Generalized Maxwell Equations in Exterior Domains III: Electro-Magneto Statics

Transcription

1 Reports of the Department of Mathematical Information Technology Series B. Scientific Computing No. B. 9/2007 Generalized Maxwell Equations in Exterior Domains III: Electro-Magneto Statics Dirk Pauly University of Jyväskylä Department of Mathematical Information Technology P.O. Box 35 (Agora) FI University of Jyväskylä FINLAND fax

2 Copyright c 2007 Dirk Pauly and University of Jyväskylä ISBN ISSN X

3 Generalized Maxwell Equations in Exterior Domains III: Electro-Magneto Statics Dirk Pauly Abstract We develop a solution theory for a generalized electro-magneto static Maxwell system in an exterior domain Ω R N with anisotropic coefficients converging at infinity with a rate r τ, τ > 0, towards the identity. Our main goal is to treat right hand side data from some polynomially weighted Sobolev spaces and obtain solutions which are up to a finite sum of special generalized spherical harmonics in another appropriately weighted Sobolev space. As a byproduct we prove a generalized spherical harmonics expansion suited for Maxwell equations. In particular, our solution theory will allow us to give meaning to higher powers of a special static solution operator. Finally we show, how this weighted static solution theory can be extended to handle inhomogeneous boundary data as well. This paper is the second one in a series of three papers, which will completely reveal the low frequency behavior of solutions to the time-harmonic Maxwell equations. Key words: exterior boundary value problems, Maxwell equations, variable coefficients, electro-magnetic theory, electro-magneto statics, spherical harmonics expansion, harmonic Dirichlet fields, inhomogeneous boundary data AMS MSC-classifications: 35Q60, 78A25, 78A30 This research was supported by the Deutsche Forschungsgemeinschaft via the project We 2394: 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 2007 as a postdoc fellow. Department of Mathematics, University of Duisburg-Essen, Campus Essen, Essen, Germany, Fon: , Fax: , dirk.pauly@uni-due.de 1

4 Contents 1 Introduction 2 2 Towers of static solutions in the whole space 6 3 Electro-magneto static operators 16 4 Generalized electro-magneto statics 29 5 Powers of a static Maxwell operator 32 6 Electro-magneto statics with inhomogeneous boundary data 39 7 Appendix: Second order operators 40 1 Introduction In the bounded domain case it is just an easy exercise to show that the solution operator for the time-harmonic Maxwell equations L ω is approximated by Neumann s series of the corresponding electro-magneto static solution operator L for small frequencies ω, i.e. L ω = ω 1 Π + ω j L j+1 Π reg, j=0 where Π and Π reg are projections onto irrotational and solenoidal fields. In the case of an exterior domain (a domain with compact complement) this low frequency asymptotic holds no longer true. We run into problems even if we formally want to define higher powers of a static solution operator since the well known electromagneto static solution theories developed e.g. by Picard in [9, 10, 11] and Kuhn and the present author in [2] treat data from some polynomially weighted Sobolev spaces and yield solutions in less weighted Sobolev spaces. In particular in [9, 10] and [2] we get from L 2 -data L 2 1-solutions by decomposing L 2 into subspaces consisting of irrotational resp. solenoidal fields. ( Here for s R we denote by L 2 s the Hilbert space of all measurable fields E, for which ρ s E is square integrable, where ρ := ρ(r) := (1 + r 2 ) 1/2 and r is the Euclidean norm in R N. Many of our notations have been previously used in [2, 5] and [4] and their explicit definitions will not be repeated here. For more details and the exact definitions we refer to these papers. ) In [11] we obtain from L 2 1-data L 2 -solutions by a second order approach and elliptization of the Maxwell system using Lax-Milgram s theorem. (The latter paper considers a more general non-linear case using a theorem suited for monotone operators, but in the linear case it is just the Lax-Milgram theorem.) So far based on these known results we can only consider first and second powers of the solution operator of the static Maxwell system. To overcome these limitations we have 2

5 to develop an electro-magneto static solution theory, which deals with arbitrarily weighted data and describes the solutions in terms of their integrability properties, such that we are able to iterate this static solution operator depending on the integrability of the data and therefore, define a generalized Neumann sum of the static solution operator. In the case of Helmholtz equation and the equations of linear elasticity theory in an exterior domain Ω R N, where comparable integrability problems for the static problem occur, Weck and Witsch, [13] and [15, 16], respectively, have shown that the time-harmonic solution operator is still approximated by a (generalized) Neumann-type expansion in terms of the corresponding generalized static solution operator for low frequencies except for some additional degenerate correction operators. In [13] they discussed the case N = 3 and in [15, 16] the case of odd space dimensions N. For even dimensions N some technical complications arise due to the appearance of logarithmic terms in the Hankel function of integer order. So in even dimensions the results still hold true but the complexity of notations and calculations increases considerably. For N = 2 (the most complicated case!) Peter showed in [8] how to do this for Helmholtz equation. So we may expect and indeed will show in [7] that a similar low frequency asymptotic holds true for Maxwell equations in an exterior domain, i.e. for small frequencies ω with non-negative imaginary part and J N 0 we will prove J 1 L ω ω 1 Π ω j L j+1 Π reg j=0 J N j=0 ω j+n 1 Γ j = O ( ω J) with projections Π and Π reg onto irrotational and solenoidal fields as well as some degenerate correction operators Γ j in the operator topology of weighted Sobolev spaces. Here the O-symbol is meant for ω 0. Motivated by these considerations and following Hermann Weyl [17] we want to discuss in this paper the generalized electro-magneto static Maxwell system rot E = G, div εe = f, ι E = 0 (1.1) in an exterior domain Ω R N using alternating differential forms. Here the electric field E is a differential form of rank q (q-form) and the data G and f are (q +1)- resp. (q 1)-forms. To invoke suggestively the applicational background of the electromagneto statics, it has become customary to denote the exterior derivative d by rot and the co-differential δ by div. Thus, we have on q-forms div = ( 1) (q 1)N rot. Here is the Hodge star operator. Furthermore, ε is a linear transformation acting on q-forms, ι : Ω Ω the natural embedding and ι the pull-back map of ι. So ι E can be considered as the restriction of the form E to the (N 1)-dimensional Riemannian submanifold Ω, the boundary. In the special (classical) cases N = 3 and q = 1 or q = 2, where we identify 1- and 2-forms with vector fields (via the Riesz representation theorem and the star operator) as well as 0- and 3-forms with scalar functions, by (1.1) the classical systems of 3

6 electro- or magneto-statics are recovered. More precisely the system (1.1) then reads for q = 1 curl E = G, div εe = f, ν E Ω = 0, (1.2) where curl = resp. div = is the classical rotation resp. divergence and ν the outward unit normal at the boundary Ω. Here we denote by the classical gradient and by resp. the vector- resp. scalar-product in R 3. Thus, in this case we get the classical electro static system for the electric field with prescribed (vanishing) tangential component at the boundary. By the vanishing tangential component of E at the boundary we model total reflection of the electric field at the boundary, i.e. the complement R 3 \ Ω is a perfect conductor. Now physically f is the charge density, ε the dielectricity of the medium Ω, εe the displacement current and G = 0. Setting q = 2 the classical magnetic case appears. In this case H := εe is the magnetic field, µ := ε 1 the permeability of our medium, µh the magnetic induction, f the current and G = 0. Now the system (1.1), i.e. turns into rot µh = G, div H = f, ι µh = 0, div µh = G, curl H = f, ν µh Ω = 0. (1.3) Thus, we obtain the classical magneto static system of a perfect conductor corresponding to (1.2). We will show in this paper that the static Maxwell problem (1.1) has a solution for data f, G taken from a closed subspace of L 2,q 1 s s > 1 N/2. Here for s R we denote by L 2,q s (Ω) Ls 2,q+1 (Ω) with weights (Ω) the Hilbert space of all measurable (Ω) we have q-forms E, for which ρ s E is square integrable over Ω, i.e. for E, H L 2,q s the scalar product E, H L 2,q s (Ω) := ρ 2s E H <. Ω (Here is the exterior product and denotes complex conjugation.) Because of the existence of a non-trivial L 2 -kernel of (1.1), the harmonic Dirichlet forms ε H q (Ω), our solution is unique, if we impose some adequate orthogonality constraints on it. We receive solutions, which lie in the naturally expected weighted Sobolev space L 2,q s 1(Ω) except of a finite sum of special generalized spherical harmonics. To obtain our results we consider linear, bounded, symmetric and uniformly positive definite transformations ε, such that the perturbations ˆε := ε Id are C 1 in the outside of an arbitrary compact set and decay with order τ > 0, i.e. α ˆε = O(r α τ ) as r for all α 1. Depending on the weight s we have to adjust the order of decay monotone increasing. 4

