, . . . ,�q�} from [3, Section 2], which are harmonic polynomials except of a multiplication


1516 D. PAULY

by some negative integer power of r . These forms are homogeneous of degree −h�� :=�−�−N ,

belong to C∞,q(RN \{0}) and satisfy the ‘tower equations’

rot−Dq,0�,m =0, div−Rq+1,0

�,m =0

div−Dq,��,m=0, rot−Rq+1,�

�,m =0

rot−Dq,k�,m =−Rq+1,k−1

�,m , div−Rq+1,k�,m =−Dq,k−1

�,m

where �=0,1,2 and k=1,2. (We note briefly that we need the positive tower forms of heightzero +Dq,0

�,m , +Rq,0�,m in our proofs as well. However, they are not required to formulate our

results.) From [3, Remark 2.5] we have for all �∈N0, m∈{1, . . . ,�q�} and all �=0,1,2 as well asall k∈N0

−Dq,��,m ∈L2,q

s (A1) ⇔ −Dq,��,m ∈H k,q

s (A1) ⇔ s<N/2+�−�

which completely determines the integrability properties of our tower forms at infinity. The sameintegrability holds true for −Rq,�

�,m . Moreover, the ground forms (forms of height 0), which onlyoccur for 1�q�N−1, are linear dependent, i.e. we have

�q� ·−Rq,0�,m+ i�q

′� ·−Dq,0

�,m =0 (5)

where �q� :=(q+�)1/2 and q ′ :=N−q . This motivates to define the harmonic tower forms

Hq�,m :=�q� ·−Rq,0

�,m =−i�q′

� ·−Dq,0�,m (6)

and the potential tower forms

Pq�,m :=�q� ·−Rq,2

�,m+ i�q′

� ·−Dq,2�,m (7)

Since �= rotdiv+divrot, we then obtain

�−Dq,��,m =�−Rq,�

�,m =�Hq�,m =�Pq

�,m =0, �=0,1

Here � denotes the componentwise scalar Laplacian in Euclidean coordinates. We note alsothat Pq

�,m =Hq�,m =0 if q∈{0,N }. Furthermore, for s∈R and �=0,1,2 we introduce the finite

dimensional vector spaces

Dq,�

s := Lin{−Dq,��,m :−Dq,�

�,m /∈L2,qs (A1)}=Lin{−Dq,�

�,m :��s−N/2+�}

Rq,�

s := Lin{−Rq,��,m :−Rq,�

�,m /∈L2,qs (A1)}=Lin{−Rq,�

�,m :��s−N/2+�}

Hqs := Lin{Hq

�,m :Hq�,m /∈L2,q

s (A1)}=Lin{Hq�,m :��s−N/2}

Pqs := Lin{Pq

�,m : Pq�,m /∈L2,q

s (A1)}=Lin{Pq�,m :��s−N/2+2}



(Here we set Lin∅:={0}.) We note

Hqs =P

qs−2=D

q,�

s−� =Rq,�

s−� ={0} ⇔ s<N/2

Unfortunately due to the fact that rotr2−N (the gradient of r2−N in classical terms) is irrotationaland solenoidal but is itself no divergence, there exist four exceptional tower forms. These are (upto constants)

P0 :=r2−N =−D0,20,1, H1 := rot P0=r1−N dr =−R1,1

0,1

and their duals P N :=∗P0=−RN ,20,1 , H N−1 :=∗H1=−DN−1,1

0,1 . We have div P N = H N−1.Following the construction of the regular tower forms, we define for s�N/2−2

P0 :=P

0s :=Lin{P0}, P

N :=PNs :=Lin{P N }

and for s�N/2−1

H1 :=H

1s :=Lin{H1}, H

N−1 :=HN−1s :=Lin{H N−1}

For all other values of s and q , we put Pqs :={0} and H

qs :={0}.

As described in [3, Section 3] for s∈R, we will consider vector spaces

V qs ��Vq

s , (� : direct sum)

where Vqs ⊂L2,q

s (�) is some Hilbert space and Vqs is some finite subset of our tower forms, e.g.

Vqs = ◦

Rqs (�)∩D

qs (�) and V

qs =H

qs . On Vq

s ��Vqs , we define a scalar product, such that

• in V qs the original scalar product is kept;

• �Vqs is an orthonormal system;

• the sum V qs ��Vq

s =Vqs ��Vq

s is orthogonal.

As already indicated we denote the orthogonal sum with respect to this new inner product by �and clearly V q

s ��Vqs is a Hilbert space since V

qs is finite.

3. RESULTS

Let �⊂RN (N�3) be an exterior domain as in the last section with the SMP or the MLCP

depending on whether the forms◦Bq(�),Bq(�) are involved in our considerations or not. Recalling

from [3, Section 3] the set of special weights I, we put

I :=I−1={N/2+n−1:n∈N0}∪{−N/2−n :n∈N0}and from now on we make the following general assumptions:

• q∈{0, . . . ,N }• s∈R\ I, i.e. s+1∈R\I, i.e. for all n∈N0

s �=n+N/2−1 and s �=−n−N/2


1518 D. PAULY

• ε is a �-C1-admissible transformation on q-forms with some �=�s+1 satisfying

�>max{0,s+1−N/2} and ��−s−1

i.e.

�

⎧⎪⎪⎨⎪⎪⎩�−s−1, s∈(−∞,−1)

>0, s∈[−1,N/2−1]>s+1−N/2, s∈(N/2−1,∞)

• and � are �-C1-admissible transformation on (q−1)-resp. (q+1)-forms with some �=�ssatisfying

�>max{0,s−N/2} and ��−s

i.e.

�

⎧⎪⎪⎨⎪⎪⎩�−s, s∈(−∞,0)

>0, s∈[0,N/2]>s−N/2, s∈(N/2,∞)

If 1−N/2<s<N/2−1 or−N/2<s<N/2 and q /∈{1,N−1} the ‘trivial’ orthogonal decomposition

L2,qs (�)= εL

2,qs (�)⊕ε εH

q(�) (8)

holds. Throughout the paper we will denote the orthogonality with respect to the 〈ε · , · 〉L2,q (�)-

scalar product or L2,qs (�)-L2,q

−s (�)-duality by ⊕ε and put ⊕=⊕Id.The first lemma shows how one may get rid of Dirichlet forms even for larger weights.

Lemma 3.1Let s>1−N/2. Then the direct decompositions

L2,qs (�)= εL

2,qs (�)�Lin

◦Bq(�), L2,q

s (�)= εL2,qs (�)�ε−1LinBq(�)

hold, where the latter is only defined for q �=1. If q /∈{1,N−1} this decompositions hold fors>−N/2 as well.

To formulate our main decomposition result, we need the operator (a perturbation of theLaplacian �)

�ε := rotdiv+ε−1divrot=�+ εdivrot

where ε−1=: Id+ ε is also �-C1-admissible. We obtain



Theorem 3.2The following decompositions hold:

(i) If s<−N/2, then

L2,qs (�)=0

◦R

qs (�)+ε−1

0Dqs (�)

and the intersection equals the finite dimensional space of Dirichlet forms εHqs (�).

Moreover,

L2,qs (�)=0

◦R

qs (�)+ε−1

0Dqs (�)

and for q �=1 even

L2,qs (�)=0

◦R

qs (�)+ε−1

0Dqs (�)

In both cases the intersection equals the finite dimensional space of Dirichlet forms

εHqs (�)∩ ◦

Bq(�)⊥ε .(ii) If −N/2<s�1−N/2, then

L2,1s (�) = 0

◦R1

s (�)�ε−10D1

s (�)

L2,N−1s (�) = 0

◦R N−1

s (�)�ε−10DN−1

s (�)�εHN−1(�)

(iii) If 1−N/2<s<N/2 or −N/2<s�1−N/2 and q /∈{1,N−1}, then

εL2,qs (�)=0

◦R

qs (�)�ε−1

0Dqs (�)

For s�0 this decomposition is even 〈ε · , · 〉L2,q (�)-orthogonal.(iv) If s>N/2, then

εL2,qs (�) = (([L2,q

s (�)��Hqs ]∩0

◦R

q<N/2(�))

⊕εε−1([L2,q

s (�)��Hqs ]∩0D

q<N/2(�)))∩L2,q

s (�)

and

εL2,qs (�)=0

◦R

qs (�)�ε−1

0Dqs (�)��ε�P

qs−2

where the first two terms in the second decomposition are 〈ε · , · 〉L2,q (�)-orthogonal as well.Furthermore,

L2,qs (�)∩εH

q−s(�)⊥ε =0

◦R

qs (�)⊕ε ε−1

0Dqs (�)

Remark 3.3

• The decompositions in (ii)–(iv) are direct and define continuous projections.

