Generalized Maxwell Equations in Exterior Domains IV: Hodge-Helmholtz Decompositions

Transcription

1 eports of the Department of Mathematical Information Technology Series B. Scientific Computing No. B. 10/007 Generalized Maxwell Euations in Exterior Domains IV: Hodge-Helmholtz Decompositions Dirk Pauly University of Jyväskylä Department of Mathematical Information Technology P.O. Box 35 (Agora) FI University of Jyväskylä FINLAND fax

2 Copyright c 007 Dirk Pauly and University of Jyväskylä ISBN ISSN X

3 Generalized Maxwell Euations in Exterior Domains IV: Hodge-Helmholtz Decompositions Dirk Pauly Abstract We study in detail Hodge-Helmholtz decompositions in nonsmooth exterior domains Ω N filled with inhomogeneous and anisotropic media. We show decompositions of alternating differential forms belonging to the weighted L - (Ω), s, into irrotational and solenoidal -forms. These decompositions are essential tools in electro-magnetic theory for exterior domains. To the best of our knowledge these decompositions in exterior domains with nonsmooth boundaries and inhomogeneous and anisotropic media are fully new results. In the appendix we translate our results to the classical framework of vector analysis N = 3 and = 1,. space L, s Key words: Hodge-Helmholtz decompositions, Maxwell euations, Electro-magnetic theory, Weighted Sobolev spaces AMS MSC-classifications: 35Q60, 58A10, 58A14, 78A5, 78A30 This research was supported by the Deutsche Forschungsgemeinschaft via the project We 394: Untersuchungen der Spektralschar verallgemeinerter Maxwell-Operatoren in unbeschränkten Gebieten and by the Department of Mathematical Information Technology of the University of Jyväskylä, where the author spent a sabbatical of three month during spring 007 as a postdoc fellow. Department of Mathematics, University of Duisburg-Essen, Campus Essen, 4517 Essen, Germany, Fon: , Fax: , dirk.pauly@uni-due.de 1

4 Contents 1 Introduction Definitions and preliminaries 6 3 esults 11 4 Proofs 16 5 Appendix A: Weighted Dirichlet forms 5 6 Appendix B: Vector fields in three dimensions Tower functions and fields esults for vector fields Introduction Hodge-Helmholtz decompositions of suare integrable fields, i.e. decompositions into irrotational and solenoidal fields, are important and strong tools for solving partial differential euations, e.g. the electro-magneto static or time-harmonic system, called the static or time-harmonic Maxwell euations. Since formally i grad and i div resp. curl and curl are adjoint to each other and curl grad = 0 and div curl = 0 holds as well, the ε-orthogonal decompositions L (Ω) = H( curl 0, Ω) ε ε 1 H(div 0, Ω) ε ε H(Ω), L (Ω) = H(curl 0, Ω) ε ε 1 (1.1) H( div 0, Ω) ε ε H(Ω), where Ω 3 is a domain, are easy conseuences of the projection theorem in Hilbert space. Here ε : Ω 3 3 is a symmetric and uniformly bounded and positive definite matrix, which models material properties, such as the dielectricity or the permeability of the medium. Moreover, and H( curl 0, Ω) = { E L (Ω) : curl E = 0 ν E Ω = 0 } H(Ω), H(div 0, Ω) = { E L (Ω) : div E = 0 } H(Ω), H(curl 0, Ω) = { E L (Ω) : curl E = 0 } H(Ω), H( div 0, Ω) = { E L (Ω) : div E = 0 ν E Ω = 0 } H(Ω) εh(ω) = { E L (Ω) : curl E = 0 div εe = 0 ν E Ω = 0 }, ε H(Ω) = { E L (Ω) : curl E = 0 div εe = 0 ν εe Ω = 0 }

5 denote the spaces of Dirichlet resp. Neumann fields, where we set H(Ω) = Id H(Ω) resp. H(Ω) = Id H(Ω). The symbol means orthogonality with respect to the L (Ω)- scalar product. (See section 6. for further definitions of all these spaces.) This problem may be generalized if we formulate Maxwell s euations in the framework of alternating differential forms of order, short -forms. This generalization not only makes things much easier, but also provides a deeper insight into the structure of the underlying problems. To remind of the electro-magnetic background it has become customary to denote the exterior derivative dby rot and the co-derivative δ = ± d by div, where denotes the Hodge star operator. Since rot and div are (formally) skew adjoint to each other as well as rot rot = 0 and div div = 0 hold, the corresponding Hodge-Helmholtz decompositions of L --forms read L, (Ω) = 0 (Ω) ε ε 1 0D (Ω) ε ε H (Ω), (1.) where Ω is a domain in N or more general a iemannian manifold of dimension N, and again are easy conseuences of the projection theorem. Here ε maps Ω to the real, linear, symmetric and uniformly bounded and positive definite transformations on -forms. Furthermore, we denote by ε the orthogonal sum with respect to the ε, L, (Ω)-scalar product, where for E, H L, (Ω) E, H L, (Ω) = E H and the bar denotes complex conjugation. Here we have 0 (Ω) = { E L, (Ω) : rot E = 0 ι E = 0 } H (Ω), 0D (Ω) = { E L, (Ω) : div E = 0 } H (Ω) and the Dirichlet forms εh (Ω) = { E L, (Ω) : rot E = 0 div εe = 0 ι E = 0 }. and the symbol means orthogonality with respect to the L, (Ω)-scalar product. (See section for more definitions.) For N = 3 and = 1 or = we obtain the two classical decompositions (1.1). In the case of unbounded domains it is often necessary and useful to work with weighted Sobolev spaces. Especially in our efforts to determine the low freuency asymptotics of the solutions of time-harmonic Maxwell euations in exterior domains [8, 9, 10] it turned out, that decompositions of weighted L -spaces are essential tools. Thus, motivated by this in the present paper we want to answer the uestion, how the weighted L -space L, s (Ω) := { F L, loc (Ω) : ρs F L, (Ω) }, s, where Ω N is an exterior domain and ρ := (1 + r ) 1/, r(x) := x, x N, may be decomposed into irrotational and solenoidal forms. 3 Ω

6 For the special case s = 0 Picard showed (1.) in [13], [11] and (in the classical framework) in [1], [6]. Moreover, for domains Ω possessing the local Maxwell compactness property MLCP (See section.) he proved the representations 0 (Ω) = rot 1 1 (Ω) = rot ( 1 1 (Ω) 0 D 1 1 (Ω) ), 0D (Ω) = div D +1 1 (Ω) = div ( D +1 1 (Ω) (Ω) ), i.e. any form from 0 (Ω) may be represented as a rotation of a solenoidal form and any form from 0 D (Ω) may be represented as a divergence of a irrotational form. Now one may expect for arbitrary s the direct decompositions L, s (Ω) = 0 s (Ω) ε 1 0D s(ω) ε H s(ω). But we will see that these only hold in a small interval around zero, i.e. 1 N/ < s < N/ 1, since for small s we lose the directness of the decomposition and for large s the right hand side is too small. For general s \ I with the discrete set of weights I = {N/ + n : n N 0 } {1 N/ n : n N 0 } Weck and Witsch [15] showed in the special case Ω = N and ε = Id, where no Dirichlet forms and no boundary exist, the decompositions 0 s + 0 D s, s (, N/) L, s = 0 s 0 D s, s ( N/, N/) 0 s 0 D s S s, s (N/, ) with 0 s = rot 1 s 1, 0D s = div D +1 s 1. Thereby S s is a finite dimensional subspace of C,( N \ {0} ), the vector space of infinitely differentiable -forms with compact supports in N \{0}, generated by the action of the commutator of the Laplacian and a cut-off function on the linear hull of some finitely many decaying potential forms in N \{0}, i.e. spherical harmonics multiplied by a negative power of r solving Laplace s euation. (Here we omit the dependence on the domain N and denote the direct sum by.) We note that Weck and Witsch even decomposed the Lebesgue-Banach spaces L p, s with p (1, ) instead of p =. The proof of their results uses heavily the corresponding results for the scalar Laplacian in N developed by McOwen in [5]. For the Hilbert space case p = these results have been generalized to smooth (C 3 ) exterior domains Ω N by Bauer in [1]. Unfortunately by their second order approach these techniues can 4