7 A solution theory for the static system (1.1) has been given by Kuhn and the present author in [2] as well as by Kress [1] and Picard [9] for homogeneous, isotropic media, i.e. ε = Id, and by Picard [11] for inhomogeneous, anisotropic media. (Here ε is even allowed to be a non-linear transformation as mentioned above.) Moreover, in the classical cases of electro- and magneto-statics Picard [10] and Milani and Picard [3] developed a solution theory for inhomogeneous, anisotropic media. In these papers the data are always taken from some closed subspaces of L 2 or L 2 1. This means in our notation that until now solution theories for the special cases of weights s = 0 and s = 1 are known. Keeping in mind that we eventually want to be able to define a generalized Neumann-type expansion, our immediate goal is to construct a special static solution operator L associated with the electro-magneto static Maxwell system rot E = G, div εe = 0, ι E = 0, div H = F, rot µh = 0, ι µh = 0, which maps data (F, G) from the closed subspace ( 0D q s(ω) H q (Ω) ) ( 0R q+1 s (Ω) H q+1 (Ω) ) of L 2,q s (Ω) L 2,q+1 s (Ω) to the forms (εe, µh) from ( εr q t (Ω) µd q+1 t (Ω) ) ( ( 0D q t (Ω) H q (Ω) ) ( 0R q+1 t (Ω) H q+1 (Ω) )) with some t s 1 and t < N/2. The main tool for this is the construction of towers of special homogeneous (iterated) static solutions in the whole space. Here for t R we introduce the weighted Sobolev spaces suited for Maxwell equations R q t (Ω) := { E L 2,q t (Ω) : rot E L 2,q+1 t+1 (Ω) }, D q+1 t (Ω) := { H L 2,q+1 t (Ω) : div H L 2,q t+1(ω) } equipped with their canonical norms. Furthermore, we generalize the homogeneous boundary condition in R q t (Ω), the closure of C,q (Ω) in the R q t (Ω)-norm. The symbol stands for orthogonality with respect to the L 2 -scalar product. A subscript 0 at the lower left corner indicates vanishing rotation resp. divergence. If Ω = R N we omit the dependence on the domain. All these spaces are Hilbert spaces. In a final section we deal with inhomogeneous boundary conditions. Using our report [2] we are able to work with traces and extensions of q-forms. This enables us to discuss the static problem rot E = G, div εe = f, ι E = λ. It turns out that the solution theory for this problem is an easy consequence of the results for homogeneous boundary conditions and the existence of an adequate extension operator for our traces. 5

8 In this report we follow closely the ideas of [13] and [16]. We note that dual results can easily be obtained utilizing the Hodge star operator, but for sake of brevity we shall refrain from stating those results explicitly. Moreover, to decrease the complexity of this paper we only deal with odd space dimensions N 3 to avoid logarithmic terms as described above. We mention that many results hold true for even dimensions N as well. Essentially we need the assumption N odd only to construct our towers and these again are used to iterate our solution operators. So the static solution theory still remains valid, if N is even. Throughout this paper we use the notations from [4, 2, 5] as mentioned above. (Especially in this paper the index loc assigned to spaces is always to be understood in the sense of Ω. Moreover, the index vox denotes compact supports.) Essentially the present paper is the second part of the author s doctoral thesis, [4]. For sake of brevity, however, some proofs are merely sketched or completely omitted. For more details and some additional results the interested reader is referred to [4]. The report at hand is the third one in a series of five reports having the aim to determine the low frequency asymptotics of the time-harmonic Maxwell equations completely. In the second report [5] we discussed the time-harmonic solution operator and showed its convergence to a suitable static solution operator as the frequency tends to zero. With the present report we provide the means to define higher powers of the static solution operators in suitably weighted Sobolev-type spaces. The fourth report [6] deals with Hodge-Helmholtz decompositions in weighted Sobolev spaces, which turn out to be necessary, since the Maxwell operator possesses a non trivial kernel. In the fifth report [7] of our series we shall then develop a generalized Neumann-type expansion to fully analyze the low-frequency behavior of timeharmonic solutions to Maxwell s equations in the topology of weighted Sobolev spaces up to arbitrary powers of the frequency. 2 Towers of static solutions in the whole space In this section we consider the homogeneous and isotropic whole space case, i.e. Ω = R N and ε = Id, and assume 3 N N to be odd. Our aim is to provide a generalized spherical harmonics expansion for differential forms. We use the spherical calculus (and its notations) developed by Weck and Witsch in [14] (a technique to use polar-coordinates for q-forms) to construct towers of homogeneous differential forms ± D q,k σ,m and solving the following system for k N 0 : ± R q,k σ,m in C,q( R N \ {0} ) rot ± Dσ,m q,0 = 0, div ± Rσ,m q+1,0 = 0 (2.1) div ± Dσ,m q,k = 0, rot ± Rσ,m q+1,k = 0 (2.2) rot ± Dσ,m q,k = ± Rσ,m q+1,k 1, div ± Rσ,m q+1,k = ± Dσ,m q,k 1 (2.3) 6

9 These towers coincide in some sense with the eigenforms S q σ,m, T q σ,m of the Laplace- Beltrami operator on the unit sphere S N 1 R N, which establish a complete orthonormal system in L 2,q (S N 1 ) and solve the Maxwell eigenvalue system Rot T q σ,m = i ω q σ S q+1 σ,m, Div S q+1 σ,m = i ω q σ T q σ,m, where ωσ q 1 := (q + σ) 1 2 (q + σ) 1 2 with q := N q. Here Rot and Div denote the exterior derivative and co-derivative on the unit sphere. For the construction of these towers we use the operators ρ, τ and their right inverses ˇρ, ˇτ introduced in [14], intensively. These towers have already been defined and discussed in [16, p. 1503]. For our purposes we have to study them more thoroughly. For k, σ N 0 we let { ± h k k + σ, ± = + σ := k σ N, ± = and with µ q σ := µ q,n σ from [14, p. 1029, Theorem 1 (iii)] we introduce Definition 2.1 Let q {0,..., N}, k, σ N 0 and m {1,..., µ q σ}. Then we define tower-forms by ( ) ± Dσ,m q,2k := ± ασ q,k r ± h 2k σ i ω q 1 σ ˇρ Tσ,m q 1 + (q + ± h 2k σ ) ˇτ Sσ,m q, ± D q 1,2k+1 σ,m := ± ασ q,k r ± hσ 2k+1 ˇτ Tσ,m q 1, ( ± Rσ,m q,2k := ± ασ q,k r ± h 2k σ (q + ± h 2k σ ) ˇρ Tσ,m q 1 + i ωσ q 1 ˇτ Sσ,m q ± R q+1,2k+1 σ,m The coefficients satisfy the recursion := ± α q,k σ r ± h 2k+1 σ ˇρ S q σ,m. ), ± α q,k σ := ± α q,k 1 σ 2k (2k ± 2σ ± N), α q,0 σ := 1, + α q,0 σ := ( 1)1+δ q,0+δ q,n 2σ + N. Moreover, we collect all these indices in an index I := (sgn, k, σ, m) taken from the set {±} N 0 N 0 N and define the notation D q I := sgn D q,k σ,m and R q I := sgn R q,k σ,m. Here we call s(i) := sgn = ± the sign, h(i) := k N 0 the height, e(i) := σ N 0 the eigenvalue index and c(i) := m N the counting index of a tower-q-form D q I or Rq I. Furthermore, we define the homogeneity degree of a tower-form by hom(d q I ) := hom(rq I ) := h I := s(i) h h(i) e(i). Finally we define the upper bound of the counting index { µ q σ, k even µ q,k σ := µ q+1 σ 7, k odd