• In (ii) we are forced to use the forms Bq(�) and◦Bq(�) in the definitions of 0

◦R

qs (�) and

0Dqs (�).


1520 D. PAULY

• To prove the last equation in (iv) we additionally assume ��N/2−1.• The coefficients of the tower forms in the first equation of (iv) are related in the followingway: If

Fr,s+∑�

hr,� ·�H�+ε−1(Fd,s+∑�

hd,� ·�H�)=F ∈ εL2,qs (�)

with Fr,s ∈◦R

qs (�), Fd,s ∈D

qs (�) and H� ∈H

qs as well as hr,�,hd,� ∈C, then

hr,�+hd,� =0

since H� are linear independent and do not belong to L2,qs (�).

• �ε�Pqs−2 is a finite dimensional subspace of

◦H

1,qs (�)∩C1,q(�), whose elements have

supports in Ar1 .• For s<−N/2 (and ��N/2−1) we have

εHqs (�)= εH

q(�)�εHqs (�)∩ ◦

Bq(�)⊥ε

• Clearly the transformation ε may be moved to the rot-free terms in our decompositions aswell.

Our decompositions and representations may be refined. For small weights, we have

Theorem 3.4

Let s<N/2+1−�q,0−�q,N . Then 0Dqs (�) and 0

◦R

qs (�) are closed subspaces of L2,q

s (�) wheneverthey exist and

(i) 0◦R

qs (�) = rot(

◦R

q−1s−1 (�)∩−1

0Dq−1s−1 (�))

= rot(◦R

q−1s−1 (�)∩−1

0Dq−1s−1 (�))= rot

◦R

q−1s−1 (�)

holds for 2�q�N as well as for q=1 and s>1−N/2;

(ii) 0Dqs (�) = div(D

q+1s−1 (�)∩�−1

0◦R

q+1s−1 (�))

holds for 1�q�N−1 as well as for q=0 and s>2−N/2;

(iii) 0Dqs (�) = div(D

q+1s−1 (�)∩�−1

0◦R

q+1s−1 (�))=divD

q+1s−1 (�)

holds for 0�q�N−1.

For large weights, we have



Theorem 3.5Let 1�q�N−1. Then for s>N/2+1

(i) 0◦R

qs (�) = rot((

◦R

q−1s−1 (�)��H

q−1s−1 )∩−1

0Dq−1<N/2(�))

= rot(◦R

q−1s−1 (�)∩−1D

q−1s−1 (�)∩ ◦

Bq−1(�)⊥)= rot◦R

q−1s−1 (�)

(ii) 0Dqs (�) = div((D

q+1s−1 (�)��H

q+1s−1 )∩�−1

0◦R

q+1<N/2(�))

= div(Dq+1s−1 (�)∩�−1 ◦

Rq+1s−1 (�)∩Bq+1(�)⊥�)=divD

q+1s−1 (�)

are closed subspaces of L2,qs (�) and for s>N/2

(iii) (L2,qs (�)��H

qs )∩0

◦R

q<N/2(�)

= rot((◦R

q−1s−1 (�)��D

q−1,0s−1 ��D

q−1,1s−1 )∩−1

0Dq−1<N/2−1(�))

=0◦R

qs (�)�rot−1�D

q−1,1s−1

(iv) (L2,qs (�)��H

qs )∩0D

q<N/2(�)

=div((Dq+1s−1 (�)��R

q+1,0s−1 ��R

q+1,1s−1 )∩�−1

0◦R

q+1<N/2−1(�))

=0Dqs (�)�div�−1�R

q+1,1s−1

are closed subspaces of L2,qs (�)��H

qs .

Remark 3.6We note div�−Dq−1,1

�,m =0 and rot�−Rq+1,1�,m =0 by [3, Remark 2.4] and thus

�Dq−1,1s−1 ⊂0D

q−1<N/2−1(�), �R

q+1,1s−1 ⊂0

◦R

q+1<N/2−1(�)

Remark 3.7Since there are no regular harmonic tower forms in the cases q∈{0,N }, i.e. H0

s ={0}, HNs ={0},

and because of �Hqs ⊂L2,q

<N/2(�) the first equations in (iii) and (iv) simplify:If s>N/2, then

(L2,1s (�)��H

1s )∩0

◦R1

<N/2(�) = rot(◦R0

s−1(�)��D0,1s−1)

(L2,N−1s (�)��H

N−1s )∩0DN−1

<N/2(�) = div(D Ns−1(�)��R

N ,1s−1)

If N/2<s<N/2+1, then

(L2,qs (�)��H

qs )∩0

◦R

q<N/2(�) = rot((

◦R

q−1s−1 (�)��D

q−1,1s−1 )∩−1

0Dq−1<N/2−1(�))

(L2,qs (�)��H

qs )∩0D

q<N/2(�) = div((D

q+1s−1 (�)��R

q+1,1s−1 )∩�−1

0◦R

q+1<N/2−1(�))


1522 D. PAULY

As mentioned above there are no harmonic tower forms in the remaining cases q∈{0,N }.Thus the equations in (i), (ii) and (iii), (iv) of the latter theorem would coincide for these values.Furthermore, in these special cases there occur the exceptional tower forms. We obtain

Theorem 3.8Let s>N/2. Then

(i) L2,Ns (�) = 0

◦R N

s (�)

= rot((◦R N−1

s−1 (�)��HN−1s−1 ��H

N−1)∩−1

0DN−1<N/2−1(�))

= rot((◦R N−1

s−1 (�)∩−1D N−1s−1 (�)∩ ◦

B N−1(�)⊥)��HN−1

)

= rot(◦R N−1

s−1 (�)∩−1D N−1s−1 (�)∩ ◦

B N−1(�)⊥)��PN

= rot◦R N−1

s−1 (�)��PN

(ii) L2,0s (�) = 0D0

s (�)

= div((D1s−1(�)��H

1s−1��H

1)∩�−1

0◦R1

<N/2−1(�))

= div((D1s−1(�)∩�−1 ◦

R1s−1(�))��H

1)

= div(D1s−1(�)∩�−1 ◦

R1s−1(�))��P

0

= divD1s−1(�)��P

0

Finally we note

Remark 3.9We always obtain easily dual results using the Hodge star operator. This would change the homo-geneous boundary condition from the tangential (electric) to the normal (magnetic) one. However,since this would multiply the number of results by two, we let their formulation to the interestedreader.