7 not be applied to handle inhomogeneities ε and the smoothness of Ω is essential as well. esults in the classical case = N 1 have been given by Specovius-Neugebauer in [14] for ε = Id and a smooth (C ) exterior domain Ω N, N 3. She considered the weaker version of (1.), which reads as resp. in the classical language L,N 1 (Ω) = 0 N 1 (Ω) div D N 1(Ω) L (Ω) = grad H 1 (grad, Ω) H( div 0, Ω). She was able to show for s \ I grad H s 1 (grad, Ω) + H s ( div 0, Ω), s (, N/) L s(ω) = grad H s 1 (grad, Ω) H s ( div 0, Ω), s ( N/, N/) grad H s 1 (grad, Ω) H s ( div 0, Ω) S s, s (N/, ), where S s corresponds to Ss N 1. We note that she proved the corresponding decompositions even for the Banach spaces L p s(ω) with 1 < p <. Since she used heavily trace operators and convolution techniues, her results can not be generalized to nonsmooth boundaries or inhomogeneities ε. Moreover, she showed no further decomposition of H s ( div 0, Ω) in 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 anisotropic media, which are only asymptotically homogeneous. To the best of our knowledge it was an open uestion, if those weighted L -decompositions hold for inhomogeneous and anisotropic media or for nonsmooth boundaries. We will allow our transformations on -forms ε to be L -perturbations of the identity, i.e. ε = Id +ˆε, where ˆε needs not to be compactly supported but decays at infinity. Moreover, ˆε is not assumed to be smooth. We only reuire ˆε C 1 outside an arbitrary large ball. Omitting some details for this introductory remarks we will show for small s (, N/) \ Ĩ L, s (Ω) = 0 s (Ω) + ε 1 0D s(ω) and for large s ( N/, ) \ Ĩ L, s (Ω) ε H (Ω) ε 0 = s(ω) ε 1 0D s(ω), s < N/ s(ω) ε 1 0D s(ω) ε η P s, s > N/ 0. 5

8 Here ε η P s is a finite dimensional subspace of H 1, s (Ω) C 1, (Ω), whose elements have supports outside of an arbitrary large ball. The forms from P s are potential forms, i.e. solve Laplace s euation in N \ {0}, and ε = rot div +ε 1 div rot. In the case ε = Id since = rot div + div rot (Here is to be understood componentwise.) we have η P s = C,η P s = S s C,( N \ {0} ), where C A,B := AB BA denotes the commutator of the operators A and B. (For details see Theorem 3..) Furthermore, L, s (Ω) decomposes for large s into the closed subspace L, s (Ω) ε H (Ω) ε and the linear hull of finitely many smooth forms, which have bounded supports. We note that for all t [ N/, N/ 1) the spaces of Dirichlet forms ε H t (Ω) coincide. Moreover, for all s \ Ĩ the irrotational forms in 0 s (Ω) resp. the solenoidal forms in 0 D s(ω) can be represented as rotations resp. divergences, i.e. 0 s(ω) = rot 1 s 1(Ω), 0D s(ω) = div D +1 s 1(Ω) hold except of some special values of s or. But contrarily to the case s = 0 for large s > 1 + N/ we lose integrability properties, if we want to represent forms in 0 s (Ω) resp. 0D s(ω) by rotations of solenoidal resp. divergences of irrotational forms. Looking at Theorem 3.5 we obtain 0 s(ω) = rot 0D s(ω) = div ( ( 1 s 1(Ω) η ( (D +1 s 1(Ω) η H 1 s 1 H +1 s 1 ) 0 D 1 < N ) 0 +1 < N ) (Ω) ) (Ω) i.e. the representing solenoidal resp. irrotational forms no longer belong to L, 1 s 1 (Ω) but only to L, 1 t (Ω) for all t < N/. (For details see Theorems 3.4, 3.5 and 3.8.) If we project onto the orthogonal complement of ε H s(ω), i.e. of more Dirichlet forms, we finally obtain even for large s > N/ L, s (Ω) ε H s(ω) ε = 0 s (Ω) ε ε 1 0D s(ω). In Appendix A we give a explicit representation of the low weighted Dirichlet spaces εh s(ω) and compute their finite dimensions. Definitions and preliminaries We consider an exterior domain Ω N, i.e. N \ Ω is compact, as a special smooth iemannian manifold of dimension 3 N N, and fix a radius r 0 and some radii 6,,

9 r n := n r 0, n N, such that N \ Ω is a compact subset of the open ball around zero U r0 := { x N : x < r 0 }. Moreover, we choose a cut-off function η, such that [4, (3.1), (3.), (3.3)] hold. Putting A r := { x N : x > r } we note that supp η A r1 U r. Throughout this paper we will use the notations from [4, 8, 9]. Considering alternating differential forms of rank Z (short -forms) we denote the exterior derivative dby rot and the co-derivative δ = ( 1) ( 1)N d ( : Hodge star operator) by div, where both of them are applied to -forms, to remind of the electro-magnetic background. On C, (Ω) ( the vector space of all C --forms with compact supports in Ω ) we have a scalar product Φ, Ψ L, (Ω) := Φ Ψ Φ, Ψ C, (Ω) Ω and hence an induced norm L, (Ω) :=, 1/ L, (Ω). Thus, we may define (taking the closure in the latter norm) L, (Ω) := C, (Ω), the Hilbert space of all suare integrable -forms on Ω. Moreover, on C, (Ω) the linear operators rot and div are formally skew adjoint to each other due to Stokes theorem, i.e. rot Φ, Ψ L,+1 (Ω) = Φ, div Ψ L, (Ω) for all (Φ, Ψ) C, (Ω) C,+1 (Ω), which gives rise to weak formulations of rot and div. Using these and the weight function ρ(r) := (1 + r ) 1/ for s we introduce the following weighted Hilbert spaces (endowed with their natural norms) of -forms L, s (Ω) := { E L, loc (Ω) : ρs E L, (Ω) }, s(ω) := { E L, s (Ω) : rot E L,+1 s+1 (Ω) }, D s(ω) := { H L, s (Ω) : div H L, 1 s+1 (Ω) }. ( ) All these spaces eual zero if / {0,..., N}. Furthermore, we introduce the Hilbert space s(ω) := C, (Ω), ( The closure is taken in s (Ω). ) which generalizes the boundary condition of vanishing tangential components of the forms at the boundary Ω. More precisely this generalizes the boundary condition ι E = 0, which means that the pull-back of E on the boundary of Ω ( considered as a (N 1)-dimensional iemannian submanifold of Ω ) vanishes. Here ι : Ω Ω denotes the natural embedding. A lower left corner index 0 indicates vanishing rotation resp. divergence. 7

10 For weighted Sobolev spaces V s, s, we define V <t := s<t V s. We only consider exterior domains Ω, which possess the local Maxwell compactness property MLCP, i.e. for all and all t < s the embeddings s(ω) D s(ω) L, t (Ω) are compact. ( See [4, Definition 3.1, emark 3.] and the literature cited there. ) We assume our transformations to be τ-admissible resp. τ-c 1 -admissible as defined in [8, Definition.1 and.]. Let ε be a τ-c 1 -admissible transformation on -forms with some τ > 0. We need the finite dimensional vector space of weighted Dirichlet forms εh t (Ω) = 0 t (Ω) ε 1 0D t (Ω), t. (Here we neglect the indices ε or t in the cases ε = Id or t = 0.) Citing [9, Lemma 3.8] we have εh (Ω) = N ε H (Ω) = ε H < N 1(Ω) and even ε H (Ω) = ε H (Ω) if / {1, N 1}. Thus, < N εh (Ω) L, s(ω) for s > 1 N/ and even for s > N/ if / {1, N 1}. Furthermore, for s > 1 N/ we introduce the (then well defined) Hilbert spaces 0D s(ω) = 0 D s(ω) ε H (Ω), 0 s(ω) = 0 s (Ω) ε H (Ω) ε, (.1) where we denote by ε the orthogonality with respect to the ε, L, (Ω)-scalar product, i.e. the duality between L, t (Ω) and L, t (Ω), t. If ε = Id we simply write := Id. The restrictions upon the weights s guarantee ε H (Ω) L, s(ω). But if / {1, N 1} also ε H (Ω) L, (Ω) holds and these definitions extend to < N s > N/. Since there are no Dirichlet forms (with t = 0!) for {0, N} in these special cases the definitions (.1) may be extended to all s and we have 0 0 s(ω) = 0 0 s (Ω) = {0}, 0 0D 0 s(ω) = 0 D 0 s(ω) = L,0 s (Ω), 0D N s (Ω) = 0 D N s (Ω) = N s (Ω) = 0 N s (Ω) = L,N s (Ω), { {0}, s N/ Lin{ 1}, s < N/ Moreover, there are some other characterizations of these spaces. We remind of the finitely many special smooth forms B (Ω) 0 (Ω) and B (Ω) 0 D (Ω) 8.

