Kristensson, Gerhard; Stratis, Ioannis; Wellander, Niklas; Yannacopoulos, Athanasios

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1 The exterior Calderón operator for non-spherical objects Kristensson, Gerhard; Stratis, Ioannis; Wellander, Niklas; Yannacopoulos, Athanasios 2017 Document Version: Publisher's PDF, also known as Version of record Link to publication Citation for published version (APA): Kristensson, G., Stratis, I., Wellander, N., & Yannacopoulos, A. (2017). The exterior Calderón operator for nonspherical objects. (Technical Report LUTEDX/(TEAT-7259)/1-43/(2017); Vol. TEAT-7259). The Department of Electrical and Information Technology. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. L UNDUNI VERS I TY PO Box L und

2 (TEAT-7259)/1-43/(2017): G. Kristensson et al., The exterior Calderón operator for non-spherical objects CODEN:LUTEDX/(TEAT-7259)/1-43/(2017) The exterior Calderón operator for non-spherical objects Gerhard Kristensson, Ioannis G. Stratis, Niklas Wellander, and Athanasios N. Yannacopoulos Electromagnetic Theory Department of Electrical and Information Technology Lund University Sweden

3 Gerhard Kristensson Department of Electrical and Information Technology Electromagnetic Theory Lund University P.O. Box 118 SE Lund Sweden Ioannis G. Stratis Department of Mathematics National and Kapodistrian University of Athens Panepistimioupolis GR Zographou Athens Greece Niklas Wellander Swedish Defense Research Agency, FOI P.O. Box 1165 SE Linköping Sweden Athanasios N. Yannacopoulos Department of Statistics Athens University of Economics and Business Patision 76 GR Athens Greece Editor: Mats Gustafsson c G. Kristensson et al., Lund, January 10, 2018

4 3 Abstract This paper deals with the exterior Calderón operator for Lipschitz surfaces. We present a new approach of nding the norm of the exterior Calderón operator for this class of surfaces. The basic tool in the treatment is the set of eigenfunctions and eigenvalues to the Laplace-Beltrami operator for the surface. The norm is obtained as an eigenvalue problem of a quadratic form containing the exterior Calderón operator. The connection of the exterior Calderón to the transition matrix for a perfectly conducting surface is analyzed. 1 Introduction The exterior Calderón operator maps the tangential scattered electric surface eld to the corresponding magnetic surface eld. This operator is also called the Poincaré- Steklov operator, and its discretization is often called the Schur complement. It has been studied intensively during many years, see e.g., [10, 22, 26]. It is related to the Dirichlet-to-Neumann map for the scalar Helmholtz equation. The exterior Calderón map is instrumental in the analysis of the solution to the exterior solution of the scattering problem. In fact, it is strongly related to the solution of the scattering problem by a perfectly conducting (PEC) obstacle, which is a subject we analyze in Section 5. The norm of the exterior Calderón operator quanties the largest amplication factor of the surface elds. This observation gives a physical interpretation of the value of the norm of the exterior Calderón operator. A new way of nding this norm is presented in this paper. The key ingredient in this analysis is the set of eigenfunctions to the Laplace-Beltrami operator of the surface. These eigenfunctions and the corresponding eigenvalues are intrinsic to the surface and constitute an excellent tool for further analysis; the literature on this subject of nding these eigenfunctions and eigenvalues is extensive, see, e.g., [3, 14, 23, 33]. Explicit values of the norm of the exterior Calderón operator have only been obtained for the sphere case [22, 26] and the planar case [2, 10]. In this paper, we present a new way to explicitly nd the norm for non-spherical obstacles. There is very extended literature containing detailed studies regarding various properties of vector potentials, bounded integral operators, trace theorems, etc, in non-smooth domains; a representative list would certainly include [1, 3, 5, 8, 9, 13, 15, 16, 24, 25]. The nal expression of the norm for a non-spherical obstacle is related to an eigenvalue problem of a quadratic form containing the Calderón matrrix. An outline of the organization of the contents in this paper is now presented. In Section 2, the statement of the problem is introduced, the exterior Calderón operator is dened, and the useful integral representation of the scattered eld is presented. The intrinsic generalized harmonics (both scalar and vector valued) are introduced in Section 3, and these functions are used in Section 4. That section also contains a constructive method to compute the norm of the exterior Calderón operator for nonspherical obstacles. The connection between the exterior Calderón operator and the transition matrix of the corresponding perfectly conducting obstacle is claried in