4. PROOFS

Let ε, and � be as in Section 3. We start with the

Proof of Lemma 3.1Let E ∈L2,q

s (�). Looking at [3, Section 4] and using the Helmholtz decompositions [2, (2.7)],we may choose b� ∈Lin

◦Bq(�), �=1, . . . ,dq , with b� =��+H� ∈ rot

◦Rq−1(�)⊕ε εH

q(�), where

{H�} is a ⊕ε-ONB of εHq(�). Then e :=E−∑�〈E,εH�〉L2,q (�)b� ∈ εL

2,qs (�). This proves one

inclusion and the other one is trivial, because the forms of◦Bq(�) are smooth and compactly



supported. Moreover, if E ∈Lin◦Bq(�)∩εH

q(�)⊥ε , then εE=∑� e�εb� ∈ εHq(�)⊥ and thus

0=〈εE,Hk〉L2,q (�) =∑�

e�〈εH�,Hk〉L2,q (�) =ek

which proves the directness of the sum. The other direct decomposition may be shown in a similarway. �

We introduce the Hilbert spaces (closed subspaces of L2,qs (�))

◦R˜

qs (�) := {E ∈ ◦

Rqs (�) : rot(�2s E)=0}

= {E ∈ ◦R

qs (�) : rotE=−2s�−2RE}

D˜qs (�) := {E ∈D

qs (�) :div(�2s E)=0}

= {E ∈Dqs (�) :divE=−2s�−2T E}

and note the important fact that the ||·||Rqs (�)-, ||·||Rq

s (�)- and ||·||L2,qs (�)

-norms resp. the ||·||Dqs (�)-,

||·||Dqs (�)- and ||·||

L2,qs (�)-norms are equivalent on

◦R˜

qs (�) resp. D˜

qs (�). Here we used the operators

R,T from [9, Definition 1]. First, we need an easy consequence of the projection theorem:

Lemma 4.1Let s∈R. Then the orthogonal decompositions

(i) L2,qs (�) = rot

◦R

q−1s−1 (�)⊕s,ε ε−1D˜

qs (�)=ε−1rot

◦R

q−1s−1 (�)⊕s,ε D˜

qs (�)

(ii) L2,qs (�) = divD

q+1s−1 (�)⊕s,ε ε−1

◦R˜

qs (�)=ε−1divD

q+1s−1 (�)⊕s,ε

◦R˜

qs (�)

hold with continuous projections. Here we denote by ⊕s,ε the orthogonal sum with respect to the

〈ε�2s · , · 〉L2,q (�)-scalar product and the closures are taken in L2,qs (�). The space rot

◦R

q−1s−1 (�) resp.

divDq+1s−1 (�) may be replaced by rot

◦C∞,q−1(�) resp. divD

q+1vox (�).

Proof

Since◦C∞,q−1(�) is dense in

◦R

q−1s−1 (�) we have E ∈L2,q

s (�)∩(rot◦R

q−1s−1 (�))⊥s,ε , if and only

if E ∈L2,qs (�)∩(rot

◦C∞,q−1(�))⊥s,ε , which means �2sεE ∈0D

q−s(�). Thus E ∈ D˜

qs (�), because

0=div(�2sεE)=�2sdivεE+2s�2s−2T εE . This shows (i) and (ii) follows analogously. �

Remark 4.2Clearly this lemma holds for 0-admissible transformations ε as well.

We need two important results, which may be formulated as follows: Defining the Hilbert space

εXqt (�) :=(

◦R

qt (�)∩ε−1D

qt (�))��H

qt ��H

qt , t ∈R

we have


1524 D. PAULY

Lemma 4.3Let s∈R\ I and q �=0 or q=0 and s>−N/2. Then

εROTqs : εX

qs (�)∩ ε−1

0Dqloc(�) −→ 0

◦R

q+1s+1 (�)

E �−→ rotE

εDIVqs : εX

qs (�)∩ 0

◦R

qloc(�) −→ 0D

q−1s+1 (�)

H �−→ divεH

are continuous and surjective Fredholm operators with kernels

N (εROTqs )=N (εDIV

qs )=

{εH

qs (�) if s<−N/2

εHq(�) if s>−N/2

Remark 4.4By adding suitable Dirichlet forms from εH

q(�) we always may obtain H⊥Bq(�), q �=1, and

εE⊥ ◦Bq(�) or εH,εE⊥εH

q(�), if s>−N/2. Then for s>−N/2 the operators

εROTqs : εXq

s (�)∩ε−10D

qloc(�)−→0

◦R

q+1s+1 (�)

εDIVqs : εXq

s (�)∩0◦R

qloc(�)−→0D

q−1s+1 (�)

are also injective and hence topological isomorphisms by the bounded inverse theorem. Further-more, we note (using the notations from [3])

Hqt =Dq(I

q,0t )=Rq(J

q,0t ), H

qt =D

q,1t =R

q,1t

and for t<N/2

Hqt =H

qt−1={0}

ProofThis lemma has been proved in [3, Corollary 3.13, Lemma 3.14] for s>−N/2. Hence we only haveto discuss the cases of small weights s<−N/2 and ranks of forms 1�q�N . Well-definedness,continuity and the assertions about the kernels are trivial in these cases. To show surjectivity for

εDIVqs , let us pick some F ∈0D

q−1s+1 (�). Following the proofs of [3, Lemmas 3.5, 3.12] and using

[9, Theorem 4], we represent the extension by zero of F to RN by

F=:FD+FR ∈0Dq−1s+1 (RN )+0R

q−1s+1 (RN )

Now FD is contained in the range of the operator B from [9, Theorem 7] and thus we obtain some

h∈Dqs (R

N )∩0Rqs (R

N ) solving divh=FD



Applying the regularity result [13, Satz 3.7] we even have h∈H1,qs (RN ). Since FR is an element

of 0Dq−1s+1 (�)∩0R

q−1s+1 (�) we may represent this form in terms of a spherical harmonics expansion

FR|Ar0 = ∑I∈Iq−1,0

f I Dq−1I + f Dq−1,1+ ∑

I∈ s+Iq−1,0

f I Dq−1I

with uniquely determined f I , f ∈C using [3, Theorem 2.6], where

s

+Iq−1,0 :={I ∈Iq−1,0 :s(I )=+ ∧ e(I )<−s−1−N/2}

Now looking at [3, Remark 2.5] the first term of the sum on the right-hand side belongs toL2,q−1

<N/2 (Ar0), the second to L2,q−1<N/2−1(Ar0) and the third to L2,q−1

s+1 (Ar0). Therefore,

FR− ∑I∈ s

+Iq−1,0

f I�Dq−1I ∈L2,q−1(RN )

This suggests the ansatz

H :=�h+ ∑I∈ s

+Iq−1,0

f I�Rq1 I

+�

to solve H ∈0◦R

qs (�)∩ε−1D

qs (�) and divεH =F . Thus, we are searching for some q-form � in

◦R

qs (�)∩ε−1D

qs (�) satisfying

rot� = −rot(�h)=−Crot,�h=: Gdivε� = F−div(�εh)− ∑

I∈ s+Iq−1,0

f I div(�εRq1 I

)=: F (9)

since rot(�Rq1 I

)=0. Clearly, we have got G∈0◦R

q+1vox (�)⊂0

◦Rq+1(�). Moreover, not only F is an

element of 0Dq−1s+1 (�) but also F ∈0Dq−1(�) holds, because

F = div(1−�)h− ∑I∈ s

+Iq−1,0

f I Cdiv,�Rq1 I

+FR− ∑I∈ s

+Iq−1,0

f I�Dq−1I

−div(ε�(h+ ∑

I∈ s+Iq−1,0

f I�Rq1 I

))

where the first two terms of the sum on the right-hand side lie in L2,q−1vox (�), the sum of the third

and fourth terms in L2,q−1(�) and the last one in L2,q−1s+1+�(�)⊂L2,q−1(�), since

�(h+ ∑I∈ s

+Iq−1,0

f I�Rq1 I

)∈H1,qs (RN )

and ��−s−1. Now we are able to apply the result [13, Satz 6.10] and obtain some q-form

�∈ ◦R

q−1(�)∩ε−1D

q−1(�) solving the system (9). Since s<−N/2<−3/2, we see that is an


1526 D. PAULY

element of◦R

qs (�)∩ε−1D

qs (�), which completes the proof. The assertion about εROT

qs follows

analogously.�

Proof of Theorem 3.4Apply Lemma 4.3 and Remark 4.4 with the modified values for q , s and ε. �

Proof of Theorem 3.5The first equations in (i)–(iv) have already been proved in [3, Theorem 5.8]. Let

G∈0◦R

qs (�)= rot((

◦R

q−1s−1 (�)��H

q−1s−1 )∩−1

0Dq−1<N/2(�))

i.e.