11 presented in [9, section 4], which have compact resp. bounded supports in Ω and the properties We note in passing εh (Ω) B (Ω) ε = ε H (Ω) B (Ω) = {0}. (.) dim ε H (Ω) = dim H (Ω) = # B (Ω) = # B (Ω) = d N 0. Using [9, Corollary 4.4] we see in fact that 0D s(ω) = 0 D s(ω) H (Ω) = 0 D s(ω) B (Ω), 0 s(ω) = 0 s (Ω) H (Ω) = 0 s (Ω) B (Ω) do not depend on the transformation ε. Since B (Ω) is only defined for 1 the last characterization in the second euation holds only for 1. Now the definitions of 0D s(ω) and 0 s (Ω) extend to arbitrary weights s because the forms B (Ω), B (Ω) have bounded supports. We say Ω possesses the SMP, if and only if Ω has the MLCP and the forms B (Ω), B (Ω) exist, which is guaranteed for Lipschitz domains Ω, for instance. ( See [9, section 4] and the literature cited there. ) We may choose r 0, such that supp η supp b = for all b B (Ω) B (Ω) and all. Finally for s > 1 N/ or s > N/ and / {1, N 1} we put εl, s (Ω) := L, s (Ω) ε H (Ω) ε. We also need the negative tower forms D,l σ,m,,l σ,m for the values l = 0, 1, and σ N 0, m {1,..., µ σ} from [9, section ], which are essentially harmonic polynomials except of a multiplication by some negative integer power of r. These forms are homogeneous of degree h l σ := l σ N, belong to C,( N \ {0} ) and satisfy rot Dσ,m,0 = 0, div σ,m +1,0 = 0, div Dσ,m,l = 0, rot σ,m +1,l = 0, rot Dσ,m,k = σ,m +1,k 1, div σ,m +1,k = Dσ,m,k 1, where l = 0, 1, and k = 1,. (We note briefly that we need the positive tower forms of height zero + Dσ,m,0, + σ,m,0 in our proofs as well. But they are not reuired to formulate our results.) From [9, emark.5] we have for all σ N 0, m {1,..., µ σ} and all l = 0, 1, as well as k N 0 D,l σ,m L, s (A 1 ) D,l σ,m H k, s (A 1 ) s < N/ + σ l, 9

12 which completely determines the integrability properties of our tower forms at infinity. The same integrability holds true for σ,m,l. Moreover, the ground forms (forms of height 0), which only occur for 1 N 1, are linearly dependent, i.e. we have ασ,0 σ,m + i ασ D,0 σ,m = 0, (.3) where α σ := ( + σ) 1/ and := N. This motivates to define the harmonic tower forms H σ,m := α σ and moreover the potential tower forms P σ,m := α σ,0 σ,m = i ασ D,0 σ,m (.4), σ,m + i ασ D, σ,m, (.5) since the tower forms of height 0 and 1 (and then clearly the harmonic tower forms as well) and then also the potential tower forms P σ,m (of height ) satisfy D,l σ,m =,l σ,m = H σ,m = P σ,m = 0, l = 0, 1, because = rot div + div rot. Here denotes the componentwise scalar Laplacian in Euclidean coordinates. We note that P σ,m = 0 if {0, N}. Furthermore, for s and l = 0, 1, we introduce the finite dimensional vector spaces D,l s := Lin { Dσ,m,l : Dσ,m,l / L, s (A 1 ) } = Lin{ Dσ,m,l : σ s N/ + l},,l s := Lin { σ,m,l : σ,m,l / L, s (A 1 ) } = Lin{ σ,m,l : σ s N/ + l}, H s := Lin { Hσ,m : Hσ,m / L, s (A 1 ) } = Lin{Hσ,m : σ s N/}, P s := Lin { Pσ,m : Pσ,m / L, s (A 1 ) } = Lin{Pσ,m : σ s N/ + }. ( ) Here we set Lin := {0}. We note H s = P s =,l,l D s l = s l = {0} s < N/. Unfortunately due to the fact that rot r N (the gradient of r N in classical terms) is irrotational and solenoidal but is itself no divergence, there exist four exceptional tower forms. These are (up to constants) ˇP 0 := r N = D 0, 0,1, Ȟ 1 := rot ˇP 0 = r 1 N dr = 1,1 0,1 and their duals ˇP N := ˇP 0 = N, 0,1, Ȟ N 1 := Ȟ1 = D N 1,1 0,1. Of course we have div ˇP N = ȞN 1. Following the construction of the regular tower forms we define for s N/ ˇP 0 := ˇP 0 s := Lin{ ˇP 0 }, ˇPN := ˇP N s := Lin{ ˇP N } and for s N/ 1 Ȟ 1 := Ȟ1 s := Lin{Ȟ1 }, Ȟ N 1 := ȞN 1 s := Lin{ȞN 1 }. 10

13 For all other values of s and we put ˇP s := {0} and Ȟ s := {0}. As described, for instance, in [9, section 3] for s we consider vector spaces V s ηv s, ( : direct sum) where Vs L, s (Ω) is some Hilbert space and V s is some finite subset of the tower forms, e.g. Vs = s(ω) D s(ω) and V s = H s. On Vs ηv s we define a scalar product, such that in V s the original scalar product is kept, ηv s is an orthonormal system, the sum V s ηv s = V s ηv s is orthogonal. As already indicated we denote the orthogonal sum with respect to this new inner product by and clearly V s ηv s is a Hilbert space since V s is finite. 3 esults Let Ω N (N 3) be an exterior domain as introduced in the last section with the SMP or the MLCP depending on whether the forms B (Ω), B (Ω) are involved in our considerations or not. ecalling from [9, section 3] the set of special weights I we put Ĩ := I 1 = {N/ + n 1 : n N 0 } { N/ n : n N 0 } and from now on we make the following general assumptions: s \ Ĩ, i.e. s + 1 \ I, i.e. for all n N 0 s n + N/ 1 and s n N/. ε : τ-c 1 -admissible transformation on -forms with some τ = τ s+1 satisfying τ > max{0, s + 1 N/} and τ s 1, i.e. τ s 1, s (, 1) > 0, s [ 1, N/ 1] > s + 1 N/, s (N/ 1, ). ν, µ : τ-c 1 -admissible transformation on ( 1)- resp ( + 1)-forms with some τ = τ s satisfying τ > max{0, s N/} and τ s, 11

14 i.e. τ s, s (, 0) > 0, s [0, N/] > s N/, s (N/, ). If 1 N/ < s < N/ 1 or N/ < s < N/ and / {1, N 1} the trivial orthogonal decomposition L, s (Ω) = ε L, s (Ω) ε ε H (Ω) (3.1) holds. We will denote the orthogonality with respect to the ε, L, (Ω)-scalar product or L, s (Ω)-L, s(ω)-duality by ε and put = Id. The first lemma shows how one may get rid of Dirichlet forms even for larger weights. Lemma 3.1 Let s > 1 N/. Then the direct decompositions L, s (Ω) = ε L, s (Ω) Lin B (Ω), L, s (Ω) = ε L, s (Ω) ε 1 Lin B (Ω) hold, where the latter is only defined for 1. If / {1, N 1} this decompositions hold for s > N/ as well. To formulate our main decomposition result we need the operator (a perturbation of the Laplacian ) ε := rot div +ε 1 div rot = + ˇε div rot, where ε 1 =: Id +ˇε is also τ-c 1 -admissible. We obtain Theorem 3. The following decompositions hold: (i) If s < N/, then L, s (Ω) = 0 s (Ω) + ε 1 0D s(ω) and the intersection euals the finite dimensional space of Dirichlet forms ε H s(ω). Moreover, and for 1 even L, s (Ω) = 0 s (Ω) + ε 1 0D s(ω) L, s (Ω) = 0 s (Ω) + ε 1 0D s(ω). In both cases the intersection euals the finite dimensional space of Dirichlet forms εh s(ω) B (Ω) ε. 1