10 and the two index sets I q := { I : s(i) {+, } h(i), e(i) N 0 1 c(i) µ q,h(i) e(i) }, J q := { J : s(j) {+, } h(j), e(j) N 0 1 c(j) µ q 1,h(J)+1 e(j) Remark 2.2 }. (i) The recursion of the coefficients is well defined because N is odd. Thus, our towerforms are well defined. For even dimensions the recursion is also well defined for towerforms with positive sign and for tower-forms with negative sign as long as k < N/2. (In the last case we would have to work with logarithmic terms of the radius r for higher k.) Therefore, for even dimensions N 4 all tower-forms with negative sign up to heights three are well defined. (ii) The tower-forms D q I and Rq I are elements of C,q( R N \ {0} ), homogeneous of degree h I and solutions of the system (2.1)-(2.3) in R N \ {0}. (iii) An index I resp. J of a tower-form D q I resp. Rq J belongs to the index set Iq resp. J q. (iv) The elements of the countable set of tower-forms { D q I, Rq J : I Iq, J J q h(j) 1 } = { D q I, Rq J : I Iq, J J q h(i) 1 } are linear independent. (v) From the defining recursion of the coefficients we get the following explicit formulas: + α q,k σ = α q,k σ = Γ(1 + N/2 + σ) 4 k k! Γ(k N/2 + σ) ( 1)1+δq,0+δq,N 2σ + N Γ(1 N/2 σ) 4 k k! Γ(k + 1 N/2 σ) Here Γ denotes the gamma-function. The coefficients ± ασ q,k converge rapidly to zero as k. Thus, for 0 < a b < the tower-forms D q I and Rq I, I = (sgn, k, σ, m), together with all their derivatives and even after multiplication with arbitrary powers of r are uniformly bounded with respect to a x b and k, σ, m N 0. (vi) The definitions of the tower-forms in Definition 2.1 have to be understood in the sense that all not defined terms are set to be zero. Thus, only for q {1,..., N 1} no problems occur and we get regular tower-forms. In the extreme cases q {0, N}, where we have only the index (σ, m) = (0, 1), the only tower-forms are ± D 0,2k ± R N,2k 0,1, ± D N 1,2k+1 0,1 and ± R 1,2k+1 0,1. We note in this cases 0,1, D 0,0 0,1 = 0, + D 0,0 0,1 Lin{1}, R N,0 0,1 = 0, + R N,0 0,1 Lin{ 1}. 8

11 To get an idea about the utilization of the spherical calculus we may compute the rotation and divergence of the forms D q I. We note that these are homogeneous and derive [ ] div ± Dσ,m q 1,2k+1 = ± ασ q,k [ˇρ ˇτ] div ρ (r ± h 2k+1 σ ˇτ Tσ,m q 1 ) τ [ ] [ ] = ± ασ q,k r ± hσ 2k+1 1 Div 0 0 [ˇρ ˇτ] (q 1) + ± h 2k+1 σ Div Tσ,m q 1 = 0, [ ] ρ rot ± Dσ,m q 1,2k+1 = ± ασ q,k [ˇρ ˇτ] rot (r ± h 2k+1 σ ˇτ Tσ,m q 1 ) τ = ± α q,k = ± α q,k σ σ r ± hσ 2k+1 = ± R q,2k σ,m, div ± Dσ,m q,2k = ± ασ q,k [ˇρ ˇτ] div 1 [ˇρ ˇτ] [ Rot q 1 + ± h 2k+1 σ 0 Rot ( r ± h 2k σ (q + ± h 2k σ ) ˇρ Tσ,m q 1 + i ωσ q 1 ˇτ Sσ,m q [ ρ τ = ± ασ q,k r ± h 2k σ 1 [ˇρ ˇτ] ] ( r ± h 2k σ = ± ασ q,k r ( ± h 2k σ 1 i ωσ q 1 = 0, rot ± Dσ,m q,2k = ± α q,k [ˇρ ˇτ] rot σ [ ρ τ = ± ασ q,k r ± h 2k σ 1 [ˇρ ˇτ] ( i ω q 1 σ [ Div 0 q + ± h 2k σ Div ] ( r ± h 2k σ ) ] [ 0 T q 1 σ,m ) ) ˇρ Tσ,m q 1 + (q + ± h 2k σ ) ˇτ Sσ,m q ] ] [ i ω q 1 σ T q 1 σ,m ] (q + ± h 2k σ )S q σ,m (q + ± h 2k σ ) + i ωσ q 1 (q + ± h 2k σ ) ) ˇτ T q 1 ( i ω q 1 σ [ Rot q + ± h 2k σ 0 Rot = ( (q + ± h 2k σ )(q + ± h 2k σ ) (ωσ q 1 ±α q,k σ ) 2) σ,m ) ) ˇρ Tσ,m q 1 + (q + ± h 2k σ ) ˇτ Sσ,m q ] ] [ i ω q 1 σ ±α q,k σ T q 1 σ,m (q + ± h 2k σ )S q σ,m r ± h 2k 1 σ = 2k(2k ± 2σ ± N) r ± h 2k 1 σ ˇρ Sσ,m q { ± ασ q,k 1 r ± h 2k 1 σ ˇρ Sσ,m q = ± Rσ,m q+1,2k 1, if k 1 = 0, if k = 0 ˇρ S q σ,m. Analogously we can prove the formulas for R q J and the exceptional towers. 9

12 Remark 2.3 Because of = rot div + div rot (Here the Laplacian acts on each Euclidean component of the differential form.) all tower-forms D q I, Rq J of heights less or equal to one satisfy D q I = Rq J = 0. Therefore, comparing these tower-forms with the potential forms discussed in [14] we obtain for q {1,..., N 1} D q,0 σ,m = (q + σ) 1 2 (2σ + N) 1 2 Q q,3 σ+2,m, R q,0 σ,m = i(q + σ) 1 2 (2σ + N) 1 2 Q q,3 σ+2,m, + D q,0 σ,m = i(q + σ) 1 2 (2σ + N) 1 2 P q,4 σ,m, + R q,0 σ,m = (q + σ) 1 2 (2σ + N) 1 2 P q,4 σ,m, D q 1,1 σ,m R q+1,1 σ,m = Q q 1,2 σ+1,m, = Q q+1,1 σ+1,m, + D q 1,1 σ,m = 1 2σ + N P q 1,2 σ+1,m, + R q+1,1 σ,m = 1 2σ + N P q+1,1 σ+1,m. In particular the tower-forms D q,0 σ,m and R q,0 σ,m resp. + D q,0 σ,m and + R q,0 σ,m are linear dependent, which we will indicate by the symbol =. In detail we have Dσ,m q,0 = ϑ q σ Rq,0 σ,m, + Rσ,m q,0 = ϑ q σ +Dq,0 σ,m, where ϑ q σ := i(q + σ) 1 2 (q + σ) 1 2. Furthermore, the potential forms Q q,4 σ,m resp. P q,3 linear combinations of the tower-forms Dσ,m q,2 and Rσ,m q,2 resp. + Dσ,m q,2 and + Rσ,m q,2, i.e. (2σ + N) σ N Qq,4 σ,m = (q + σ) 1 2 R q,2 σ,m + i(q + σ) 1 2 D q,2 σ,m, i (2σ + N) σ + N P q,3 σ+2,m = (q + σ) R q,2 σ,m i(q + σ) D q,2 σ,m. For q {0, N} we see σ+2,m are D 0,2 0,1 = i 2 N Q0,4 0,1, R N,2 0,1 = 1 2 N QN,4 0,1, + D 0,0 0,1 = i P 0,4 0,1, + R N,0 0,1 = P N,4 0,1, D N 1,1 0,1 = Q N 1,2 1,1, R 1,1 0,1 = Q 1,1 1,1, + D N 1,1 0,1 = 1 N P N 1,2 1,1, + R 1,1 0,1 = 1 N P 1,1 1,1. 10

13 The following picture explains the denotation tower : div rot 3. floor ± Dσ,m q 1,3 rot div 2. floor ± Rσ,m q,2 ± Dσ,m q,2 div rot 1. floor ± Dσ,m q 1,1 rot div ± R q+1,3 σ,m ± R q+1,1 σ,m ground ± Rσ,m q,0 = ± Dσ,m q,0 rotation-tower divergence-tower Remark 2.4 For an index I I q resp. J J q with odd height we get ρd q I = 0 resp. τrq J = 0 since ρ ˇτ = 0 resp. τ ˇρ = 0. Thus, by dr dr = 0 T D q I = 0 resp. RRq J = 0 with the operators T = ± R, R = r dr = x n dx n from [14]. Because of (2.2) and with the commutator formula C div,ϕ(r) = ϕ (r)r 1 T resp. C rot,ϕ(r) = ϕ (r)r 1 R ( ) see e.g. [14] or [2, (2.24)] we obtain div ( ϕ(r)d q I for any (!) ϕ C 1 (R). ) = 0 resp. rot ( ϕ(r)r q J) = 0 To shorten the formulas we write for I I q resp. J J q D q (I) := Lin{D q I : I I} resp. Rq (J) := Lin{R q J : J J} 11