G= rotGs−1+ ∑I∈Iq−1,0

s−1

gI rot�Dq−1I

with Gs−1∈ ◦R

q−1s−1 (�)∩−1D

q−1s−1 (�)∩ ◦

Bq−1(�)⊥ and gI ∈C. Since

rot�Dq−1I = rot��Dq−1

2 I−rotCdivrot,�D

q−12 I

and clearly

rot��Rq−12 I

= rotdivCrot,�Rq−12 I

we obtain

i�q′

e(I ) rot�Dq−1I = rot��Pq−1

e(I ),c(I )− i�q′

e(I )rotCdivrot,�Dq−12 I

−�qe(I ) rotdivCrot,�Rq−12 I

Now ��Pq−1e(I ),c(I ) =C�,�P

q−1e(I ),c(I ) has compact support and therefore

G∈ rot(◦R

q−1s−1 (�)∩−1D

q−1s−1 (�)∩ ◦

Bq−1(�)⊥)

which proves (i). (ii) is shown analogously. The last equation in (iii) follows from (i), since wecan split off a term

�−Dq−1,1�,m =−1�−Dq−1,1

�,m =−1�−Dq−1,1�,m +−1�−Dq−1,1

�,m

and the tower forms are smooth and decays as well as div�−Dq−1,1�,m =0 holds by Remark 3.6.

Finally the last equation in (iv) is a direct consequence of (ii) and a similar argument like the latterone. �

Proof of Theorem 3.8Lemma 4.3 yields

L2,Ns (�)= rot((

◦R N−1

s−1 (�)��HN−1s−1 ��H

N−1)∩−1

0DN−1<N/2−1(�))

and the same arguments used in the latter proof show

rot((◦R N−1

s−1 (�)��HN−1s−1 )∩−1

0DN−1<N/2−1(�))= rot(

◦R N−1

s−1 (�)∩−1D N−1s−1 (�)∩ ◦

B N−1(�)⊥)



Because div�H N−1=0, we have

L2,Ns (�)= rot((

◦R N−1

s−1 (�)∩−1D N−1s−1 (�)∩ ◦

B N−1(�)⊥)��HN−1

)

Now div P N = H N−1 and thus rot�H N−1=��P N −rotCdiv,� P N , which proves

L2,Ns (�)= rot(

◦R N−1

s−1 (�)∩−1D N−1s−1 (�)∩ ◦

B N−1(�)⊥)��PN

Finally, we have to show that the sum is direct. To do this, let F= f ��P N = rotE with some

f ∈C and E ∈ ◦R N−1

s−1 (�). We want to use the notations from [9], i.e. the forms PN ,40,1 and QN ,4

0,1 , as

well. By definition there exists a constant c �=0, such that P N =c−RN ,20,1 =c/(2−N )QN ,4

0,1 using

[3, Remark 2.3]. Since s>N/2 and PN ,40,1 ∈L2,N

<−N/2(�) partial integration yields

〈rotE, PN ,40,1 〉L2,q (�) =0

because PN ,40,1 ∈Lin{∗1} is constant. However, on the other hand, we obtain

〈F, PN ,40,1 〉L2,q (�) = f 〈��P N , PN ,4

0,1 〉L2,q (�) =c f

2−N〈C�,�Q

N ,40,1 , PN ,4

0,1 〉L2,q (�) =c f

by [9, (73)], i.e. f =0. �

Now we turn to the main idea of our decompositions and the

Proof of Theorem 3.2Let 1�q�N−1 and s be as in Section 3 as well as F ∈L2,q

s (�). Using Lemma 4.1 we decompose

F=Fr +ε−1 Fd with Fr ∈ rot◦C∞,q−1(�) and Fd ∈ D˜

qs (�). A second application of this lemma

yields the decomposition ε−1 Fd =ε−1Fd + F with Fd ∈divDq+1vox (�) and F ∈ ◦

Rqs (�)∩ε−1D

qs (�).

Furthermore, there exists a constant c>0 independent of F , such that

||Fr ||L2,qs (�)+||Fd ||L2,qs (�)

+||F ||Rqs (�)∩ε−1D

qs (�)�c||F ||

L2,qs (�)

Now F is more regular than F and this enables us to solve

divεH =divε F ∈0Dq−1s+1 (�), rotE= rot F ∈0

◦R

q+1s+1 (�)

with some H ∈ εXqs (�)∩ 0

◦R

qloc(�) and E ∈ εX

qs (�)∩ ε−1

0Dqloc(�) by Lemma 4.3. We note that

E,H ∈L2,qt (�) for all t with t�s and t<N/2−1. Then F := F−E−H ∈ εH

qt (�) and

F=Fr +H+ε−1(Fd +εE)+ F (10)

where Fr +H ∈0◦R

qt (�) and Fd +εE ∈0D

qt (�).

For s>−N/2 we may refine this representation of F . In fact for these values of s we haveF ∈ εH

q>−N/2(�)= εH

q(�) by [3, Lemma 3.8]. Using Remark 4.4 or [3, Theorem 5.1] addition-ally we may obtain εE⊥ ◦

Bq(�) and H⊥Bq(�), if q �=1, or εE⊥εHq(�) and εH⊥εH

q(�), if


1528 D. PAULY

s>1−N/2. Therefore, Fd +εE ∈0Dqt (�) holds for s>−N/2 and Fr +H ∈0

◦R

qt (�) for s>1−N/2

or 1−N/2>s>−N/2 and q �=1. Moreover, [3, Theorem 5.1] yields not only E,H ∈ εXqs (�)

but also

E ∈ (◦R

qs (�)∩ε−1D

qs (�))��Dq(I

q,0s ) if q �=N−1

H ∈ (◦R

qs (�)∩ε−1D

qs (�))��Rq(J

q,0s ) if q �=1

i.e. the exceptional forms do not appear in these cases.The stronger assumption F ∈ εL

2,qs (�) for s>1−N/2 or s>−N/2 and 2�q�N−2 implies

F ∈ εHq(�)⊥ε and thus F=0. Hence in these cases (10) turns to

F=Fr +H+ε−1(Fd +εE) (11)

Until now we have shown the assertions of Theorem 3.2 (i), (ii) and also (iii) for s<N/2−1.Considering larger weights s>N/2−1 and F ∈ εL

2,qs (�), the tower forms occur in the repre-

sentation (11). More precisely we have

E = Es+ ∑I∈Iq,0

s

eI�DqI +e�

{−DN−1,10,1 if q=N−1

0 otherwise

H = Hs+ ∑J∈Jq,0

s

h J�RqJ +h�

{−R1,10,1 if q=1

0 otherwise

with uniquely determined Es,Hs ∈◦R

qs (�)∩ε−1D

qs (�) and eI ,e,hJ ,h∈C. We note I

q,0s =J

q,0s .

For s<N/2 we have Iq,0s =∅ and for s>N/2 we observe �qe(I )�R

qI =−i�q

′e(I )�D

qI ∈L2,q

<N/2(�) for

all I ∈Iq,0s by (5). Thus, we obtain

F=Fr +Hs+ε−1Fd +Es+ ∑I∈Iq,0

s

(hI − eI )�RqI +�

⎧⎪⎪⎨⎪⎪⎩h−R1,1

0,1 if q=1

e−DN−1,10,1 if q=N−1

0 otherwise

where we define eI :=−ieI�qe(I )/�

q ′e(I ). Looking, for example, at the case q=1 we observe that

�−R1,10,1 /∈L2,q

�N/2−1(�). Then for integrability reasons we obtain h=0, such that the exceptionaltower form does not appear. Clearly also e=0 holds true for q=N−1. Moreover, hI = eI since

RqI are linear independent and �Rq

I /∈L2,qs (�) for all I ∈I

q,0s and s>N/2.