15 (ii) If N/ < s 1 N/, then L,1 s (Ω) = 0 1 s (Ω) ε 1 0D 1 s(ω), L,N 1 s (Ω) = 0 N 1 s (Ω) ε 1 0D N 1 s (Ω) ε H N 1 (Ω). (iii) If 1 N/ < s < N/ or N/ < s 1 N/ and / {1, N 1}, then εl, s (Ω) = 0 s (Ω) ε 1 0D s(ω). For s 0 this decomposition is even ε, L, (Ω)-orthogonal. (iv) If s > N/, then and εl, s (Ω) = ( ([L, s (Ω) η H s] 0 (Ω) ) < N ε ε 1( [L, s (Ω) η H s] 0 D < N εl, s (Ω) = 0 s (Ω) ε 1 0D s(ω) ε η P s, (Ω) )) L, s (Ω) where the first two terms in the second decomposition are ε, L, (Ω)-orthogonal as well. Furthermore, emark 3.3 L, s (Ω) ε H s(ω) ε = 0 s (Ω) ε ε 1 0D s(ω). The decompositions in (ii)-(iv) are direct and define continuous projections. In (ii) we are forced to use the forms B (Ω) and B (Ω) in the definitions of 0 s (Ω) and 0 D s(ω). To prove the last euation in (iv), we additionally assume τ N/ 1. The coefficients of the tower forms in the first euation of (iv) are related in the following way: If F r,s + l h r,l ηh l + ε 1( F d,s + l ) h d,l ηh l = F ε L, s (Ω) with F r,s s(ω), F d,s D s(ω) and H l H s as well as h r,l, h d,l C, then h r,l + h d,l = 0, since the H l are linear independent and do not belong to L, s (Ω). 13

16 ε η P s is a finite dimensional subspace of H 1, s (Ω) C 1, (Ω), whose elements have supports in A r1. For s < N/ (and τ N/ 1) we have εh s(ω) = ε H (Ω) ε H s(ω) B (Ω) ε. Clearly the transformation ε may be moved to the irrotational terms in our decompositions as well. Our decompositions and representations may be refined. For small weights we get Theorem 3.4 Let s < N/+1 δ,0 δ,n. Then 0 D s(ω) and 0 s (Ω) are closed subspaces of L, s (Ω) whenever they exist and (i) 0 s(ω) = rot ( 1 s 1(Ω) ν 1 0D 1 s 1(Ω) ) = rot ( 1 s 1(Ω) ν 1 0D 1 s 1(Ω) ) = rot 1 s 1(Ω) holds for N as well as for = 1 and s > 1 N/, (ii) 0D s(ω) = div ( D +1 s 1(Ω) µ s 1(Ω) ) holds for 1 N 1 as well as for = 0 and s > N/, (iii) 0D s(ω) = div ( D +1 s 1(Ω) µ holds for 0 N 1. s 1(Ω) ) = div D +1 s 1(Ω) 14

17 For large weights we have Theorem 3.5 Let 1 N 1. Then for s > N/ + 1 ( ( ) ) (i) s(ω) = rot s 1(Ω) η H s 1 ν 1 0 D 1 (Ω) < N (ii) = rot ( 1 s 1(Ω) ν 1 D 1 s 1(Ω) B 1 (Ω) ν) = rot 1 s 1(Ω), ( (D ) 0D ) s(ω) = div s 1(Ω) η H s 1 µ (Ω) < N = div ( D +1 s 1(Ω) µ 1 +1 s 1(Ω) B +1 (Ω) µ) = div D +1 s 1(Ω) are closed subspaces of L, s (Ω) and for s > N/ (iii) ( L, s (Ω) η H s ) 0 (Ω) < N ( ( 1 1,0 = rot s 1(Ω) η D s 1 η = 0 s (Ω) rot ν 1 1,1 η D s 1, ) ) 1,1 D s 1 ν 1 0 D 1 < N 1(Ω) (iv) are closed subspaces of L, s (Ω) η H s. emark 3.6 We note div η D 1,1 σ,m 1,1 η D s 1 ( L, s (Ω) η H s) 0 D (Ω) < N ( (D ) ) +1 +1,0 +1,1 = div s 1(Ω) η s 1 η s 1 µ < N 1(Ω) = 0 D s(ω) div µ 1 +1,1 η s 1 = 0 and rot η +1,1 σ,m = 0 by [9, emark.4] and thus 0 D 1 +1,1 < N 1(Ω), η s < N 1(Ω). emark 3.7 Since there are no regular harmonic tower forms in the cases {0, N}, i.e. H s 0 = {0}, HN s = {0}, and because of η H s L, (Ω) the first euations in (iii) and (iv) < N simplify: If s > N/, then ( L,1 s ( L,N 1 s (Ω) η (Ω) η H ) s (Ω) = rot ( ) 0 0,1 < s 1(Ω) η D N s 1 ) 0 D N 1 (Ω) = div ( ) D N s 1(Ω) η H N 1 s < N N,1 s 1,. If N/ < s < N/ + 1, then ( L, s (Ω) η H ) s 0 (Ω) = rot < N ( L, s (Ω) η H s) 0 D < N (Ω) = div ( ( 1 s 1(Ω) η ( (D +1 s 1(Ω) η ) 1,1 D s 1 ν 1 0 D 1 ) µ ,1 s 1 < N 1(Ω) ) < N 1(Ω) ),. 15

18 As mentioned above there are no harmonic tower forms in the remaining cases {0, N}. Thus, the euations 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 3.8 Let s > N/. Then (i) L,N s (Ω) = 0 N s (Ω) ( ( N 1 = rot s 1 (Ω) η = rot H N 1 s 1 ηȟn 1) ν 1 0D N 1 < N 1(Ω) ) ( ( N 1 s 1 (Ω) ν 1 D N 1 s 1 (Ω) B N 1 (Ω) ν) ηȟn 1 ) = rot ( N 1 s 1 (Ω) ν 1 D N 1 s 1 (Ω) B N 1 (Ω) ν) ηˇp N = rot N 1 s 1 (Ω) ηˇp N, (ii) L,0 s (Ω) = 0 D 0 s(ω) ( (D 1 = div s 1(Ω) η H s 1 1 ηȟ1) µ 1 ) 0 1 < N 1(Ω) ( (D 1 = div s 1(Ω) µ 1 1 s 1(Ω) ) ) ηȟ1 = div ( D 1 s 1(Ω) µ 1 1 s 1(Ω) ) ηˇp 0 = div D 1 s 1(Ω) ηˇp 0. Finally we want to note that we always get dual results using the Hodge star operator. (This will change the homogeneous boundary condition from the electric to the magnetic one.) Since this will multiply the number of results by two we let their formulation to the interested reader. 4 Proofs Let ε, ν and µ be as in section 3. We start with the Proof of Lemma 3.1: Let E L, s (Ω). Looking at [9, section 4] and using the Helmholtz decompositions [4, (3.5)] we may choose b l Lin B (Ω), l = 1,..., d, with b l = Φ l +H l rot 1 (Ω) ε ε H (Ω), where {H l } is a ε -ONB of ε H (Ω). Then e := E l E, εh l L, (Ω)b l ε L, s (Ω). This proves one inclusion and the other one is trivial, because the forms of B (Ω) are smooth and compactly supported. Moreover, if E Lin B (Ω) ε H (Ω) ε, then εe = l e lεb l ε H (Ω) and thus 0 = εe, H k L, (Ω) = l e l εh l, H k L, (Ω) = e k, 16

19 which proves the directness of the sum. The other direct decomposition may be shown in a similar way. We introduce the Hilbert spaces ( closed subspaces of L, s (Ω) ) s (Ω) := { E e s(ω) : rot(ρ s E) = 0 } = { E s(ω) : rot E = sρ E }, D e s (Ω) := { E D s(ω) : div(ρ s E) = 0 } = { E D s(ω) : div E = sρ T E } and note that the s (Ω) -, s (Ω) - and L, s (Ω) -norms resp. the D s (Ω) -, D s (Ω) - and L, s (Ω) -norms are euivalent on s (Ω) resp. D s (Ω). Here we e e used the operators, T from [4, (.0) and (.1)] and the commutator formulas [4, (.4)]. First we need an easy conseuence of the projection theorem: Lemma 4.1 Let s. Then the orthogonal decompositions (i) L, s (Ω) = rot 1 s 1(Ω) s,ε ε 1 D s (Ω) = ε 1 rot 1 e s 1(Ω) s,ε D s (Ω), e (ii) L, s (Ω) = div D +1 s 1(Ω) s,ε ε 1 e s (Ω) = ε 1 div D +1 s 1(Ω) s,ε e s (Ω) hold with continuous projections. Here we denote by s,ε the orthogonal sum with respect to the ερ s, L, (Ω)-scalar product and the closures are taken in L, s (Ω). The space rot 1 s 1(Ω) resp. div D +1 s 1(Ω) may be replaced by rot C, 1 (Ω) resp. div D +1 vox (Ω). Proof: Since C, 1 (Ω) is dense in 1 s 1(Ω) we have E L, s (Ω) ( rot 1 s 1(Ω) ) s,ε, if and only if E L, s (Ω) ( rot C, 1 (Ω) ) s,ε, which means ρ s εe 0 D s(ω). Thus, E D e s (Ω), because 0 = div(ρ s εe) = ρ s div εe + sρ s T εe. This shows (i) and (ii) follows analogously. emark 4. Clearly 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 we have εx t (Ω) := ( t (Ω) ε 1 D t (Ω) ) η H t ηȟ t, t, 17