14 ( with the convention D q ( ) := {0} resp. R q ( ) := {0} ). For s R let I s := { I I : D q I / L2,q s (A 1 ) }, J s := { J J : R q J Furthermore, for s R and k, K N 0 we present the index sets I q,k := { I I q : h(i) = k }, I q, K := I q,k s := ( I q,k ) s, I q, K s := Ī q,k := { I I q,k : s(i) = }, Ī q, K := Ī q,k s := ( Ī q,k ) s, Ī q, K s := / L2,q s (A 1 ) }. K I q,k, k=0 K k=0 I q,k s, K Ī q,k, and replacing I by J similar sets for J q. Moreover, we introduce for indices the negative indices k=0 K k=0 Ī q,k s I := (sgn, k, σ, m) I q resp. J := (sgn, k, σ, m) J q+1 I := ( sgn, k, σ, m) I q resp. J := ( sgn, k, σ, m) J q+1 and with j Z for the shifted indices the notation { J q+1 ji := (sgn, k + j, σ, m) resp. jj := (sgn, k + j, σ, m) For subsets I resp. J of I q resp. J q+1 we set { J q+1 ji := { j I : I I} resp. jj := { j J : J J} With these definitions we then have I q { I q J q+1 I q { I q J q+1, j odd, j even, j odd, j odd, j even, j odd, j even, j even rot D q 1J = Rq+1 J, div R q+1 = D q 1I I, rot D q I = Rq+1 1I, div R q+1 J = D q 1J. 12..

15 Remark 2.5 Let m N 0 and I I q. Since our tower-forms are smooth and homogeneous we have for s R D q I L2,q s (A 1 ) D q I Hm,q s (A 1 ) h I < s N/2 s < s(i) ( e(i) + N/2 ) h(i). If in particular I Īq,k, then D q I is an element of Hm,q s (A 1 ), if and only if e(i) > s + k N/2. Thus, for k N 0 we can characterize our special index sets by We note that D q (Īq,k s the spaces D q (Īq,k s L 2,q Ī q,k s = { I Īq,k : e(i) s + k N/2 }, Ī q, k s = { I Īq, k : e(i) s + h(i) N/2 }. ) = D q (Īq, k s ) and D q (Īq, k s ) = {0}, if and only if s < N/2 k. Thus, for s N/2 k ) are subspaces of H m,q < N k(a 1) but by definition even not of 2 s (A 1 ). Clearly all these assertions also hold true for tower-forms R q J with J Jq. Let us now introduce the matrix -differential operator [ ] 0 div M := rot 0 (2.4) acting on pairs of (q, q + 1)-forms (E, H) by M(E, H) := (div H, rot E). Now we are able to prove the main result of this section, a generalized spherical harmonics expansion suited for Maxwell equations. To this end we have to define for K N and s R some exceptional forms: D 0,K 0,1, q = 0 K even ˆD R 1,1 q,k 0,1, q = 1 := (2.5) D N 1,K 0,1, q = N 1 K odd 0, otherwise ˆR q+1,k := R 1,K 0,1, q = 0 K odd D N 1,1 R N,K 0,1, q = N 2 0,1, q = N 1 K even 0, otherwise (2.6) 13

16 ˆD q,k s := ˆR q+1,k s := D 0,K R 1,1 0,1, q = 0 K even s < N/2 K 0,1, q = 1 s < N/2 1 D N 1,K 0,1, q = N 1 K odd s < N/2 K 0, otherwise R 1,K 0,1, q = 0 K odd s < N/2 K D N 1,1 R N,K 0,1, q = N 2 s < N/2 1 0,1, q = N 1 K even s < N/2 K 0, otherwise (2.7) (2.8) Theorem 2.6 With K N and 0 r < r let (E, H) denote a solution of the iterated Maxwell system M K (E, H) = (0, 0) and div E = 0, rot H = 0 in Z r, r. Then (E, H) C,q (Z r, r ) C,q+1 (Z r, r ) and in Z r, r the representations E = e q,k I D q I + ê q,k ˆD q,k, (2.9) H = I I q, K 1 J J q+1, K 1 h q+1,k J R q+1 J + ĥ q+1,k ˆR q+1,k (2.10) hold with unique constants e q,k, h q+1,k C and ê q,k, ĥq+1,k C, provided that the exceptional forms do not vanish. These series converge in C (Z r, r ), i.e. uniformly together with all their derivatives in compact subsets of Z r, r. In the case 0 < r < r = we have with some s R (E, H) H m,q s (A r ) H m,q+1 s (A r ) for all m N 0, if and only if all coefficients e q,k I and h q+1,k J with h I, h J s N/2 vanish. This holds true, if and only if h(i) + e(i), h(j) + e(j) s N/2 for indices I, J with positive sign and h(i) e(i), h(j) e(j) s + N/2 for I, J with negative sign. Then the series converge for all ˆr > r uniformly together with all derivatives even after multiplication with arbitrary powers of r in Aˆr. Thus, in particular they converge in H m s (Aˆr ). Especially for s N/2 there appear only tower-forms with negative sign. In this case (2.9) and (2.10) turn to E = e q,k I D q I + ê q,k ˆD q,k s, H = I Īq, K 1 \Īq, K 1 s J J q+1, K 1 \ J q+1, K 1 s h q+1,k J R q+1 J + ĥq+1,k ˆR q+1,k s. 14

17 Proof: The smoothness of (E, H) follows by the regularity result [2, Theorem 3.6 (i)]. Remark 2.5 yields the integrability properties of each single term in the stated expansion. Concerning the mode of convergence we refer to [14, p. 1033] and [16, p. 1508, Theorem 1], where similar expansions have been discussed. In particular for r = the series converge in L 2 s(aˆr ), if and only if all terms in the expansion belong to L 2 s(aˆr ). Thus, we only have to show the representation formulas (2.9) and (2.10). Let us look at E in the case K = 1. We have rot E = 0 and div E = 0. Thus, E is a potential form, i.e. E = 0, and we obtain from [14, p. 1033] the representation E = k,σ,m α q k,σ,m P σ,m q,k + β q k,σ,m Qq,k σ,m, α q k,σ,m, βq k,σ,m C. (2.11) k,σ,m By testing the equation rot E = 0 with ϕ(r) ˇρ T q σ 1,m for any ϕ C ( ( r, r) ), i.e. computing 0 = rot E, ϕ(r) ˇρ T q σ 1,m L 2,q+1 with partial integration and (2.11), we see α q 2,σ,m = β q 2,σ,m = 0 except of β2,1,1 N 1. Testing with ϕ(r) ˇρ S q σ 2,m yields α q 3,σ,m = β q 4,σ 2,m = 0. Analogously we obtain from the equation div E = 0 by testing with ϕ(r) ˇρ T q 2 σ 1,m that α q 1,σ,m = β q 1,σ,m must vanish except of β1,1,1 1. Finally only E = α q 4,σ,m Pσ,m q,4 + β1,1,1 1 Q 1,1 1,1, q = 1 β q 3,σ,m Q q,3 σ,m + β2,1,1 N 1 Q N 1,2 1,1, q = N 1 σ,m σ,m 0, otherwise remains from (2.11). With Remark 2.3 we can replace these potential forms by our tower-forms and we receive the asserted representation. Because H solves the same system as E (replacing q by q + 1) we obtain the representation for H in the case K = 1 as well. Assuming now that our representations hold for some K 1, we consider E solving the system M K+1 (E, 0) = (0, 0) and div E = 0. Then H := rot E satisfies M K (0, H) = (0, 0) and rot H = 0. Our assumptions for K yield H = h q+1,k J R q+1 J + ĥq+1,k q+1,k ˆR J J q+1, K 1 and with the ansatz Ẽ := J J q+1, K 1 h q+1,k J D q 1J + ĥq+1,k we see that e := E Ẽ solves the system D 0,K+1 0,1, q = 0 K + 1 even D N 1,K+1 0,1, q = N 1 K + 1 odd 0, otherwise div e = 0, rot e = ĥq+1,k { D N 1,1 0,1, q = N 2 0, otherwise. 15