By the smoothness of �DqI as well as the decay and differentiability properties of ε, we obtain

furthermore ε�DqI ∈H1,q

s (�). Thus, for all s>1−N/2 we obtain the representation

F= Fr + ∑I∈Iq,0

s

h I�RqI +ε−1(Fd + ∑

I∈Iq,0s

eI�DqI

)Copyright q 2008 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2008; 31:1509–1543

DOI: 10.1002/mma


where Fr :=Fr +Hs and Fd :=Fd +εEs+∑I∈Iq,0s

eI ε�DqI as well as

Fr + ∑I∈Iq,0

s

h I�RqI ∈(L2,q

s (�)��Hqs )∩0

◦R

q<N/2(�)

Fd + ∑I∈Iq,0

s

eI�DqI ∈(L2,q

s (�)��Hqs )∩0D

q<N/2(�)

with �q′

e(I )hI + i�qe(I )eI =0 for all I ∈Iq,0s . This proves the remaining assertions of (iii) and the

first equation in (iv).To show the second equation in (iv), we observe

�DqI = �divrotDq

2 I=divrot�Dq

2 I−Cdivrot,�D

q2 I

�RqI = � rotdivRq

2 I= rotdiv�Rq

2 I−Crotdiv,�R

q2 I

and therefore

F= ˜Fr +ε−1 ˜Fd + ∑I∈Iq,0

s

h I

�qe(I )�ε�P

qe(I ),c(I )

where

˜Fr := Fr − ∑I∈Iq,0

s

h ICrotdiv,�Rq2 I

− ∑I∈Iq,0

s

eI rotCdiv,�Dq2 I

∈0◦R

qs (�)

˜Fd := Fd − ∑I∈Iq,0

s

eICdivrot,�Dq2 I

− ∑I∈Iq,0

s

h I divCrot,�Rq2 I

∈0Dqs (�)

∑I∈Iq,0

s

h I

�qe(I )�ε�P

qe(I ),c(I ) ∈�ε�P

qs−2

Clearly all sums are direct resp. orthogonal as stated. Only in the second equation of (iv) one maysee this not directly. Hence, for example, if

E= ∑I∈Iq,0

s

eI�ε�Pqe(I ),c(I ) =G+ε−1F ∈0

◦R

qs (�)�ε−1

0Dqs (�)

with some s>N/2, then

H :=G− ∑I∈Iq,0

s

eI rotdiv�Pqe(I ),c(I ) =

∑I∈Iq,0

s

eI ε−1 divrot�Pq

e(I ),c(I )−ε−1F

is not only a Dirichlet form, i.e. H ∈ εHq(�), but also an element of

◦Bq(�)⊥ε . Hence, H must

vanish and thus

G= ∑I∈Iq,0

s

eI rotdiv�Pqe(I ),c(I ) ∈L2,q

s (�)


1530 D. PAULY

which is only possible, if eI =0 for all I ∈Iq,0s , since rotdiv�Pq

e(I ),c(I ) /∈L2,qs (�) are linear inde-

pendent.It remains to prove the last equation of (iv). Before we start with this we observe that by the

closed graph theorem all projections in (ii)–(iv) are continuous. We note

0◦R

qs (�),ε−1

0Dqs (�)⊂L2,q

s (�)∩εHq−s(�)⊥ε =:Yq

s (�)

and thus

Xqs (�) :=0

◦R

qs (�)⊕ε ε−1

0Dqs (�)⊂Y

qs (�)

Furthermore, Xqs (�) and Y

qs (�) are closed subspaces of L2,q

s (�). By the first equation of (iv) andLemma 3.1, we have

codimXqs (�)=dim εH

q(�)+dim �ε�Pqs−2=dq + ∑

0��<s−N/2�q�

since dim �ε�Pqs−2=dim P

qs−2=dim H

qs and s−N/2 /∈N0 because s /∈ I. With the identity

codimYqs (�)=dim εH

q−s(�) we get by Appendix A that X

qs (�) and Y

qs (�) possess the same

finite codimension in L2,qs (�). Consequently we obtain X

qs (�)=Y

qs (�). �

APPENDIX A: WEIGHTED DIRICHLET FORMS

Let �>0. As already mentioned in Section 2 for the space of Dirichlet forms we have

εHq−N/2(�)= εH

q(�)= εHq<t (�), t :=N/2−�q,1−�q,N−1

and its dimension equals dq =�q ′ , the q ′th Betti number of �. Furthermore, we have for all t ∈R

εH0t (�)={0}, εH

Nt (�)=

{{0}, t�−N/2

ε−1Lin{∗1}, t<−N/2

(This holds even for �=0.) We repeat some notations and results from [3, 9, 13]. Let us introducethe ‘special growing Dirichlet forms’ E+

�,m from [13, Lemma 7.11] or [4] as the unique solutionsof the problems

E+�,m ∈ εH

q<−N/2−�(�)∩ ◦

Bq(�)⊥ε , E+�,m−+Dq,0

�,m ∈L2,q>−N/2(�)

where �∈N0 and 1�m��q� with

�q� =(N

q

)(N−1+�

�

)qq ′(N+2�)

N (q+�)(q ′+�)



from [9, Theorem 1 (iii)]. To guarantee their existence we have to impose the decay conditions

�>� and ��N/2−1. We note �q0 =(Nq

)and thus �00=�N

0 =1. Moreover, +D0,00,1 resp. +DN ,0

0,1 is

a multiple of 1 resp. ∗1.Lemma A.1Let 1�q�N−1 and s∈(−∞,−N/2)\ I as well as �>−s−N/2 and ��N/2−1. Then

εHqs (�)= εH

q(�)�εHqs (�)∩ ◦

Bq(�)⊥ε

holds. Moreover,

εHqs (�)∩ ◦

Bq(�)⊥ε =Lin{Eq�,m :�<−s−N/2}

Corollary A.2The dimension dqs of εH

qs (�) is finite and independent of ε. More precisely

dqs =dq + ∑�<−s−N/2

�q�

Furthermore, the mapping

dq( ·)(−∞,−N/2)\ I∪(−N/2,N/2−1) −→ N0

s �−→ dqs

is locally constant and monotone decreasing. It jumps exactly at the points s∈ I, i.e. −s−N/2∈N0.

ProofThe directness of the sum follows by (4) and the inclusions

εHq(�)�εH

qs (�)∩ ◦

Bq(�)⊥ε ⊂ εHqs (�)

Lin{Eq�,m :�<−s−N/2} ⊂ εH

qs (�)∩ ◦

Bq(�)⊥ε

are trivial. Hence, it remains to prove

εHqs (�)⊂ εH

q(�)�Lin{Eq�,m :�<−s−N/2}

Therefore, we pick some E ∈ εHqs (�). We observe E ∈H1,q

s (A�) by the regularity result [13,Korollar 3.8] and even

rotE=0, divE |A�=− div εE

∣∣A�

∈L2,qs+�+1(A�)

for all r0<�<r1. Thus, we have �E ∈ ◦R

qs (�)∩D

qs (�) with

rot�E =Crot,�E ∈0◦R

q+1vox (�)∩Bq+1(�)⊥

div�E =Cdiv,�E−�divE ∈0Dq−1s+�+1(�)∩ ◦

Bq−1(�)⊥


1532 D. PAULY

The assumptions on � yield s+�+1>1−N/2. Thus, by Lemma 4.3 there exists some q-form

e∈ ◦R

qt (�)∩D

qt (�) with some t>−N/2 solving

rote= rot�E, dive=div�E

Therefore H :=�E−e∈Hqs (�). Thus, rotH =0 and divH =0 in Ar0 and we may represent H in

terms of a spherical harmonics expansion

H |Ar0 =∑�,n

h−�,n

−Dq,0�,n + h Dq,1+ ∑

�<−s−N/2

�q�∑m=1

h+�,m

+Dq,0�,m

with uniquely determined h−�,n, h,h+

�,m ∈C using [3, Theorem 2.6]. By [3, Remark 2.5] the first

term of the sum on the right-hand side belongs to L2,q<N/2(Ar0), the second to L2,q

<N/2−1(Ar0) and

the third to L2,qs (Ar0). We obtain

H− ∑�<−s−N/2

�q�∑m=1

e�,m+Dq,0

�,m ∈L2,q<N/2−1(Ar0)

which yields

h :=H− ∑�<−s−N/2

�q�∑m=1

h+�,mE

+�,m ∈L2,q

>−N/2(�)

Finally, we obtain

E− ∑�<−s−N/2

�q�∑m=1

h+�,mE

+�,m =(1−�)E+e+h∈ εH

q>−N/2(�)= εH

q(�) �

APPENDIX B: VECTOR FIELDS IN THREE DIMENSIONS

Now we will translate our results to the classical framework of vector analysis. Thus, we switchto some (maybe) more common notations.