5 4 Ω e ˆν Ω Figure 1: Typical geometry of the scattering problem in this paper. The domain Ω, its boundary and the exterior Ω e. Section 5. The paper is concluded with some nal remarks in Section 6 and several appendices with some relevant details of the analysis presented in this paper. 2 Formulation of the scattering problem In this section, we present the geometry of the problem and the solution of the scattered eld in the exterior region. 2.1 Statement of problem (E) Let Ω be an open, bounded, domain in R 3 with simply connected 1 Lipschitz boundary. The outward pointing unit normal is denoted by ˆν. We denote the exterior of the domain Ω by Ω e = R 3 \ Ω, which is assumed to be simply connected. See Figure 1 for a typical geometry. The Maxwell equations in the exterior region are given by 2 (we adopt the time convention e iωt ) E(x) = ikh(x) x Ω e. (2.1) H(x) = ike(x) The wave number k = ω/c is assumed to be a positive constant, where ω is the angular frequency of the elds, and c is the speed of light in the exterior medium. 1 For non simply connected boundary see Remark We use scaled electric and magnetic elds, i.e., the SI-unit elds E SI and H SI are related to the elds E and H used in this paper by E SI (x) = E(x) ɛ0 ɛ, H SI(x) = H(x) µ0 µ, where the permittivity and permeability of vacuum are denoted ɛ 0 and µ 0, respectively, and the relative permittivity and permeability of the exterior material are denoted ɛ and µ, respectively.

6 5 Throughout this paper vectors in C 3 are typed in italic boldface. Unit vectors are denoted by a caret (ˆ). In the region Ω e, the (scattered) elds satisfy the time-harmonic Maxwell equations (2.1) and the Silver-Müller radiation condition at innity, and we are looking for solutions E sc and H sc in the space H loc (curl, Ω e ). The trace operators π and γ on C(Ω e ; C 3 ) are given by π(u) = ˆν (u Ω ˆν) and γ(u) = ˆν u Ω, respectively, and in the case that u belongs to H loc (curl, Ω e ), the elds have traces on Ω belonging to H 1/2 (div, ), more precisely we have (γ(e sc ), γ(h sc )) H 1/2 (div, ) H 1/2 (div, ), see [27] for the denition and the properties of the trace operators in H loc (curl, Ω e ). The exterior Calderón operator, or admittance operator C e, is dened as the mapping of the tangential component of the scattered electric eld to the tangential component of the scattered magnetic eld on the boundary of Ω [10]. We use the solution of a specic exterior problem to make the denition precise. Consider the following exterior problem where the trace of the scattered electric eld on the boundary is given by a xed vector m H 1/2 (div, ), 3 1) (E sc, H sc ) H loc (curl, Ω e ) H loc (curl, Ω e ) Esc (x) = ikh sc (x) 2) x Ω e H sc (x) = ike sc (x) ˆx E sc (x) H sc (x) = o(1/x) 3) or as x ˆx H sc (x) + E sc (x) = o(1/x) uniformly w.r.t. ˆx 4) γ(e sc ) = m H 1/2 (div, ) (Problem (E)), (2.2) where x = x. This problem has a unique solution [2, 10, 18], and a brief sketch of the proof is found in Appendix C. The following theorem represents the solution to Problem (E): Theorem 2.1. Let E sc and H sc be the solution of Problem (E). Then the elds satisfy the integral representations 1 ik g(k, x x )γ(h sc )(x ) ds } + g(k, x x )γ(e sc )(x ) ds E sc (x), x Ω e = 0, x Ω, 3 The source m can be interpreted as a magnetic current density.

7 6 and 1 ik g(k, x x )γ(e sc )(x ) ds } + where the scalar Green function is g(k, x x )γ(h sc )(x ) ds H sc (x), x Ω e = 0, x Ω, g(k, x x ) = eik x x 4π x x. The proof of this theorem is found in e.g., [18]. The second (lower) term of the integral representation, i.e., x Ω, is usually called the extinction part of the integral representation. 2.2 Denition of the exterior Calderón operator We now dene the exterior Calderón operator C e. Denition 2.1. The exterior Calderón operator C e is dened as C e : m γ(h sc ), H 1/2 (div, ) H 1/2 (div, ), where m = γ(e sc ) and the elds E sc and H sc satisfy Problem (E) in (2.2). We notice that the exterior Calderón operator C e is uniquely dened for all m H 1/2 (div, ), since Problem (E) has a unique solution in H loc (curl, Ω e ) H loc (curl, Ω e ) for any m H 1/2 (div, ). Details on the space H 1/2 (div, ) and its dual space H 1/2 (curl, ) are given in [10] and [26]. Theorem 2.2. The exterior Calderón operator dened in Denition 2.1 has the following properties [10]: 1. Re C e (m) (ˆν m ) ds > 0 for all m H 1/2 (div, ), m 0, (2.3) where ds denotes the surface measure of, and the star denotes the complex conjugation. 2. (C e ) 2 = I on H 1/2 (div, ), 3. The exterior Calderón operator is a boundedly invertible linear map in the space H 1/2 (div, ), and consequently there exist constants 0 < Θ C Θ C, such that Θ C m H 1/2 (div,) C e (m) H 1/2 (div,) Θ C m H 1/2 (div,).