18 If q N 2 the conclusion for K = 1 gives e = I D q I + êq,1 I I q,0 e q,1 ˆD q,1 and thus E = e + Ẽ = I I q, 1 h(i) K h q+1,k 1I D q I + e q,1 I I I q,0 D q I ĥ q+1,k R 1,1 D 0,K+1 0,1, q = 0 K + 1 even ê q,1 0,1, q = 1 + ê q,1 DN 1,1 0,1, q = N 1 ĥ q+1,k DN 1,K+1 0,1, q = N 1 K + 1 odd 0, otherwise, which had to be shown. If q = N 2 we only have div e = 0 and e = 0. Following the arguments used in the case K = 1 we obtain for e the expansion (2.11) and reduce this representation similarly with the additional information div e = 0 to e = k=2,4, σ,m α N 2 k,σ,m P σ,m N 2,k + { β 1 β N 2 k,σ,m 1,1,1 Q 1,1 QN 2,k 1,1, q = 1 σ,m + k=2,3, 0, otherwise σ,m. Remark 2.3 yields with new constants e = e N 2,2 I I I q, 1 D N 2 I + ê 1,2 { R 1,1 0,1, q = 1 0, otherwise. Analogously we show the representation for H. Our proof is complete. 3 Electro-magneto static operators From now on we want to discuss the inhomogeneous, anisotropic (generalized) static Maxwell equations in an exterior domain Ω R N, 3 N N, 16

19 and fix a radius r 0 > 0 and some radii r n := 2 n r 0, n N, such that R N \ Ω U r0. We assume Ω to possess the Maxwell local compactness property MLCP [2, Definition 3.1], i.e. the inclusions R q (Ω) D q (Ω) L 2,q loc (Ω) are compact for all q. Furthermore, we remind of the cut-off functions η, ˆη and η from [2, (3.1), (3.2), (3.3)]. Let ε = Id +ˆε and µ = Id +ˆµ be two τ-c 1 -admissible ( see [5, Definition 2.1 and 2.2] ) transformations on q- and (q + 1)-forms with some τ 0, which will vary throughout this paper. The greek letter τ always stands for the order of decay of the perturbations ˆε and ˆµ. Moreover, we introduce the Dirichlet forms defined in [2, (3.4)] εh q t (Ω) = 0 R q t (Ω) ε 1 0D q t (Ω), t R and obtain our first result on the integrability of Dirichlet forms: Lemma 3.1 H q (Ω) = H q (Ω) = H q N < N 1(Ω) and even Hq (Ω) = H q (Ω) holds, if < N q / {1, N 1}. Remark 3.2 In particular if s > 1 N/2 then H q (Ω) is a closed subspace of L 2,q s(ω), the dual space of L 2,q s (Ω). If q / {1, N 1} this remains valid for s > N/2. Proof: Applying Theorem 2.6 with s = N/2 we have for E H q (Ω) N 2 E Ar0 = I Īq,0 e q,1 I D q I + êq,1 because of Īq,0 s =. By Remark 2.5 the sum is an element of L 2,q (Ω) and the second < N 2 term, which vanishes for q / {1, N 1}, lies in L 2,q < N 1(Ω). 2 ˆD q,1 N 2 One easily concludes with [5, Lemma 2.13 and Remark 2.14]: Corollary 3.3 Let s > 1 N/2. Then with closures taken in L 2,q s (Ω) rot R q 1 s 1(Ω) rot R q 1 s (Ω) div D q+1 s 1(Ω) div D q+1 s (Ω) H q (Ω) holds true. Here we denote by the orthogonality in L 2,q (Ω), i.e. in the sense of the usual L 2,q s (Ω)-L 2,q s(ω) duality. 17

20 Let us now closely follow the constructions in [13, p. 1631] or [16, p. 1511]. For some t R we consider vector spaces of the form U q t (Ω) := V q t (Ω) + ηd q (I), I I q with some Hilbert spaces V q t (Ω) L 2,q t (Ω), e.g. V q t (Ω) = R q t (Ω) ε 1 D q t (Ω). Because of the smoothness and integrability properties of our tower-forms we have U q t (Ω) = V q t (Ω) ηd q (I t ) ( : direct sum). We define an inner product in U q t (Ω), such that in V q t (Ω) the original scalar product is kept; ηd q (I t ) = {ηd q I : I I t} is an orthonormal system; the sum V q t (Ω) ηd q (I t ) = V q t (Ω) ηd q (I t ) is orthogonal. Clearly we extend these definitions replacing ηd q (I) by ηr q (J). U q t (Ω) is a Hilbert space, if and only if I t is finite, and independent of the cut-off function η in the following sense: If ξ is another cut-off function with the same properties, then the two Hilbert spaces V q t (Ω) ξd q (I t ) and V q t (Ω) ηd q (I t ) have different scalar products but coincide as sets and the identity mapping is a topological isomorphism between them with norm depending on η and ξ. ( We note E + ηt = E + (η ξ)t + ξt and supp(η ξ) is a compact subset of Ω. ) We introduce the special forms Ďs q,k and Řs q+1,k ˆD q,k s ˆR q+1,k s replacing < by in the definitions of and in (2.7) and (2.8). In particular we have for K = 1 R 1,1 0,1, q = 1 s N/2 1 Ďs q,1 = Řq,1 s = D N 1,1 0,1, q = N 1 s N/2 1 0, otherwise and for K = 2 Ď q,2 s = D 0,2 R 1,1 0,1, q = 0 s N/2 2 0,1, q = 1 s N/2 1 0, otherwise, Ř q+1,2 s = D N 1,1 R N,2 0,1, q = N 2 s N/2 1 0,1, q = N 1 s N/2 2 0, otherwise. Moreover, we set Ďq,K s := Lin Ďq,K s and Řq+1,K s := Lin Řq+1,K s. 18

21 We need one more technical lemma: Lemma 3.4 Let ˆr r 0, s N/2 and E L 2,q (Ω) be a solution of E = 0 in N Aˆr. 2 Then E is represented in Aˆr by = h q,2 J J J q, 1 s e q,2 I I Īq, 1 s e q,1 I I Īq,0 s k,σ,m, σ 2+s N/2 E k,σ,m, σ>2+s N/2 Rq J + ȟq,2 Řq,2 s D q I + ěq,2 Ďq,2 s D q I + ěq,1 Ďq,1 s β k,σ,m Q q,k σ,m with unique constants β k,σ,m, e q,1 I, e q,2 I, h q,2 J exceptional forms do not vanish. Thus, E L 2,q s (Ω) ηr q ( J q, 1 s ) ηřq,2 s ) ηďq,2 ηd q (Īq, 1 s ηd q (Īq,0 s β k,σ,m Q q,k σ,m, if rot E L 2,q+1 s+1 (Aˆr ), if div E L 2,q 1 s+1 (Aˆr ), if rot E L 2,q+1 s+1 (Aˆr ) and div E L 2,q 1, otherwise s+1 (Aˆr ) C and ě q,1, ě q,2, ȟq,2 C, provided that the s ) ηďq,1 holds. We note ηd q (Īq,0 s ) ηďq,1 s = ηr q ( J q,0 s ) ηřq,1 s, if rot E L 2,q+1 s+1 (Aˆr ), if div E L 2,q 1 s+1 (Aˆr ), if rot E L 2,q+1 s+1 (Aˆr ) and div E L 2,q 1 s+1 (Aˆr ) s. Proof: Similarly to (2.11) we have E Aˆr = k,σ,m β k,σ,m Q q,k σ,m L 2,q < N 2 2(Aˆr) because E L 2,q (Ω) and therefore, no potential forms P N σ,m q,k occur. Thus, only the 2 case s N/2 2 is interesting and may be assumed from now on. With the properties of Q q,k σ,m we obtain ( ) except of the forms Q 0,4 0,1, Q 1,1 1,1, Q N 1,2 1,1 and Q N,4 0,1 rot E Aˆr div E Aˆr = σ,m = σ,m β 2,σ,m Q q+1,3 σ+1,m + σ,m β 1,σ,m Q q 1,3 σ+1,m + σ,m β 4,σ,m Q q+1,1 σ+1,m, β 4,σ,m Q q 1,2 σ+1,m 19