20 Lemma 4.3 Let s \ Ĩ and 0 or = 0 and s > N/. Then εot s : εx s(ω) ε 1 0D loc (Ω) 0 +1 s+1(ω) E rot E εdiv s : εx s(ω) 0 loc (Ω) 0D 1 s+1(ω) H div εh are continuous and surjective Fredholm operators with kernels { N( ε OT s) = N( ε DIV εh s) = s(ω), if s < N/ εh (Ω), if s > N/,. emark 4.4 By adding suitable Dirichlet forms from ε H (Ω) we always may obtain the constraints H B (Ω), 1, and εe B (Ω) or εh, εe ε H (Ω), if s > N/. Then for s > N/ the operators εot s : ε X s(ω) ε 1 0D loc (Ω) 0 +1 s+1(ω), εdiv s : ε X s(ω) 0 loc (Ω) 0D 1 s+1(ω) are also injective and hence topological isomorphisms by the bounded inverse theorem. Furthermore, we note ( using the notations from [9] ) H t = D (Ī,0 t ) = ( J,0 t ), Ȟ t = Ď,1 t = Ř,1 t and for t < N/ H t = Ȟ t 1 = {0}. Proof: This lemma has been proved in [9, Corollary 3.13, Lemma 3.14] for s > N/. So we only have to discuss the cases of small weights s < N/ and ranks of forms 1 N. Welldefinedness, continuity and the assertion about the kernels are trivial in these cases. So for example to show surjectivity for ε DIV s, let us pick some F 0 D 1 s+1(ω). Following the proofs of [9, Lemma 3.5, Lemma 3.1] and using [15, Theorem 4] we represent the extension by zero of F to N ˆF =: F D + F 0 D 1 s+1( N ) s+1( N ). Now F D is contained in the range of the operator B from [15, Theorem 7] and thus we get some h D s( N ) 0 s( N ) solving div h = F D. Applying [4, Theorem 3.6 (ii)] we even have h H 1, s ( N ). Since F 0 D 1 s+1(ω) 0 1 s+1(ω) 18

21 we may represent this form in terms of a spherical harmonics expansion F Ar0 = f I D 1 I + f ˆD 1,1 + I Ī 1,0 I s + I 1,0 f I D 1 I with uniuely determined f I, f C using [9, Theorem.6], where s+ I 1,0 := { I I 1,0 : s(i) = + e(i) < s 1 N/ }. Now looking at [9, emark.5] the first term of the sum on the right hand side belongs to L, 1 (A < N r0 ), the second to L, 1 < N 1(A r 0 ) and the third to L, 1 s+1 (A r0 ). Therefore, F f I ηd 1 I L, 1 ( N ). I s + I 1,0 This suggests the ansatz H := ηh + f I η 1I + Φ I s + I 1,0 to solve H 0 s (Ω) ε 1 D s(ω) and div εh = F. We note that the middle term of the right hand side of the ansatz belongs to s(ω) and even to 0 s (Ω), for instance. Thus, we are searching for some Φ s(ω) ε 1 D s(ω) satisfying rot Φ = rot(ηh) = C rot,η h =: G, div εφ = F div(ηεh) f I div(ηε 1I ) =: F, (4.1) I s + I 1,0 since rot(η 1I ) = 0. Clearly we have G 0 +1 vox (Ω) 0 +1 (Ω). Moreover, not only F 0 Ds+1(Ω) 1, but also F 0 D 1 (Ω) holds, because F = div(1 η)h f I C div,η 1I + F f I ηd 1 I div (ˆεη(h + I s + I 1,0 I s + I 1,0 f I η 1I )), I s + I 1,0 where the first two terms of the sum on the right hand side lie in Lvox, 1 (Ω), the sum of the third and fourth terms in L, 1 (Ω) and the last one in L, 1 s+1+τ(ω) L, 1 (Ω), since η(h + I s + I 1,0 f I η 1I ) H1, s ( N ) 19

22 and τ s 1. Now we are able to apply [4, Theorem 3.1] and get some Φ 1(Ω) ε 1 D 1(Ω) solving the system (4.1). Since s < N/ < 3/ we have Φ s(ω) ε 1 D s(ω), which completes the proof. The assertion about ε OT s follows analogously. Proof of Theorem 3.4: Apply Lemma 4.3 and emark 4.4 with the modified values of, s and ε. Proof of Theorem 3.5: The first euations in (i)-(iv) have already been proved in [9, Theorem 5.8]. Let ( G 0 ( 1 s (Ω) = rot s 1(Ω) η H 1 s 1 ) ν 1 0 D 1 < N ) (Ω), i.e. G = rot G s 1 + I Ī 1,0 s 1 g I rot ηd 1 I with G s 1 1 s 1(Ω) ν 1 D 1 s 1(Ω) B 1 (Ω) ν and g I C. Since and clearly we get rot ηd 1 I rot η 1 I = rot ηd 1 I = rot div C rot,η 1 I rot C div rot,η D 1 I i α e(i) rot ηd 1 I = rot ηp 1 e(i),c(i) i α e(i) rot C div rot,ηd 1 I α e(i) rot div C rot,η 1. I Now ηp 1 e(i),c(i) = C,ηP 1 e(i),c(i) has compact support and therefore G rot ( 1 s 1(Ω) ν 1 D 1 s 1(Ω) B 1 (Ω) ν), which proves (i). (ii) is shown analogously. The last euation in (iii) follows from (i), since we can split off a term η D 1,1 σ,m = ν 1 νη D 1,1 σ,m = ν 1 η D 1,1 σ,m + ν 1ˆνη D 1,1 σ,m and the tower forms are smooth and ˆν decays as well as div η Dσ,m 1,1 = 0 holds by emark 3.6. Finally the last euation in (iv) is a direct conseuence of (ii) and a similar argument like the latter one. 0

23 Proof of Theorem 3.8: Lemma 4.3 yields ( ( ) L,N N 1 N 1 s (Ω) = rot s 1 (Ω) η H s 1 ηȟn 1) ν 1 0D N 1 < N 1(Ω) and the same arguments used in the latter proof show ( ( ) ) N 1 N 1 rot s 1 (Ω) η H s 1 ν 1 0 D N 1 < N 1(Ω) = rot ( N 1 s 1 (Ω) ν 1 D N 1 s 1 (Ω) B N 1 (Ω) ν). Because div ηȟn 1 = 0 we have ( ( L,N N 1 s (Ω) = rot s 1 (Ω) ν 1 D N 1 s 1 (Ω) B N 1 (Ω) ) ) ηȟn 1. Now div ˇP N = ȞN 1 and thus rot ηȟn 1 = η ˇP N rot C div,η ˇP N, which proves L,N s (Ω) = rot ( s 1 N 1 (Ω) ν 1 D N 1 s 1 (Ω) B N 1 (Ω) ν) ηˇp N. Finally we have to show that the sum is direct. To do this, let F = f η ˇP N = rot E with some f C and E N 1 s 1 (Ω). We want to use the notations from [15], i.e. the forms P N,4 0,1 and Q N,4 0,1, as well. By definition there exists a constant c 0, such that ˇP N = c N, 0,1 = c N QN,4 0,1 using [9, emark.3]. Since s > N/ and P N,4 0,1 L,N (Ω) < N partial integration yields rot E, P N,4 0,1 L, (Ω) = 0, because P N,4 0,1 Lin{ 1} is constant. But on the other hand we obtain F, P N,4 0,1 L, (Ω) = f η ˇP N, P N,4 0,1 L, (Ω) = c f N C,ηQ N,4 0,1, P N,4 0,1 L, (Ω) = c f by [15, (73)], i.e. f = 0. Now we turn to the main idea of our decompositions and the Proof of Theorem 3.: Let 1 N 1 and s be as in section 3 as well as Using Lemma 4.1 we decompose F L, s (Ω). F = F r + ε 1 ˆFd with F r rot C, 1 (Ω) and ˆF d D e s (Ω). A second application of this lemma yields the decomposition ε 1 ˆFd = ε 1 F d + F 1