8 7 4. The exterior Calderón operator is independent of the material properties inside the domain Ω. From Item 2 we conclude that the norm of the exterior Calderón operator satises C e H 1/2 (div,) 1, and also that the constants in Item 3 can be chosen as Θ C = 1/ C e H 1/2 (div,) and Θ C = C e H 1/2 (div,). Notice, that if we dene the Calderón operator with an extra imaginary unit (i), the Calderón operator becomes its own inverse. 2.3 Integral equation approach The result in Theorem 2.1 and the exterior Calderón operator can also be put in a surface integral equation setting. The following theorem is important for the analysis in this paper and proved in [18, Th. 5.52]: Theorem 2.3. Let Q be a bounded domain such that Q. 1. Dene the operators L and M. ) ( Lf (x) = g(k, x x )f(x ) ds ) x Q. ( Mf (x) = g(k, x x )f(x ) ds These operators are well dened and bounded from the space H 1/2 (div, ) into the space H(curl, Q). 2. For f H 1/2 (div, ), the elds F = Mf and F = Lf satisfy γ(f ) + γ(f ) = f, γ( F ) + γ( F ) = 0. The notation ± refers to the trace of the eld taken from the outside (+) or the inside ( ) of, respectively. In particular, F C (R 3 \ ; C 3 ), and F satises ( F ) k 2 F = 0 in R 3 \. Furthermore, the functions F and F satises the Silver-Müller radiation conditions, i.e., for F we get ikˆx F F = o(1/x) or as x, ˆx ( F ) + ikf = o(1/x) uniformly w.r.t. ˆx. 3. The traces L and M dened by Lf = γ( Lf) Mf = 1 ( ) γ( Mf) + γ( Mf) 2 + f H 1/2 (div, ), are bounded from H 1/2 (div, ) into itself.

9 8 4. For f H 1/2 (div, ), the elds F = Mf and F = Lf have traces γ(f ) ± = ± 1 2 f + Mf γ( F ) ± = Lf. 5. The operator L is the sum L = Î + K of an isomorphism Î from H 1/2 (div, ) onto itself and a compact operator K. 6. The operator L can be written as Lf = Sdiv f + k 2 Sf, f H 1/2 (div, ), where the scalar single layer potential operator S is dened as (Sf) (x) = g(k, x x )f(x ) ds, x, where the surface integral is interpreted as a generalized integral (punctured surface by a circle). The corresponding vector-valued operator S is denoted (Sf) (x) = g(k, x x )f(x ) ds, x, which is interpreted as the operator S applied to each Cartesian components of the tangential vector eld f. The surface divergence, div, is dened in e.g., [10, 27, 31]. Theorem 2.4. The exterior Calderón operator satises 1 2 Ce (m) MC e (m) = 1 ik Lm, for each m = γ(e sc ) H 1/2 (div, ), where Lm = γ ( (Sdiv m)) + k 2 γ (Sm). Proof. From the second representation in Theorem 2.1, we get by letting m = γ(e sc ) and C e (m) = γ(h sc ), g(k, x x )C e (m)(x ) ds H sc (x), x Ω e 0, x Ω = 1 ik g(k, x x )m(x ) ds }. We intend to take the trace γ of this equation. In this limit process, the left-hand side becomes 1 2 Ce (m) + MC e (m), by the the result of Theorem 2.3. This result holds, irrespectively from which side the limit is taken. The right-hand side has the limit 1 ik Lm = 1 γ ( (Sdiv m)) + k 2 γ (Sm) }, m H 1/2 (div, ), ik and the result of the theorem follows.