22 with new constants satisfying β k,σ,m = 0 β k,σ,m = 0 βk,σ,m = 0. By Remarks 2.3 and 2.5 and rot E L 2,q+1 s+1 (Aˆr ) resp. div E L 2,q 1 s+1 (Aˆr ) we see that all coefficients β k,σ,m with σ 2 + s N/2 and k = 2, 4 resp. k = 1, 4 vanish except of β 4,0,1 (for q = 0), β 1,1,1 (for q = 1), β 2,1,1 (for q = N 1) and β 4,0,1 (for q = N). Let us discuss the case div E L 2,q 1 s+1 (Aˆr ). Then we get by Remark 2.3 in Aˆr E β k,σ,m Q q,k σ,m Because = = k,σ,m, k=2,3, σ 2+s N/2 I Īq, 1 s k,σ,m, σ>2+s N/2 k,σ,m, σ>2+s N/2 β k,σ,m Q q,k σ,m + e q,2 I D q I + ěq,2 Ďq,2 s. β k,σ,m ηq q,k σ,m L 2,q s (Ω) we obtain ηe I Īq, 1 s e q,2 I β 4,0,1 Q 0,4 0,1, q = 0 β 1,1,1 Q 1,1 1,1, q = 1 s N/2 1 0, otherwise ηd q I ěq,2 ηďq,2 s L 2,q (Ω) s and thus E L 2,q s (Ω) ηd q (Īq, 1 s ) ηďq,2 The other two cases may be shown in the same way. s. We recall from [14, p. 1034] the set of exceptional weights I := { n + N/2 : n N 0 } { 1 n N/2 : n N0 }. (3.1) (There this set is denoted by J and here we only need it in the Hilbert space case p = 2.) Moreover, we define for s > 1 N/2 0D q s(ω) := 0 D q s(ω) H q (Ω), 0R q s(ω) := 0R q s (Ω) H q (Ω) (3.2) and put as usual 0 D q (Ω) := 0 D q 0(Ω) and 0 R q (Ω) := 0 R q 0(Ω). Now we are ready to establish our electro-magneto static solution theory and formulate a first result for homogeneous, isotropic media: 20

23 Lemma 3.5 Let 1 N/2 < s / I. Then DIV q+1 s 1 : D(DIV q+1 s 1) 0 D q s(ω) H div H is a continuous and surjective Fredholm operator on its domain of definition ( ( R D(DIV q+1 q+1 s 1) := s 1(Ω) D q+1 with kernel H q+1 (Ω). 0R q+1 (Ω) D q+1 (Ω) > N > N 2 2 s 1(Ω) ) ηr q+1 ( J q+1,0 s 1 ) ηřq+1,1 s 1 ) 0 R q+1 loc (Ω) Proof: Let us abbreviate DIV := DIV q+1 s 1. div and rot map tower-forms from ηr q+1 ( J q+1,0 s 1 ) ηřq+1,1 s 1 to compactly supported forms. Thus, with Corollary 3.3 and Remark 2.5 DIV is well defined, linear and continuous because ηr q+1 ( J q+1,0 s 1 ) ηřq+1,1 s 1 is finite-dimensional. Lemma 3.1 yields N(DIV) H q+1 (Ω) = H q+1 (Ω). > N 2 On the other hand let H H q+1 (Ω). Then by Lemma 3.4 we obtain and therefore, H L 2,q+1 s 1 H ( R q+1 s 1(Ω) D q+1 (Ω) ηr q+1 ( J q+1,0 s 1 ) ηřq+1,1 s 1 s 1(Ω) ) ηr q+1 ( J q+1,0 s 1 ) ηřq+1,1 s 1, which implies H N(DIV), i.e. N(DIV) = H q+1 (Ω). So only the surjectivity of DIV demands a proof. For this let F 0 D q s(ω) and ˆF be its extension by zero in R N. Applying [14, p. 1037, Theorem 4] we decompose ˆF =: F D + F R + F S 0 D q s 0 R q s S q s (3.3) ( Here S q s = C,η Lin{Q q,4 σ,m : σ < s N/2} is a finite-dimensional subspace of C,q( R N \ {0} ) and C,η denotes the commutator of the Laplacian and the multiplication with η. ) and set f := F D I Īq,1 s 2 F D, D q I L 2,q C,ηD q I. (3.4) The duality products are well defined because by Remark 2.5 I Īq,1 s 2 e(i) < s 1 N/2 D q I = + D q,1 e(i),c(i) L2,q s. 21

24 Using Remark 2.4 we get for I Īq,1 C,η D q I = div rot(ηdq I ) + rot div(ηdq I }{{} ) 0 D q vox =0 and hence f 0 D q s. Moreover, we compute for all I, Ĩ Īq,1 with Remark 2.3 and [14, p. 1035, (73)] C,η D q I, Dq = C Ĩ L 2,q,η D q,1 e(i),c(i), + D q,1 L e(ĩ),c(ĩ) 2,q 1 = 2 e(ĩ) + N C,η Q q,2 e(i)+1,c(i), P q,2 L e(ĩ)+1,c(ĩ) 2,q = α( N, e(i) + 1 ) 2 } e(ĩ) {{ + N δ e(i),e( Ĩ) δc(i),c(ĩ). } =1 Thus, for all I Īq,1 s 2 we have and finally by Remark 2.3 f, D q I L 2,q = 0 f 0 D q s ( ) P q,2. <s N 2 In particular f belongs to the range of the operator B from [14, p. 1039, Theorem 7], if 1 q N 1 or s N/2. To get this likewise in the case q = 0 and s > N/2, additionally f has to be perpendicular to 1 = + D 0,0 0,1. This can be achieved replacing f in (3.4) by f := f F D, + D0,1 0,0 L 2,0 C,η D 0,2 0,1. Utilizing [14, p. 1039, Theorem 7] we then obtain some h D q+1 s 1 0 R q+1 s 1 solving div h = f. By the regularity result [2, Theorem 3.6 (ii)] we even have h H 1,q+1 s 1. Thus, the ansatz H := η h + Φ (3.5) transforms the system under consideration div H = F, rot H = 0 with our assumptions and Corollary 3.3 into the system rot Φ = rot(ηh) = C rot,η h 0R q+2 vox (Ω), div Φ = F div(ηh) 0 D q s(ω) 0 D q (Ω). >1 N 2 (3.6) 22

25 So to solve this system using the classical solution theory developed in [9] we only have to show F div(ηh) L 2,q (Ω). For 1 q N 1 we have in Ω F div(ηh) = div ( (1 η)h ) + F R + F S + F D, D q I L 2,q C,ηD q I I Īq,1 s 2 and therefore, F div(ηh) F R L 2,q vox(ω). This remains true even in the case q = 0, s > N/2. Furthermore, (3.3) and the vanishing divergence of F yield F R 0 R q s and div F R = 0 in Ω \ supp F S. By Theorem 2.6 and Remark 2.5 or as in the proof of Lemma 3.1 we obtain F R L 2,q < N 2 1(Ω) L2,q (Ω). Now we are able to apply [9, Satz 2] and get some Φ R q+1 1 (Ω) D q+1 1 (Ω) solving the system (3.6). Moreover, in A r2 hold and thus rot Φ = 0, rot div Φ = rot F R Φ Ar2 = 0, (div Φ, rot Φ) L 2,q s (Ω) Lvox 2,q+2 (Ω). Lemma 3.4 yields Φ L 2,q+1 s 1 (Ω) ηr q+1 ( J q+1,0 s 1 ) ηřq+1,1 s 1 and hence Φ ( R q+1 s 1(Ω) D q+1 s 1(Ω) ) ηr q+1 ( J q+1,0 s 1 ) ηřq+1,1 s 1. This shows H D(DIV) and div H = F, which completes the proof. Lemma 3.6 Let 1 N/2 < s / I. Then ROT q s 1 : D(ROT q s 1) 0R q+1 s (Ω) E rot E is a continuous and surjective Fredholm operator on its domain of definition D(ROT q s 1) := with kernel H q (Ω). ( ( R q s 1(Ω) D q s 1(Ω) ) ) ηd q (Īq,0 s 1) ηďq,1 s 1 0 D q loc (Ω) R q (Ω) > N 0 D q (Ω) > N