Let N :=3. We identify 1-forms with vector fields via Riesz’ representation theorem and 2-formswith 1-forms via the Hodge star operator and thus with vector fields as well. Using Euclideancoordinates {x1, x2, x3} this means in detail we identify the vector field

E=⎡⎢⎣E1

E2

E3

⎤⎥⎦Copyright q 2008 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2008; 31:1509–1543

DOI: 10.1002/mma


with the 1-form

E1 dx1+E2 dx2+E3 dx3

resp. with the 2-form

E1∗dx1+E2∗dx2+E3∗dx3=E1 dx2∧dx3+E2 dx3∧dx1+E3 dx1∧dx2

Moreover, we identify the 3-form E dx1∧dx2∧dx3 with the 0-form and/or function E . We willdenote these identification isomorphisms by ∼=. Then the exterior derivative and co-derivative turnto the classical differential operators

grad=∇ =

⎡⎢⎢⎣�1

�2

�3

⎤⎥⎥⎦ , curl=∇×, div=∇·

from vector analysis, where × resp. · denotes the vector resp. scalar product in R3. In particularwe have the following identification table:

q=0 q=1 q=2 q=3

rot=d grad curl div 0div=� 0 div −curl grad

B.1. Tower functions and fields

Let us briefly construct our tower forms once more in classical terms. Using polar coordinates{r, ,ϑ}, i.e.

x=�(r, ,ϑ)=r

⎡⎢⎣cos cosϑ

sin cosϑ

sinϑ

⎤⎥⎦we have with an obvious notation⎡⎢⎣

dx1

dx2

dx3

⎤⎥⎦= J�

⎡⎢⎣dr

d

dϑ

⎤⎥⎦=Q

⎡⎢⎣dr

r cosϑd

r dϑ

⎤⎥⎦where Q :=[er e eϑ] is an orthonormal matrix and

er :=⎡⎢⎣cosϑcos

cosϑsin

sinϑ

⎤⎥⎦ , e :=⎡⎢⎣

−sin

cos

0

⎤⎥⎦ , eϑ :=⎡⎢⎣

−sinϑcos

−sinϑsin

cosϑ

⎤⎥⎦Copyright q 2008 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2008; 31:1509–1543

DOI: 10.1002/mma

1534 D. PAULY

the corresponding orthonormal basis of R3. Since {dx1,dx2,dx3} is an orthonormal basis of 1-forms, {dr,r cosϑd ,r dϑ} is an orthonormal basis as well. Moreover, we have again with anobvious notation ⎡⎢⎣

dr

r cosϑd

r dϑ

⎤⎥⎦=Qt

⎡⎢⎣dx1

dx2

dx3

⎤⎥⎦=

⎡⎢⎢⎣etr

et

etϑ

⎤⎥⎥⎦⎡⎢⎣dx1

dx2

dx3

⎤⎥⎦=⎡⎢⎣er ·dxe ·dxeϑ ·dx

⎤⎥⎦which shows

dr ∼=er , r cosϑd ∼=e , r dϑ∼=eϑ

We will denote the representations of grad, curl and div in polar coordinates by grad, curl anddiv as well as their realizations on S := S2 by gradS , curlS and divS . These may be derived bythe formula

(∇xu)◦�= J−t� ∇r, ,ϑ(u◦�)=[er (r cosϑ)−1e r−1eϑ]∇r, ,ϑ(u◦�)

and we then have the following representations:

gradu = er�r u+ 1

rgradS u, gradS u= 1

cosϑe � u+eϑ�ϑu

curlv = er ×�rv+ 1

rcurlS v, curlS v= 1

cosϑe ×� v+eϑ×�ϑv

divv = er ·�rv+ 1

rdivS v, divS v= 1

cosϑe ·� v+eϑ ·�ϑv

We note that we do not distinguish between u resp. v and u◦� resp. v◦� anymore. Moreover, inpolar coordinates the Laplacian reads

D=�2r + 2

r�r + 1

r2DS

where

DS = 1

cos2ϑ�2 +�2ϑ− sinϑ

cosϑ�ϑ

is the Laplace–Beltrami operator.Let us introduce the classical spherical harmonics of order n

yn,m, n∈N0, m=1, . . . ,2n+1

which form a complete orthonormal system in L2(S), i.e. 〈yn,m, y�,k〉L2(S) =�n,��m,k and satisfy

(DS+�n)yn,m =0, �n :=n(n+1)

as well as the corresponding potential functions

z±,n,m :=r�±,n yn,m, n∈N0, m=1, . . . ,2n+1



which are homogeneous of degree

�±,n :={n if ±=+−n−1 if ±=−

and solve

�z±,n,m =Dz±,n,m =0

(See, for example, [14, Kapitel VII, Section 4; 15, Chapter 2.3].)Moreover, for k,n∈N0, m=1, . . . ,2n+1, we define

zk±,n,m :=�k±,nr2k z±,n,m =�k±,nr

�2k±,n yn,m

where

�k±,n := �(1±n±1/2)

4k ·k! ·�(k+1±n±1/2)

and � denotes the gamma-function. The functions zk±,n,m are homogeneous of degree �2k±,n with

��±,n :=�+�±,n

and satisfy

�zk±,n,m = zk−1±,n,m

where z−1±,n,m :=0. With the aid of these functions, which we will call a ‘�-tower’, we constructfor k∈N0 the functions and fields

U 2k±,n,m := zk±,n,m

U 2k−1±,n,m := gradU 2k±,n,m

which we will call a ‘divgrad-tower’, as well as the fields

V 2k±,n,m := rer ×grad zk±,n,m =er ×gradS zk±,n,m =�k±,nr

�2k±,ner ×Yn,m

V 2k−1±,n,m := −curlV 2k±,n,m

which we will call a ‘−curlcurl-tower’. Here Yn,m :=gradS yn,m . The fields U2k−1±,n,m are irrotational

and the fields V �±,n,m solenoidal. Moreover, we have

divU 2k+1±,n,m =U 2k±,n,m, curlV 2k+1±,n,m =V 2k±,n,m

as well as divU−1±,n,m =0 and curlV−1±,n,m =0 and thus

�U 2k±,n,m = divgradU 2k±,n,m =U 2k−2±,n,m

�U 2k+1±,n,m = graddivU 2k+1±,n,m =U 2k−1±,n,m

�V �±,n,m = −curlcurlV �±,n,m =V �−2±,n,m


1536 D. PAULY

where U−2±,n,m :=0, V−2±,n,m :=0. We mention that U �±,n,m and V �±,n,m are homogeneous of degree

��±,n . Moreover,

U−1±,n,m =V−1±,n,m

Thus, we define

P±,n,m :=U 1±,n,m−V 1±,n,m

Then U �±,n,m , V�±,n,m , �=−1,0, and even P±,n,m are potential fields resp. functions.

The next picture may illustrate the denotations tower:

In the exceptional case (n,m)=(0,1), the function zk±,0,1 is a multiple of r2k+�±,0 and thus we

obtain V �±,0,1=0 for all � as well as an exceptional divgrad-tower

U 2k±,0,1 = �k±,0r

2k−�−,±

U 2k−1±,0,1 = gradU 2k

±,0,1=�k±,0�r r2k−�−,±er =�k±,0(2k−�−,±)r2k−1−�−,±er

where U−1+,0,1=0.