10 9 3 Generalized harmonics In this section we give a review of two introduced dierential operators that act on scalars and vectors, respectively. The eigenfunctions of these operators provide bases for L 2 () and T L 2 (; C 2 ), respectively. They are well suited for expansion of the traces of solutions to the Maxwell equations. The spherical surface case yields the well known vector spherical harmonics, see Appendix B. The scalar Laplace-Beltrami operator on acting on a scalar eld f is dened as [27] f def = div grad f = curl curl f, (3.1) and the vector Laplace-Beltrami operator on acting on a tangential vector eld f is dened as f def = grad div f curl curl f. The dierential operators operating on the boundary are dened in Appendix A.1. The scalar Laplace-Beltrami operator has a countable set of eigenfunctions in L 2 (), which we denote Y n } n=1, and they satisfy, see Appendix D and [27] Y n = k 2 λ n Y n. (3.2) The eigenvalues are all real, positive, and the only possible accumulation point of the eigenvalues is at innity [20, 27]. We order the eigenvalues as λ 1 λ 2..., and normalizing the eigenfunctions Y n } n=1 in L 2 (), i.e., Y n Yn ds = δ nn, (3.3) we obtain an orthonormal basis in L 2 (), where, as above, a star denotes complex conjugation. Notice that the eigenvalues are scaled with the wave number k 2 in order to have a dimensionless quantity, and moreover the functions Y n have dimension inverse length, i.e., [m 1 ]. The following lemma is easily veried with the denitions of the scalar and vector Laplace-Beltrami operators. Lemma 3.1. If f satises for some Λ R, then f = Λf, curl f = Λ curl f, grad f = Λ grad f. Proof. Start with curl f = grad div curl f + curl curl curl f = curl curl curl f = curl f = Λ curl f.

11 10 since div curl f 0. We also have grad f = grad div grad f + curl curl grad f since curl grad f 0, and the lemma is proved. = grad div grad f = grad f = Λ grad f. By the use of this lemma, we can construct a set of eigenfunctions to the vector Laplace-Beltrami operator. Denition 3.1. The vector generalized harmonics are dened as Y 1n = 1 k λ n curl Y n, Y 2n = 1 k λ n grad Y n. These functions have dimension inverse length, i.e., [m 1 ]. Remark 3.1. Note that Y 1n and Y 2n are eigenfunction to the curl curl and grad div operators, respectively. We also observe that Y 1n belongs to the kernel of the grad div operator, and that Y 2n belongs to the kernel of the curl curl operator. The following lemma proves that the set Y τn, n = 1, 2,..., τ = 1, 2 is an orthonormal system on T L 2 (; C 2 ). Lemma 3.2. The vector functions Y 1n and Y 2n dened in Denition 3.1 constitute an orthonormal basis on T L 2 (; C 2 ), i.e., Y τn Y τ n ds = δ ττ δ nn. The vector functions satisfy where the dual index τ is 1 = 2 and 2 = 1. Moreover, ˆν Y τn = ( 1) τ+1 Y τn, τ = 1, 2, (3.4) curl Y τn = kδ τ,1 λn Y n, div Y τn = kδ τ,2 λn Y n, and Y τn = k 2 λ n Y τn. Proof. We start by noticing that Y 1n and Y 2n both are tangential to, by the denition of the operators curl and grad. Equations (3.1), (3.2), (3.3), the relations curl u, v T L 2 () = u, curl v L 2 (), div u, φ L 2 () = u, grad φ T L 2 (;C 2 ),

12 11 curl grad φ = 0, and curl u = grad u ˆν imply Y 1n Y 1n ds = 1 k 2 λ n λ n 1 = k 2 λ n λ n curl Y n curl Y n Y n curl curl Y n ds = 1 k 2 λ n λ n ds = λ n λn λ n Y n Y n ds Y n Y n ds = δ nn, and Y 2n Y 2n ds = 1 k 2 λ n λ n 1 = k 2 λ n λ n grad Y n grad Y n Y n div grad Y n ds = 1 k 2 λ n λ n = λ n λn λ n ds Y n Y n ds Y n Y n ds = δ nn, and Y 1n Y 2n ds = 1 k 2 λ n λ n curl Y n grad Y n = 1 k 2 λ n λ n ds Y n curl grad Yn ds = 0. Moreover, and ˆν Y 1n = 1 k λ n ˆν curl Y n = 1 k λ n ˆν (grad Y n ˆν) ˆν Y 2n = ˆν (ˆν Y 1n ) = Y 1n. The nal statements are easily proven by curl Y τn = 1 k λ n δ τ,1 Y n = kδ τ,1 λn Y n, = 1 k λ n grad Y n = Y 2n, and div Y τn = 1 k λ n δ τ,2 Y n = kδ τ,2 λn Y n,