26 Proof: The proof is analogous to the last one but more simple because the extension by zero into R N of G 0R q+1 s (Ω) is still an element of 0 R q+1 s, such that we do not need a Helmholtz decomposition like in (3.3). The roles of D q I are now played by R q+1 q+1,1 J, J J s 2, and we use [14, p. 1037, Theorem 5] instead of [14, p. 1039, Theorem 7]. In the special case q = N 1 ( formerly q = 0 ), s > N/2 we have to guarantee the orthogonality to 1 = + R N,0 0,1 with the help of R N,2 0,1. We can generalize Lemma 3.4 to Lemma 3.7 Let ˆr r 0 and E L 2,q (Ω). If N 2 hold with some N/2 < s / I, then E L 2,q s div E L 2,q 1 s+1 (Aˆr ) and rot E L 2,q+1 s+1 (Aˆr ) (Ω) ηd q (Īq,0 s ) ηďq,1 s = L 2,q s (Ω) ηr q ( J q,0 s ) ηřq,1 Proof: Let ϕ := η ( r/(2ˆr) ). Then ϕe R q (A N r0 ) D q (A N r0 ) and 2 2 div(ϕe) 0 D q 1 s+1(a r0 ), rot(ϕe) 0 R q+1 s+1(a r0 ). By Lemma 3.5 and Lemma 3.6 there exists some e ( R q s(a r0 ) D q s(a r0 ) ) ηd q (Īq,0 s ) ηďq,1 s, s. such that rot e = rot(ϕe) and div e = div(ϕe). Thus, e ϕe H q (A N r0 ) is a Dirichlet form and therefore, e ϕe H q (A r0 ) by Lemma 3.1. Extending e by zero into 2 Ω we get with Lemma 3.4 e, e ϕe L 2,q s (Ω) ηd q (Īq,0 s ) ηďq,1 s. Thus, ϕe and E = (1 ϕ)e + ϕe are elements of L 2,q s (Ω) ηd q (Īq,0 s ) ηďq,1 s. Now we consider inhomogeneous, anisotropic media. First we generalize Lemma 3.1: Lemma 3.8 Let τ > 0. Then εh q (Ω) = N ε H q (Ω) = ε H q < N 1(Ω). 2 2 For q / {1, N 1} even ε H q (Ω) = ε H q (Ω) holds. < N 2 Remark 3.9 In particular ε H q (Ω) L 2,q s(ω), if s > 1 N/2. Moreover, in the case q / {1, N 1} this inclusion remains valid for s > N/2. 24

27 Proof: Let E ε H q (Ω). By regularity, e.g. [2, Corollary 3.8 (ii)], E belongs to N 2 H 1,q (A N r0 ) and thus in A r0 2 rot E = 0, div E = div ˆεE L 2,q 1 N 2 +1+τ(A r 0 ) hold true. We assume w. l. o. g. τ N/2 / I and obtain by Lemma 3.7 E L 2,q (Ω) ηd q τ N (Īq,0 τ ) N ηďq,1 2 2 τ N 2. By Remark 2.5 we get ηd q (Īq,0 τ ) N ηďq,1 2 τ N 2 L 2,q <s q (Ω) with s q := N/2 δ q,1 δ q,n 1. If τ N/2 s q, we get E L 2,q <s q (Ω), i.e. E ε H<s q q (Ω). If τ N/2 < s q, we only have E L 2,q (Ω), i.e. E τ N ε H q (Ω). Repeating this τ N 2 2 argument leads us after finitely many τ-steps to E ε H<s q q (Ω). Using Helmholtz decompositions, e.g. [2, (3.5)], it is easy to show that the dimension d q of the Dirichlet forms ε H q (Ω) does not depend on the transformation ε, i.e. d q = dim H q (Ω). Furthermore, d q is finite since Ω has the MLCP. Moreover, we obtain Corollary 3.10 Let τ > 0 and N/2 t < N/2 1. Then dim ε H q t (Ω) = d q <. If q / {1, N 1} the first equation holds even for N/2 t < N/2. Lemma 3.8 yields a generalization of Corollary 3.3: Corollary 3.11 Let τ > 0 and s > 1 N/2. Then with closures in L 2,q s (Ω) rot R q 1 s 1(Ω) rot R q 1 s (Ω) ε H q (Ω) ε, div D q+1 s 1(Ω) div D q+1 s (Ω) ε H q (Ω) hold. Here we denote by ε the orthogonality with respect to the ε, L 2,q (Ω)-duality. 25

28 Lemma 3.12 Let 1 N/2 < s / I and τ > max{0, s N/2}, τ s. Then µdiv q+1 s 1 : D( µ DIV q+1 s 1) 0 D q s(ω) H div H is a continuous and surjective Fredholm operator on its domain of definition D( µ DIV q+1 s 1) ( (µ 1 := R q+1 s 1(Ω) D q+1 s 1(Ω) ) ) ηr q+1 ( J q+1,0 s 1 ) ηřq+1,1 s 1 µ 1 0R q+1 loc (Ω) µ 1 0 R q+1 > N 2 with kernel µ 1 µ 1Hq+1 (Ω). (Ω) D q+1 (Ω) > N 2 Proof: We set DIV := µ DIV q+1 s 1 and follow in close lines the proof of Lemma 3.5. A form ηh ηr q+1 ( J q+1,0 s 1 ) ηřq+1,1 s 1 belongs to H 1,q+1 and the assumptions on τ yield < N 1 2 rot(µηh) = C rot,η H + rot(ˆµηh) L 2,q+2 < N +τ(ω) L2,q+2 s (Ω). 2 Thus, DIV is well defined and clearly linear and continuous. By Lemma 3.8 we obtain µn(div) µ 1H q+1 (Ω). Applying the regularity result [2, Corollary 3.8 (ii)] we get H µ 1 µ 1Hq+1 (Ω) H 1,q+1 < N 2 1(A r 1 ) and therefore, div H = 0, rot H = rot(ˆµh) L 2,q+2 < N 2 +τ(a r 2 ) L 2,q+2 s (A r2 ), which implies µ 1 µ 1Hq+1 (Ω) N(DIV) by Lemma 3.7. So it remains to show that DIV is surjective. Let F 0 D q s(ω). We follow exactly the arguments in Lemma 3.5 leading to the ansatz (3.5). By Corollary 3.3 the system is transformed into div H = F, rot µh = 0 rot µφ = rot(µηh) = C rot,η h rot(ˆµηh) 0R q+2 s+τ(ω), div Φ = F div(ηh) 0 D q s(ω) 0 D q (Ω). >1 N 2 (3.7) As in the proof of Lemma 3.5 we compute F div(ηh) L 2,q < N 1(Ω) and with τ s 2 we get additionally ( ) F div(ηh), rot(µηh) L 2,q (Ω) L 2,q+2 (Ω). (3.8) Thus, the generalized classical static solution theory from [2, Theorem 3.21 and Remark 3.22] yields some Φ µ 1 R q+1 1 (Ω) D q+1 1 (Ω) 26