Let us briefly compare these classical towers with the q-form towers: On S we identify 1-formswith linear combinations of e and eϑ as well as 2-forms with scalar functions. More precisely a1-form � cosϑd +�ϑ dϑ will be identified with the tangential vector field � e +�ϑeϑ and a



2-form �cosϑd ∧ dϑ with the function �. Then our operators �,� and �, � from [9] turn to

q=0 q=1 q=2 q=3

�v∼= 0 v ·er v×er v

�v∼= v −(v×er )×er v ·er 0�v∼= ver −v×er v —�v∼= v v ver —

where

−(v×er )×er = v ·e e +v ·eϑeϑ =v−v ·ererv×er = v ·eϑe −v ·e eϑ

We note

y0,1 ∼= S00,1 is constant

yn,m ∼= T 0n−1,m

Yn,m =gradS yn,m ∼= in1/2(n+1)1/2S1n−1,m

for n∈N in the terminology of [9]. Then for n∈N we get up to constants

U 2k±,n,m∼= D0

(±,2k+1,n−1,m) =∗R3(±,2k+1,n−1,m)

U 2k−1±,n,m∼= R1

(±,2k,n−1,m) =∗D2(±,2k,n−1,m)

V �±,n,m∼= D1

(±,�+1,n−1,m) =∗R2(±,�+1,n−1,m)

and for n=0

U 2k+,0,1

∼= D0(+,2k,0,1) =∗R3

(+,2k,0,1)

U 2k−1+,0,1

∼= R1(+,2k−1,0,1) =∗D2

(+,2k−1,0,1)

U 2k−,0,1

∼= D0(−,2k+2,0,1) =∗R3

(−,2k+2,0,1)

U 2k−1−,0,1

∼= R1(−,2k+1,0,1) =∗D2

(−,2k+1,0,1)

Finally for s∈R and �=−1,0, we put (with Lin∅:={0})V

�

s := Lin{V �−,n,m :V �−,n,m /∈L2s (A1)}=Lin{V �−,n,m :n��+s+1/2}

U�

s := Lin{U �−,n,m :U �−,n,m /∈L2s (A1)}=Lin{U �−,n,m :n��+s+1/2}

Ps := Lin{P−,n,m : P−,n,m /∈L2s (A1)}=Lin{P−,n,m :n�s+3/2}

U� :=U

�

s := Lin{U �−,0,1 :U

�−,0,1 /∈L2

s (A1)}=Lin{U �−,0,1 :0��+s+1/2}


1538 D. PAULY

B.2. Results for vector fields

For some operator ♦∈{grad,curl,div} and s∈R, we define the Hilbert spaces

Hs(♦,�) :={u∈L2s (�) :♦u∈L2

s+1(�)}, Hs(◦♦,�) := ◦

C∞(�)

Hs(♦0,�) :={Hs(♦,�) :♦u=0}, Hs(◦♦0,�) :={Hs(

◦♦,�) :♦u=0}

where the closure is taken in Hs(♦,�). Then the spaces◦R

qs (�) and D

qs (�) turn to the usual

Sobolev spaces, i.e.

q=0 q=1 q=2 q=3

◦R

qs (�) Hs(

◦grad,�)= ◦

H1s (�) Hs(

◦curl,�) Hs(

◦div,�) L2

s (�)

Dqs (�) L2

s (�) Hs(div,�) Hs(curl,�) Hs(grad,�)=H1s (�)

For two operators ♦,�∈{(◦)

grad(0),(◦)

curl(0),(◦)

div(0)}, we define

Hs(♦,�,�) :=Hs(♦,�)∩Hs(�,�)

The generalized boundary condition �∗E=0 for a q-form E from◦R

qloc(�) turns to the usual

boundary conditions �E= E |�� =0, �t E= ×E |�� =0 and �n E= ·E |�� =0 (for q=0,1,2)

weakly formulated in the spaces H(◦

grad,�), H(◦

curl,�) and H(◦div,�), where denotes the outward

unit normal at �� and � the trace as well as �t resp. �n the tangential resp. normal trace of the vectorfield E . The linear transformations ε, , � ( and � may be identified!) can be considered as real-valued, variable, symmetric and uniformly positive-definite matrices with L∞(�)-entries, which

satisfy the asymptotics at infinity assumed in Sections 2 and 3. Moreover, for ♦,�∈{(◦)

curl(0),(◦)

div(0)}we define

Hs(�ε,�) :=ε−1Hs(�,�), Hs(♦,�ε,�) :=Hs(♦,�)∩Hs(�ε,�)

Now we have two kinds of Dirichlet fields. The first ones, the classical Dirichlet fields,

εHs(�) :=Hs(◦

curl0,div0 ε,�)∼= εH1s (�), s∈R

correspond to q=1 and the second ones, the classical Neumann fields,

εHs(�) :=Hs(curl0,◦div0 ε,�)∼=ε−1

ε−1H2s (�), s∈R

correspond to q=2. Moreover, we have the compactly supported fields

◦B1(�)∼=: ◦

B1(�)⊂H(◦

curl0,�),◦B2(�)∼=: ◦

B2(�)⊂H(◦div0,�)

and the fields with bounded supports

B2(�)∼=:B(�)⊂H(curl0,�)



Let s>− 12 . Using the Dirichlet and Neumann fields, we put

Hs(♦,�) :=Hs(♦,�)∩H(�)⊥, Hs(♦,�) :=Hs(♦,�)∩ H(�)⊥

and define in the same way Hs(♦,�,�), Hs(♦,�ε,�) and Hs(♦,�,�), Hs(♦,�ε,�). Thenwe have

Hs(div0,�) =Hs(div0,�)∩εH(�)⊥ =Hs(div0,�)∩ ◦B1(�)⊥

Hs(◦div0,�) =Hs(

◦div0,�)∩εH(�)⊥ =Hs(

◦div0,�)∩B(�)⊥

Hs(curl0,�) =Hs(curl0,�)∩εH(�)⊥ε =Hs(curl0,�)∩ ◦B2(�)⊥

Hs(◦

curl0,�) =Hs(◦

curl0,�)∩εH(�)⊥ε

and thus except of the last one the definitions of these spaces extend to all s∈R. Moreover, weset for s>− 1

2

εL2s (�) :=L2

s (�)∩εH(�)⊥ε , εL2s (�) :=L2

s (�)∩εH(�)⊥ε

We obtain

Lemma B.1Let s>− 1

2 . Then the direct decompositions

L2s (�) = εL2

s (�)�Lin◦B1(�)

L2s (�) = εL2

s (�)�ε−1Lin◦B2(�)= εL2

s (�)�LinB(�)

hold. If additionally s< 12 , then

L2s (�)= εL2

s (�)⊕ε εH(�), L2s (�)= εL2

s (�)⊕ε εH(�)

Let us note that the operator �ε reads as follows:

q �ε

0 ε−1 divgrad1 graddiv−ε−1 curlcurl2 −curlcurl+ε−1 graddiv3 divgrad

Thus we define

�ε :=graddiv−ε−1 curlcurl=�− ε curlcurl

Always assuming s /∈ I, i.e. for all n∈N0

s �=n+ 12 and s �=−n− 3

2


1540 D. PAULY

we obtain

Theorem B.2The following decompositions hold:

(i) If s<− 32 , then

L2s (�) =Hs(

◦curl0,�)+ε−1Hs(div0,�)=Hs(curl0,�)+ε−1Hs(

◦div0,�)

=Hs(◦

curl0,�)+ε−1Hs(div0,�)=Hs(curl0,�)+ε−1Hs(◦div0,�)

In the first line the intersections equal the finite dimensional space of Dirichlet fieldsresp. Neumann fields εHs(�) resp. εHs(�) and in the second line the intersections equal

the finite dimensional space of Dirichlet fields resp. Neumann fields εHs(�)∩ ◦B1(�)⊥ε

resp. εHs(�)∩ ◦B2(�)⊥.