24 with F d div D +1 vox (Ω) and F s(ω) ε 1 D s(ω). Furthermore, there exists a constant c > 0 independent of F, such that F r L, s (Ω) + F d L, s (Ω) + F s (Ω) ε 1 D s(ω) c F L, s (Ω). Now F is more regular than F and this enables us to solve div εh = div ε F 0 D 1 s+1(ω), rot E = rot F 0 +1 s+1(ω) with some H ε X s(ω) 0 loc (Ω) and E εx s(ω) ε 1 0D loc (Ω) by Lemma 4.3. We note E, H L, t (Ω) for all t with t s and t < N/ 1. Now ˆF := F E H ε H t (Ω) and F = F r + H + ε 1 (F d + εe) + ˆF, (4.) where F r + H 0 t (Ω) and F d + εe 0 D t (Ω). emark 4.5 This short and simple argument is the main idea. The rest is just refinement utilizing the result from [9]. We note that the idea of using the more regular F replaces the old idea of working in H 1 (Ω), the dual of H 1 (Ω), in the case of the classical Helmholtz decomposition. For s > N/ we may refine the representation (4.) of F. For these s we have ˆF ε H (Ω) = > N ε H (Ω) by [9, Lemma 3.8]. Using emark 4.4 or [9, Theorem 5.1] additionally we may obtain εe B (Ω) and H B (Ω), if 1, or εe ε H (Ω) and εh ε H (Ω), if s > 1 N/. Therefore, F d + εe 0 D t (Ω) holds for s > N/ and F r + H 0 t (Ω) for s > 1 N/ or 1 N/ > s > N/ and 1. Moreover, [9, Theorem 5.1] yields not only E, H ε X s(ω) but also E ( s(ω) ε 1 D s(ω) ) ηd (Ī,0 s ), if N 1, H ( s(ω) ε 1 D s(ω) ) η ( J,0 s ), if 1, i.e. the exceptional forms do not appear in these cases. The stronger assumption on F ε L, s (Ω) for weights s > 1 N/ or s > N/ and N implies ˆF ε H (Ω) ε and thus ˆF = 0. So in these cases (4.) turns to F = F r + H + ε 1 (F d + εe). (4.3) Until now we have shown the assertions of Theorem 3. (i), (ii) and also (iii) for s < N/ 1. Considering larger weights s > N/ 1 and F ε L, s (Ω) the tower forms occur in the representation (4.3). More precisely we have

25 E = E s + I Ī,0 s H = H s + J J,0 s e I ηd I + eη { D N 1,1 0,1, = N 1 0, otherwise h J η J + hη { 1,1 0,1, = 1 0, otherwise, with uniuely determined E s, H s s(ω) ε 1 D s(ω) and e I, e, h J, h C. We note Ī,0,0 s = J s. For s < N/ we have Ī,0 s = and for s > N/ we see for all I Ī,0 s α e(i) η I = i α e(i) ηd I L, (Ω) < N by (.3). Thus, we obtain F = F r + H s + ε 1 F d + E s + h 1,1 0,1, = 1 (h I ẽ I )η I + η e D N 1,1 0,1, = N 1 I Ī,0 s 0, otherwise, where ẽ I := i e I α e(i) /α e(i). Now looking, for example, at the case = 1 we see η 1,1 0,1 / L, N 1(Ω). Then for integrability reasons we get h = 0, such that the exceptional tower form does not appear. Clearly also e = 0 holds true for = N 1. Moreover, h I = ẽ I since I are linear independent and η I / L, s (Ω) for all I Ī,0 s and s > N/. By the smoothness of ηd I as well as the decay and differentiability properties of ˆε we obtain furthermore ˆεηD I H1, s (Ω). Thus, for all s > 1 N/ we get the representation F = F r + h I η I + ε 1( Fd + ) e I ηd I, I Ī,0 s where F r := F r + H s and F d := F d + εe s + I Ī,0 s e I ˆεηD I as well as F r + I Ī,0 s F d + I Ī,0 s I Ī,0 s h I η I ( L, s (Ω) η H ) s 0 (Ω), < N e I ηd I ( L, s (Ω) η H s) 0 D (Ω) < N with α e(i) h I + i α e(i) e I = 0 for all I Ī,0 s. 3

26 This proves the remaining assertions of (iii) and the first euation in (iv). To show the second euation in (iv) we observe and therefore where ηd I = η div rot D I = div rot ηd I C div rot,ηd I, η I = η rot div I = rot div η I C rot div,η I F r := F r F = Fr + ε 1 Fd + I Ī,0 s F d := F d I Ī,0 s h I C rot div,η I I Ī,0 s e I C div rot,η D I h I α e(i) ε ηp e(i),c(i), e I rot C div,η D I 0 s(ω), h I div C rot,η I 0D s(ω), I Ī,0 s I Ī,0 s h I I Ī,0 s α e(i) ε ηp e(i),c(i) εη P s. Clearly all sums are direct resp. orthogonal as stated. Only in the second euation of (iv) one may see this not directly. So for example if E = I Ī,0 s e I ε ηp e(i),c(i) = G + ε 1 F 0 s (Ω) ε 1 0D s(ω) with some s > N/, then H := G e I rot div ηp e(i),c(i) = e I ε 1 div rot ηp e(i),c(i) ε 1 F I Ī,0 s I Ī,0 s is a Dirichlet form, i.e. H ε H (Ω), but also an element of B (Ω) ε. Hence, H must vanish and thus G = e I rot div ηp e(i),c(i) L, s (Ω), I Ī,0 s which is only possible, if e I = 0 for all I Ī,0 s, since rot div ηp e(i),c(i) / L, s (Ω) are linear independent. It remains to prove the last euation of (iv). Before we start with this, we observe, that by the closed graph theorem all projections in (ii)-(iv) are continuous. To show the last euation of (iv) we note 0 s(ω), ε 1 0D s(ω) L, s (Ω) ε H s(ω) ε =: Y s(ω) 4

27 and thus X s(ω) := 0 s (Ω) ε ε 1 0D s(ω) Y s(ω). Furthermore, X s(ω) and Y s(ω) are closed subsets of L, s (Ω). By the first euation of (iv) and Lemma 3.1 we have codim X s(ω) = dim ε H (Ω) + dim ε η P s = d + µ σ, 0 σ<s N since dim ε η P s = dim P s = dim H s and s N/ / N 0 because s / Ĩ. With codim Y s(ω) = dim ε H s(ω) we get by Appendix A that X s(ω) and Y s(ω) possess the same finite codimension in L, s (Ω). Conseuently we obtain X s(ω) = Y s(ω). 5 Appendix A: Weighted Dirichlet forms Let τ > 0. As already mentioned in section for the space of Dirichlet forms we have εh (Ω) = N ε H (Ω) = ε H<t(Ω), t := N/ δ,1 δ,n 1, and its dimension euals d = β, the -th Betti number of Ω. Furthermore, we have for all t εht 0 (Ω) = {0}, εht N (Ω) = { {0}, t N/ ε 1 Lin{ 1}, t < N/. (This holds even for τ = 0.) We repeat some notations and results from [7], [9] and [15]. Let us introduce the special growing Dirichlet forms E + σ,m from [10, Lemma.4 and emark.5] as the uniue solutions of the problems E σ,m + ε H < N σ(ω) B (Ω) ε, E σ,m + + Dσ,m,0 L, (Ω), > N where σ N 0 and 1 m µ σ with ( )( ) N N 1 + σ µ (N + σ) σ = σ N( + σ)( + σ) from [15, Theorem 1 (iii)]. To guarantee their existence we have to impose the decay conditions τ > σ and τ N/ 1. We note µ 0 = ( ) N and thus µ 0 0 = µ N 0 = 1. Moreover, + D 0,0 0,1 resp. + D N,0 0,1 is a multiple of 1 resp. 1. 5