29 solving the system (3.7). Clearly ηh H 1,q+1 s 1 implies ηh D(DIV). So our proof is complete, if we can show Φ D(DIV). Because of div Φ L 2,q s (Ω) this decomposition of Φ follows by Lemma 3.7 and the assumptions on τ, if e.g. rot Φ L 2,q+2 s (A r1 ). (3.9) By regularity, e.g. [2, Corollary 3.8 (ii)], Φ H 1,q+1 1 (A r1 ), i.e. rot Φ L 2,q+2 (A r1 ). Thus, we may assume s > 0 in (3.9). Because of rot Φ = rot(ˆµφ) rot(µηh) L 2,q+2 min{τ,s+τ} (A r 1 ) = L 2,q+2 τ (A r1 ) (3.10) we only have to discuss the case 0 < τ < s. From Lemma 3.7 (w. l. o. g. τ / I) we obtain Φ L 2,q+1 τ 1 (Ω) ηr q+1 ( J q+1,0 τ 1 ) ηřq+1,1 τ 1. If N/2 τ < s we get Φ L 2,q+1 < N 1(Ω) and with (3.10) rot Φ L2,q+2 N (A r1 ), i.e. 2 2 Φ H 1,q+1 < N 1(A r 1 ). (3.10) and the assumption τ > s N/2 show (3.9). In the other 2 case τ < min{s, N/2} we have ηr q+1 ( J q+1,0 τ 1 Once more we obtain Φ H 1,q+1 τ 1 ) ηřq+1,1 τ 1 = {0} and thus Φ L 2,q+1 τ 1 (Ω). (A r1 ) and with (3.10) rot Φ L 2,q+2 min{2τ,s+τ} (A r 1 ). After finitely many repetitions of this argument either lτ s or lτ N/2 with l N holds. By the arguments given above we achieve (3.9) in this case as well. Corollary 3.13 Let 1 N/2 < s / I and τ > max{0, s N/2}, τ s. Then µ 1 D( µ 1DIV q+1 s 1) ( ( R q+1 = s 1(Ω) µ 1 D q+1 s 1(Ω) ) ) ηr q+1 ( J q+1,0 s 1 ) ηřq+1,1 s 1 and with D( µ DIV q+1 s 1) := µ 1 D( µ 1DIV q+1 s 1) µ DIV q+1 s 1 : D( µ DIV q+1 s 1) 0 D q s(ω) H div µh is a continuous and surjective Fredholm operator with kernel µ H q+1 (Ω). 0 R q+1 loc (Ω) Proof: The assertions follow from the previous lemma, if we can show the first assertion of this corollary. But with the properties of τ this is clear because of µ 1( ηr q+1 ( J q+1,0 s 1 ( R q+1 s 1(Ω) µ 1 D q+1 ) ) ηřq+1,1 s 1 s 1(Ω) ) ηr q+1 ( J q+1,0 s 1 ) ηřq+1,1 s 1. 27

30 Analogously we obtain Lemma 3.14 Let 1 N/2 < s / I and τ > max{0, s N/2}, τ s. Then εrot q s 1 : D( ε ROT q s 1) 0R q+1 s (Ω) E rot E is a continuous and surjective Fredholm operator on its domain of definition with kernel ε H q (Ω). D( ε ROT q s 1) ( ( R q := s 1(Ω) ε 1 D q s 1(Ω) ) ) ηd q (Īq,0 s 1) ηďq,1 s 1 ε 1 0D q loc (Ω) R q (Ω) ε 1 > 0D q (Ω) N > N 2 2 Corollary 3.15 Let 1 N/2 < s / I and τ > max{0, s N/2}, τ s. Then ε 1 D( ε 1ROT q s 1) ( (ε 1 = R q s 1(Ω) D q s 1(Ω) ) ) ηd q (Īq,0 s 1) ηďq,1 s 1 0 D q loc (Ω) and with D( ε ROT q s 1) := ε 1 D( ε 1ROT q s 1) ε ROT q s 1 : D( ε ROT q s 1) 0R q+1 s (Ω) E rot εe is a continuous and surjective Fredholm operator with kernel ε 1 ε 1Hq (Ω). Remark 3.16 Let the assumptions of Lemma 3.12 be fulfilled and additionally ε, µ be two τ-c 1 -admissible transformations with τ > 0. Then we can characterize the ranges of µdiv q+1 s 1, µ DIV q+1 s 1 resp. ε ROT q s 1, ε ROT q s 1 by as well. 0D q s(ω) = 0 D q s(ω) ε H q (Ω) resp. 0 R q+1 s (Ω) = 0R q+1 (Ω) µ H q+1 (Ω) µ Proof: By Corollary 3.11 all operators are still well defined. Let us consider e.g. µdiv q+1 s 1 from Lemma Only the argument showing surjectivity has to be changed. Now (3.7) and (3.8) are replaced by ( ) ( F div(ηh), rot(µηh) 0D q (Ω) ε H q (Ω) ) ( 0R q+2 (Ω) H q+2 (Ω) ) but with [2, (3.5)] we see that the latter set equals div D q+1 (Ω) rot R q+1 (Ω) s and thus is independent of ε. Similarly we prove the assertion concerning the range of ε ROT q s 1. 28

31 4 Generalized electro-magneto statics For s > 1 N/2 we put W q s(ω) := 0 D q 1 s (Ω) 0R q+1 s (Ω) C dq and choose d q continuous linear functionals Φ l ν on ( R q s 1(Ω) ν 1 D q s 1(Ω) ) ηd q (Īq,0 s 1) ηďq,1 s 1 with d q νh q (Ω) N(Φ l ν) = {0} for some given 0-admissible transformation ν. We set Φ ν obtain l=1 := (Φ 1 ν,..., Φ dq ν ) and Theorem 4.1 Let s (1 N/2, ) \ I and τ > max{0, s N/2}, τ s as well as Then the operator D( ε Max q s 1) := ( R q s 1(Ω) ε 1 D q s 1(Ω) ) ηd q (Īq,0 s 1) ηďq,1 s 1. is a topological isomorphism. εmax q s 1 : D( ε Max q s 1) W q s(ω) E ( div εe, rot E, Φ ε (E) ) Remark 4.2 Let ν be a 0-admissible and λ be a τ-c 1 -admissible transformation on q-forms as well as { } d q θh l for θ {ε, λ} be some basis of l=1 θh q (Ω). Then for s > 2 N/2 we can choose the d q continuous linear functionals Φ l ε, for instance, as Φ l ε(e) := νe, ε h l L 2,q (Ω) or Φ l ε(e) := εe, λ h l L 2,q (Ω) or Φ l ε(e) := λe, λ h l L 2,q (Ω). Corollary 4.3 Let the assumptions of Theorem 4.1 be satisfied. Then holds and D( ε Max q s 1) := ε 1 D( ε 1Max q s 1) is a topological isomorphism. = ( ε 1 R q s 1(Ω) D q s 1(Ω) ) ηd q (Īq,0 s 1) ηďq,1 s 1 ε Max q s 1 : D( ε Max q s 1) W q s(ω) E ( div E, rot εe, Φ ε 1(εE) ) 29

32 Proof: By Corollary 3.13 and Lemma 3.14 ε Max q s 1 is continuous and with our assumptions clearly injective. Thus, by the bounded inverse theorem ε Max q s 1 is a topological isomorphism, if it is surjective. Let (f, G, γ) W q s(ω). Then a combination of Corollary 3.13 and Lemma 3.14 yields some Ê D( εmax q s 1) solving rot Ê = G and div εê = f and we are free in adding a Dirichlet form from εh q (Ω) to Ê. By our assumptions φ : εh q (Ω) C dq E Φ ε (E) is a topological isomorphism. Therefore, E := Ê + φ 1( γ Φ ε (Ê)) is the unique solution of the problem ε Max q s 1 E = (f, G, γ). From the properties of τ we get easily ε Max q s 1 = ε 1Max q s 1 ε, such that this operator is also a topological isomorphism, which proves the corollary. If s > 2 N/2 we have D( ε Max q s 1) L 2,q (Ω). Then by Lemma 3.8 and Re- >1 N 2 mark 3.9 the scalar products in Remark 4.2 are well defined and possible choices for Φ l ε. Actually we are interested in a (electro-magneto) static solution theory suited for the operator M from (2.4). Moreover, we want to define higher powers of a special static solution operator. The main tool for the iteration process are the tower-forms from section 2. (Until now essentially we only needed the ground-forms of height zero to establish our solution theory.) Thus, we expect that the heights of the towerparts of our solutions will grow in each step of the iteration, which implies decreasing integrability features of these solutions. But to guarantee uniqueness of the solutions, we always have to project along the Dirichlet forms. Therefore, it makes no sense to proceed with orthogonality constraints with respect to the Dirichlet forms anymore and we are forced to work with orthogonality constraints utilizing forms with compact supports in R N, such that the duality products with our tower-forms are still well defined. To this end we introduce from [11, p. 41] for all q finitely many smooth forms B q (Ω) := { b q 1,..., b q d q }, B q (Ω) := {b q 1,..., b q d q }, where the latter set is only defined for q 1, with compact resp. bounded support in Ω and the following properties: For all 0-admissible transformations ν B q (Ω) 0 R q vox (Ω) is linearly independent modulo rot R q 1 (Ω) and νh q (Ω) B q (Ω) ν = {0} holds. The orthogonal projections of B q (Ω) in 0 R q (Ω) along rot R q 1 (Ω) on νh q (Ω) form a basis of the Dirichlet forms ν H q (Ω) ; 30