(ii) If − 32<s�− 1

2 , then

L2s (�) =Hs(

◦curl0,�)�ε−1Hs(div0,�)

= Hs(curl0,�)�ε−1Hs(◦div0,�)�εH(�)

(iii) If − 12<s< 3

2 , then

εL2s (�) = Hs(

◦curl0,�)�ε−1Hs(div0,�)

εL2s (�) = Hs(curl0,�)�ε−1Hs(

◦div0,�)

For s�0 this decomposition is even 〈ε · , · 〉L2(�)-orthogonal.

(iv) If s> 32 , then

εL2s (�) = (([L2

s (�)��V−1s ]∩H<3/2(

◦curl0,�))

⊕εε−1([L2

s (�)��V−1s ]∩H<3/2(div0,�)))∩L2

s (�)

εL2s (�) = (([L2

s (�)��V−1s ]∩H<3/2(curl0,�))

⊕εε−1([L2

s (�)��V−1s ]∩H<3/2(

◦div0,�)))∩L2

s (�)

and

εL2s (�) = Hs(

◦curl0,�)�ε−1Hs(div0,�)��ε�Ps−2

εL2s (�) = Hs(curl0,�)�ε−1Hs(

◦div0,�)��ε�Ps−2



where the first two terms in the latter two decomposition are 〈ε · , · 〉L2(�)-orthogonal aswell. Furthermore,

L2s (�)∩εH−s(�)⊥ε = Hs(

◦curl0,�)⊕ε ε−1Hs(div0,�)

L2s (�)∩εH−s(�)⊥ε = Hs(curl0,�)⊕ε ε−1Hs(

◦div0,�)

Remark B.3Here Remark 3.3 holds analogously. In particular the matrix ε may be moved to the curl-free termsin our decompositions as well and for s<− 3

2 and �� 12 we have

εHs(�) = εH(�)�εHs(�)∩ ◦B1(�)⊥ε

εHs(�) = εH(�)�εHs(�)∩ ◦B2(�)⊥

Theorem B.4Let s∈R. Then

(i) Hs(curl0,�) = gradHs−1(grad,�)

and for s>− 12

(i′) Hs(◦

curl0,�) = gradHs−1(◦

grad,�)

If s< 52 , then

(ii) Hs(◦div0,�) = curlHs−1(

◦curl,div0�,�)

= curlHs−1(◦

curl,div0�,�)=curlHs−1(◦

curl,�)

Hs(div0,�) = curlHs−1(curl,◦div0 �,�)

= curlHs−1(curl,◦div0 �,�)=curlHs−1(curl,�)

All these spaces are closed subspaces of L2s (�). For s< 3

2

(iii) L2s (�) = divHs−1(

◦div,curl0�,�)

= divHs−1(◦div,curl0�,�)=divHs−1(

◦div,�)

L2s (�) = divHs−1(div,

◦curl0�,�) if s> 1

2

L2s (�) = divHs−1(div,

◦curl0�,�)=divHs−1(div,�)


1542 D. PAULY

Theorem B.5

(i) For s> 52

Hs(◦div0,�) = curl((Hs−1(

◦curl,�)��V

−1s−1)∩H<3/2(div0�,�))

= curl(Hs−1(◦

curl,div�,�)∩ ◦B1(�)⊥�)=curlHs−1(

◦curl,�)

Hs(div0,�) = curl((Hs−1(curl,�)��V−1s−1)∩H<3/2(

◦div0�,�))

= curl(Hs−1(curl,◦div �,�)∩B(�)⊥�)=curlHs−1(curl,�)

are closed subspaces of L2s (�).

(ii) For s> 32

(L2s (�)��V

−1s )∩H<3/2(

◦curl0,�)

=grad(Hs−1(◦

grad,�)��U0s−1)=Hs(

◦curl0,�)�grad�U

0s−1

(L2s (�)��V

−1s )∩H<3/2(curl0,�)

=grad(Hs−1(grad,�)��U0s−1)=Hs(curl0,�)�grad�U

0s−1

(L2s (�)��V

−1s )∩H<3/2(

◦div0,�)

=curl((Hs−1(◦

curl,�)��V−1s−1��V

0s−1)∩H<1/2(div0�,�))

=Hs(◦div0,�)�curl�−1�V

0s−1

(L2s (�)��V

−1s )∩H<3/2(div0,�)

=curl((Hs−1(curl,�)��V−1s−1��V

0s−1)∩H<1/2(

◦div0�,�))

=Hs(div0,�)�curl�−1�V0s−1

are closed subspaces of L2s (�)��V

−1s . Moreover, div�V 0−,n,m =0.

Theorem B.6Let s> 3

2 . Then

(i) L2s (�) = div((Hs−1(

◦div,�)��V

−1s−1��U

−1)∩H<1/2(curl0�,�))

= div((Hs−1(◦div,curl�,�)∩ ◦

B2(�)⊥�)��U−1

)

= div(Hs−1(◦div,curl�,�)∩ ◦

B2(�)⊥�)��U0

= divHs−1(◦div,�)��U

0



(ii) L2s (�) = div((Hs−1(div,�)��V

−1s−1��U

−1)∩H<1/2(

◦curl0�,�))

= div(Hs−1(div,◦

curl�,�)��U−1

)

= divHs−1(div,◦

curl�,�)��U0

= divHs−1(div,�)��U0

ACKNOWLEDGEMENTS

The author is grateful to Sebastian Bauer and Michael Trebing for many helpful discussions.

REFERENCES

1. Weyl H. Die naturlichen Randwertaufgaben im Außenraum fur Strahlungsfelder beliebiger Dimension undbeliebigen Ranges. Mathematische Zeitschrift 1952; 56:105–119.

2. Pauly D. Low frequency asymptotics for time-harmonic generalized Maxwell equations in nonsmooth exteriordomains. Advances in Mathematical Sciences and Applications 2006; 12(2):591–622.

3. Pauly D. Generalized electro-magneto statics in nonsmooth exterior domains. Analysis 2007; 27(4):425–464.4. Pauly D. Complete low frequency asymptotics for time-harmonic generalized Maxwell equations in nonsmooth

exterior domains. Asymptotic Analysis 2008; accepted.5. Picard R. Some decomposition theorems and their applications to non-linear potential theory and Hodge theory.

Mathematical Methods in the Applied Sciences 1990; 12:35–53.6. Picard R. Randwertaufgaben der verallgemeinerten Potentialtheorie. Mathematical Methods in the Applied Sciences

1981; 3:218–228.7. Picard R. On the boundary value problems of electro- and magnetostatics. Proceedings of the Royal Society of

Edinburgh 1982; 92(A):165–174.8. Picard R, Milani A. Decomposition Theorems and Their Applications to Non-linear Electro- and Magneto-

static Boundary Value Problems. Lecture Notes in Mathematics: Partial Differential Equations and Calculus ofVariations, vol. 1357. Springer: Berlin, New york, 1988; 317–340.

9. Weck N, Witsch KJ. Generalized spherical harmonics and exterior differentiation in weighted Sobolev spaces.Mathematical Methods in the Applied Sciences 1994; 17:1017–1043.

10. McOwen RC. Behavior of the Laplacian in weighted Sobolev spaces. Communications on Pure and AppliedMathematics 1979; 32:783–795.

11. Bauer S. Eine Helmholtzzerlegung gewichteter L2-Raume von q-Formen in Außengebieten des RN , Diplomarbeit,Essen, 2000. (Available from: http://www.uni-duisburg-essen.de/˜mat201.)

12. Specovius-Neugebauer M. The Helmholtz decomposition of weighted Lr -spaces. Communications in PartialDifferential Equations 1990; 15(3):273–288.

13. Pauly D. Niederfrequenzasymptotik der Maxwell–Gleichung im inhomogenen und anisotropen Außengebiet,Dissertation, Duisburg-Essen, 2003. (Available from: http://duepublico.uni-duisburg-essen.de.)

14. Courant R, Hilbert D. Methoden der Mathematischen Physik I. Springer: Berlin, 1924.15. Colton D, Kress R. Inverse Acoustic and Electromagnetic Scattering Theory (2nd edn). Springer: Berlin,

Heidelberg, New York, 1998.


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