28 Lemma 5.1 Let 1 N 1 and s (, N/) \ Ĩ as well as τ > s N/ and τ N/ 1. Then holds. Moreover, εh s(ω) = ε H (Ω) ε H s(ω) B (Ω) ε εh s(ω) B (Ω) ε = Lin{E σ,m : σ < s N/}. Corollary 5. The dimension d s of ε Hs(Ω) is finite and independent of ε. More precisely d s = d + µ σ. Furthermore, the mapping σ< s N d ( ) : (, N/) \ Ĩ ( N/, N/ 1) N 0 s d s is locally constant and monotone decreasing. It jumps exactly at the points s Ĩ, i.e. s N/ N 0. Proof: The directness of the sum follows by (.) and the inclusions εh (Ω) ε H s(ω) B (Ω) ε ε H s(ω), Lin{E σ,m : σ < s N/} ε H s(ω) B (Ω) ε are trivial. So it remains to prove εh s(ω) ε H (Ω) Lin{E σ,m : σ < s N/}. Therefore, we pick some E ε Hs(Ω). We observe E H 1, s (A ρ ) by the regularity result [4, Corollary 3.8 (ii)] and even rot E = 0, div E Aρ = div ˆεE Aρ L, s+τ+1(a ρ ) for all r 0 < ρ < r 1. Thus, we have ηe s(ω) D s(ω) with rot ηe = C rot,η E 0 +1 vox (Ω) B +1 (Ω), div ηe = C div,η E η div E 0 D 1 s+τ+1(ω) B 1 (Ω). The assumptions on τ yield s + τ + 1 > 1 N/. Thus, by Lemma 4.3 there exists e t (Ω) D t (Ω) with some t > N/ solving rot e = rot ηe, div e = div ηe. 6

29 Therefore, H := ηe e H s(ω). Thus, rot H = 0 and div H = 0 in A r0 and we may represent H in terms of a spherical harmonics expansion H Ar0 = γ,n h γ,n D,0 γ,n + ĥ ˆD,1 + σ< s N µ σ m=1 h + σ,m + D,0 σ,m with uniuely determined h γ,n, ĥ, h+ σ,m C using [9, Theorem.6]. By [9, emark.5] the first term of the sum on the right hand side belongs to L, (A < N r0 ), the second to L, < N 1(A r 0 ) and the third to L, s (A r0 ). We get which yields Finally we obtain E σ< s N H h := H µ σ m=1 σ< s N σ< s N µ σ m=1 µ σ m=1 e σ,m + D,0 σ,m L, < N 1(A r 0 ), h + σ,me σ,m + L, (Ω). > N h + σ,me σ,m + = (1 η)e + e + h ε H (Ω) = > N ε H (Ω). 6 Appendix B: Vector fields in three dimensions Now we will translate our results to the classical framework of vector analysis. Thus, we switch to some (maybe) more common notations. Let N := 3. We identify 1-forms with vector fields via iesz representation theorem and -forms with 1-forms via the Hodge star operator and thus with vector fields as well. Using Euclidean coordinates {x 1, x, x 3 } this means in detail we identify the vector field with the 1-form E 1 E = E E 3 E 1 dx 1 + E dx + E 3 dx 3 7

30 resp. with the -form E 1 dx 1 + E dx + E 3 dx 3 = E 1 dx dx 3 + E dx 3 dx 1 + E 3 dx 1 dx. Moreover, we identify the 3-form E dx 1 dx dx 3 with the 0-form and/or function E. We will denote these identification isomorphisms by =. Then the exterior derivative and co-derivative turn to the classical differential operators 1 grad = =, curl =, div = 3 from vector analysis, where resp. denotes the vector resp. scalar product in 3. In particular we have the following identification table: = 0 = 1 = = 3 rot = d grad curl div 0 div = δ 0 div curl grad 6.1 Tower functions and fields Let us briefly construct our tower forms once more in classical terms. Using polar coordinates {r, ϕ, ϑ}, i.e. cos ϕ cos ϑ x = Φ(r, ϕ, ϑ) = r sin ϕ cos ϑ, sin ϑ we have with an obvious notation dx 1 dr dr dx = J Φ dϕ = Q r cos ϑ dϕ, dx 3 dϑ r dϑ where Q := [e r e ϕ e ϑ ] is an orthonormal matrix and cos ϑ cos ϕ e r := cos ϑ sin ϕ, sin ϕ e ϕ := cos ϕ, sin ϑ cos ϕ e ϑ := sin ϑ sin ϕ sin ϑ 0 cos ϑ the corresponding orthonormal basis of 3. Since {dx 1, dx, dx 3 } is an orthonormal basis of 1-forms, {dr, r cos ϑ dϕ, r dϑ} is an orthonormal basis as well. Moreover, we have again with an obvious notation dr dx 1 r cos ϑ dϕ = Q t dx = r dϑ dx 3 e t r e t ϕ e t ϑ 8 dx 1 e r dx dx = e ϕ dx, dx 3 e ϑ dx

31 which shows dr = e r, r cos ϑ dϕ = e ϕ, r dϑ = e ϑ. We will denote the representations of grad, curl and div in polar coordinates by grad, curl and div as well as their realizations on S := S by grad S, curl S and div S. These may be derived by the formula ( x u) Φ = J t Φ r,ϕ,ϑ(u Φ) = [e r (r cos ϑ) 1 e ϕ r 1 e ϑ ] r,ϕ,ϑ (u Φ) and we then have the following representations: grad u = e r r u + 1 r grad S u, grad S u = 1 cos ϑ e ϕ ϕ u + e ϑ ϑ u curl v = e r r v + 1 r curl S v, curl S v = 1 cos ϑ e ϕ ϕ v + e ϑ ϑ v div v = e r r v + 1 r div S v, div S v = 1 cos ϑ e ϕ ϕ v + e ϑ ϑ v We note that we do not distinguish between u resp. v and u Φ resp. v Φ anymore. Moreover, in polar coordinates the Laplacian reads = r + r r + 1 r S, where S = 1 cos ϑ ϕ + ϑ sin ϑ cos ϑ ϑ is the Laplace-Beltrami operator. Let us introduce the classical spherical harmonics of order n y n,m, n N 0, m = 1,..., n + 1, which form a complete orthonormal system in L (S), i.e. y n,m, y l,k L (S) = δ n,l δ m,k, and satisfy ( S + λ n ) yn,m = 0, λ n := n(n + 1), as well as the corresponding potential functions z ±,n,m := r θ ±,n y n,m, n N 0, m = 1,..., n + 1, which are homogeneous of degree θ ±,n := { n, if ± = + n 1, if ± = and solve z ±,n,m = z ±,n,m = 0. 9

32 ( See for example [3, Kapitel VII, 4] or [, chapter.3]. ) Moreover, for k, n N 0, m = 1,..., n + 1 we define z k ±,n,m := ξ k ±,nr k z ±,n,m = ξ k ±,nr θk ±,n yn,m, where ξ±,n k Γ(1 ± n ± 1/) := and Γ denotes the gamma-function. The 4 k k! Γ(k + 1 ± n ± 1/) z±,n,m k are homogeneous of degree θ±,n k with and satisfy θ l ±,n := l + θ ±,n z k ±,n,m = z k 1 ±,n,m, where z 1 ±,n,m := 0. With the aid of these functions, which we will call a -tower, we construct for k N 0 the functions and fields U±,n,m k := z±,n,m k, U±,n,m k 1 := grad U±,n,m k, which we will call a div grad-tower, as well as the fields V±,n,m k := r e r grad z±,n,m k = e r grad S z±,n,m k = ξ±,nr k θk ±,n er Y n,m, V±,n,m k 1 := curl V±,n,m k, which we will call a curl curl-tower. Here Y n,m := grad S y n,m. The fields U k 1 ±,n,m are irrotational and the fields V l ±,n,m solenoidal. Moreover, we have div U k+1 ±,n,m = U k ±,n,m, curl V k+1 ±,n,m = V k ±,n,m as well as div U 1 ±,n,m = 0 and curl V 1 ±,n,m = 0 and thus U±,n,m k = div grad U±,n,m k = U±,n,m k, U±,n,m k+1 = grad div U±,n,m k+1 = U±,n,m k 1, V±,n,m l = curl curl V±,n,m l = V±,n,m l, where U±,n,m := 0, V±,n,m := 0. We mention that U±,n,m l and V±,n,m l are homogeneous of degree θ±,n l. Moreover, U±,n,m 1 = V±,n,m 1. Thus, we define P ±,n,m := U 1 ±,n,m V 1 ±,n,m. Then U l ±,n,m, V l ±,n,m, l = 1, 0 and even P ±,n,m are potential fields resp. functions. 30

33 The next picture may illustrate the denotations tower: div curl 3. floor U±,n,m V±,n,m grad curl curl. floor 0 U±,n,m 1 V±,n,m 1 div curl 1. floor U±,n,m 0 V±,n,m 0 grad curl ground 0 curl div 0 div 0 div 0 U±,n,m 1 = V±,n,m 1 div 0 div curl,div 0 0 div grad-tower curl curl-tower In the exceptional case (n, m) = (0, 1) the function z k ±,0,1 is a multiple of r k+θ ±,0 and thus we get V l ±,0,1 = 0 for all l as well as an exceptional div grad-tower U±,0,1 k = ξ±,0r k k δ,±, U±,0,1 k 1 = grad U±,0,1 k = ξ±,0 k r r k δ,± e r = ξ±,0(k k δ,± )r k 1 δ,± e r, where U 1 +,0,1 = 0. Let us briefly compare these classical towers with the -form towers: On S we identify 1-forms with linear combinations of e ϕ and e ϑ as well as -forms with scalar functions. More precisely a 1-form ω ϕ cos ϑ dϕ + ω ϑ dϑ will be identified with the tangential vector field ω ϕ e ϕ +ω ϑ e ϑ and a -form ω cos ϑ dϕ dϑ with the function ω. Then our operators ρ, τ and ˇρ, ˇτ from [15] turn to = 0 = 1 = = 3 ρv = 0 v e r v e r v τv = v (v e r ) e r v e r 0 ˇρ v = v e r v e r v ˇτ v = v v v e r, where (v e r ) e r = v e ϕ e ϕ +v e ϑ e ϑ = v v e r e r, v e r = v e ϑ e ϕ v e ϕ e ϑ. 31

34 We note y 0,1 = S 0 0,1 is constant, y n,m = T 0 n 1,m, Y n,m = grad S y n,m = i n 1/ (n + 1) 1/ S 1 n 1,m for n N in the terminology of [15]. Then for n N we get up to constants and for n = 0 U k ±,n,m = D 0 (±,k+1,n 1,m) = 3 (±,k+1,n 1,m), U k 1 ±,n,m = 1 (±,k,n 1,m) = D (±,k,n 1,m), V l ±,n,m = D 1 (±,l+1,n 1,m) = (±,l+1,n 1,m) U k +,0,1 = D 0 (+,k,0,1) = 3 (+,k,0,1), U k 1 +,0,1 = 1 (+,k 1,0,1) = D (+,k 1,0,1), U,0,1 k = D(,k+,0,1) 0 = (,k+,0,1) 3, U,0,1 k 1 = (,k+1,0,1) 1 = D(,k+1,0,1). Finally for s and l = 1, 0 we put ( with Lin := {0} ) V l s := Lin { V l,n,m : V l,n,m / L s(a 1 ) } = Lin{V l,n,m : n l + s + 1/}, Ū l s := Lin { U l,n,m : U l,n,m / L s(a 1 ) } = Lin{U l,n,m : n l + s + 1/}, P s := Lin { P,n,m : P,n,m / L s(a 1 ) } = Lin{P,n,m : n s + 3/}, Ǔ l := Ǔl s := Lin { U l,0,1 : U l,0,1 / L s(a 1 ) } = Lin{U l,0,1 : 0 l + s + 1/}. 6. esults for vector fields For every operator {grad, curl, div} and s we define the Hilbert spaces H s (, Ω) := { u L s(ω) : u L s+1(ω) }, H s (, Ω) := C (Ω), H s ( 0, Ω) := { H s (, Ω) : u = 0 }, H s ( 0, Ω) := { H s (, Ω) : u = 0 }, where the closure is taken in H s (, Ω). Then the spaces s(ω) and D s(ω) turn to the usual Sobolev spaces, i.e.: = 0 = 1 = = 3 s(ω) H s ( grad, Ω) = H 1 s(ω) H s ( curl, Ω) H s ( div, Ω) L s(ω) D s(ω) L s(ω) H s (div, Ω) H s (curl, Ω) H s (grad, Ω) = H 1 s(ω) 3

35 For every two operators, { () grad (0), () curl (0), () div (0) } we define The operator ε reads as: H s (,, Ω) := H s (, Ω) H s (, Ω). ε 0 ε 1 div grad 1 grad div ε 1 curl curl curl curl +ε 1 grad div 3 div grad The generalized boundary condition ι E = 0 for a -form E from loc (Ω) turns to the usual boundary conditions γe = E Ω = 0, γ t E = ν E Ω = 0, γ n E = ν E Ω = 0 (for = 0, 1, ) weakly formulated in the spaces H( grad, Ω), H( curl, Ω) and H( div, Ω), where ν denotes the outward unit normal at Ω and γ the trace as well as γ t resp. γ n the tangential resp. normal trace of the vector field E. The linear transformations ε, ν, µ (ν and µ may be identified!) can be considered as variable, real, symmetric and uniformly positive definite matrices with L (Ω)-entries, which satisfy the asymptotics at infinity assumed in sections and 3. Moreover, for any two operators, { () curl (0), div () } (0) we define H s ( ε, Ω) := ε 1 H s (, Ω), H s (, ε, Ω) := H s (, Ω) H s ( ε, Ω). Now we have two kinds of Dirichlet fields. The first ones, the classical Dirichlet fields, εh s (Ω) := H s ( curl 0, div 0 ε, Ω) = ε Hs(Ω) 1, s correspond to = 1 and the second ones, the classical Neumann fields, H ε s (Ω) := H s (curl 0, div 0 ε, Ω) = ε 1 ε 1H s(ω), s correspond to =. Moreover, we have the compactly supported fields B 1 (Ω) =: B 1 (Ω) H( curl 0, Ω), B (Ω) =: B (Ω) H( div 0, Ω) and the fields with bounded supports B (Ω) =: B(Ω) H(curl 0, Ω). Let s > 1/. Using the Dirichlet and Neumann fields we put H s (, Ω) := H s (, Ω) H(Ω), Hs (, Ω) := H s (, Ω) H(Ω) 33

36 and define in the same way H s (,, Ω), H s (, ε, Ω) and H s (,, Ω), H s (, ε, Ω). Then we have H s (div 0, Ω) = H s (div 0, Ω) ε H(Ω) = H s (div 0, Ω) B 1 (Ω), H s ( div 0, Ω) = H s ( div 0, Ω) ε H(Ω) = H s ( div 0, Ω) B(Ω), H s (curl 0, Ω) = H s (curl 0, Ω) ε H(Ω) ε = H s (curl 0, Ω) B (Ω), H s ( curl 0, Ω) = H s ( curl 0, Ω) ε H(Ω) ε, and thus except of the last one the definitions of these spaces extend to all s. Moreover, we set for s > 1/ We get εl s(ω) := L s(ω) ε H(Ω) ε, ε L s(ω) := L s(ω) ε H(Ω) ε. Lemma 6.1 Let s > 1/. Then the direct decompositions L s(ω) = ε L s(ω) Lin B 1 (Ω), L s(ω) = ε L s (Ω) ε 1 Lin B (Ω) = ε L s (Ω) Lin B(Ω) hold. If additionally s < 1/, then L s(ω) = ε L s(ω) ε ε H(Ω), L s(ω) = ε L s (Ω) ε ε H(Ω). With ε := grad div ε 1 curl curl = ˇε curl curl and always assuming s / Ĩ, i.e. for all n N 0 we obtain s n + 1/ and s n 3/, Theorem 6. The following decompositions hold: (i) If s < 3/, then L s(ω) = H s ( curl 0, Ω) + ε 1 H s (div 0, Ω) = H s (curl 0, Ω) + ε 1 H s ( div 0, Ω) = H s ( curl 0, Ω) + ε 1 H s (div 0, Ω) = H s (curl 0, Ω) + ε 1 Hs ( div 0, Ω). In the first line the intersections eual the finite dimensional space of Dirichlet resp. Neumann fields ε H s (Ω) resp. ε Hs (Ω) and in the second line the intersections eual the finite dimensional space of Dirichlet resp. Neumann fields ε H s (Ω) B 1 (Ω) ε resp. ε H s (Ω) B (Ω). 34