13 12 and Y τn = grad div Y τn + curl (curl Y τn ) = kδ τ,2 λn grad Y n + kδ τ,1 λn curl Y n = k 2 λ n Y τn, and the lemma is proved. 4 Expansions in the orthonormal basis In this section, for simplicity we assume that the surface is simply connected. 4 Any m H 1/2 (div, ) has a convergent Fourier expansion on the surface : m = e τn Y τn, e τn = m, Y τn T L 2 (;C 2 ) = m Y τn ds, (4.1) τn where e τn l 1/2 (div). With the solution of Problem (E), the image of the exterior Calderón map C e (m) H 1/2 (div, ) has an expansion C e (m) = γ(h sc ) = i τn h τn Y τn, h τn = i γ(h sc ), Y τn T L 2 (;C 2 ), (4.2) and h τn l 1/2 (div). Note the bar over the index τ, which denotes the dual index in τ (1 = 2 and 2 = 1), and an extra factor of i. The reason for this choice is that the expansion coecients of the magnetic surface eld then has a simple relation to the corresponding coecients of the electric surface eld. Remark 4.1. We note that the expansion in (4.1) is a Helmholtz-Hodge decomposition of the elements m in H 1/2 (div, ) (and similarly of H 1/2 (curl, )) and that the L 2 -projection can be interpreted as a duality pairing between H 1/2 (div, ) and H 1/2 (curl, ), see Remark A.2. The mapping l 1/2 (div) e τn h τn l 1/2 (div) is a realization of the exterior Calderón operator. To every set of coecients e τn there exists a unique set of coecients h τn, and this association denes a linear relation between e τn h τn manifested by a matrix C (the exterior Calderón matrix) and h τn = C τn,τ n e τ n. (4.3) τ n The explicit form of the matrix is C τn,τ n = i Ce (Y τ n ), Y τn T L 2 (;C 2 ). (4.4) By the use of Lemma E.1 in Appendix E, we conclude that the exterior Calderón matrix C is invertible in l 1/2 (div). 4 For the generalization of the analysis to not simply connected surfaces, see Remark 4.2.

14 13 Lemma 4.1. The exterior Calderón matrix C τn,τ n = i Ce (Y τ n ), Y τn T L 2 (;C 2 ) dened by (4.3) and (4.4) satises τ n C τn,τ n C τ n,τ n = δ ττ δ nn, and its inverse is C 1 τn,τ n = C τn,τ n. Proof. The lemma is a consequence of (C e ) 2 = I on H 1/2 (div, ) and the expansions in (4.1), (4.2), and the map (4.3). We have m = C e (C e (m)), m H 1/2 (div, ), or due to continuity of the exterior Calderón operator e τn Y τn = i h τn C e (Y τn ) = i C τn,τ n e τ n Ce (Y τn ) τn τn τn τ n = C τn,τ n e τ n C τ n,τny τ n = C τ n,τ n e τ n C τn,τ n Y τn, τn τ n τ n τn τ n τ n since by (4.2) and (4.3) C e (Y τn ) = i C τ n,τny τ n. (4.5) τ n Orthogonality then implies e τn = C τ n,τ n C τn,τ n e τ n, τ n τ n or, since e τn is arbitrary C τn,τ n C τ n,τ n = C τ n,τ n C τn,τ n = δ τ,τ δ n,n. τ n τ n which ends the proof. Moreover, we have Lemma 4.2. The matrix is positive denite. 1 2i } ( 1) τ C τn,τ n ( 1)τ Cτ n,τn, Proof. The exterior Calderón operator satises (2.3) Re C e (m) (ˆν m ) ds > 0 for all m H 1/2 (div, ) m 0.

15 14 Insert the expansions of m and C e (m), see (4.1) and (4.2). We obtain Re i h τn e τ n Y τn (ˆν Y τ n ) ds = Re i ( 1) τ+1 h τn e τn > 0, τn τ n τn }} =δ ττ δ nn ( 1) τ +1 where we used ˆν Y τ n = ( 1)τ +1 Y τ n, see (3.4) in Lemma 3.2. This implies Im e τn( 1) τ C τn,τ n e τ n > 0, e τn l 1/2 (div) not all e τn = 0. τn τ n Rewrite the imaginary part explicitly and change summation indices. We get 1 } ( 1) τ C τn,τ 2i n ( 1)τ Cτ n,τn e τ n > 0 τn τ n e τn which proves the lemma. e τn l 1/2 (div) not all e τn = 0, Theorem 4.1. The norm of the exterior Calderón operator in H 1/2 (div, ) is determined by the square root of the largest eigenvalue of the Hermitian matrix P = D 1/2 C D 1 CD 1/2, i.e., the matrix P τn,τ n = ( ) τ 3/2 ( ) τ 1 C τ n 1 + τ n,τn (1 + λ n ) τ +3/2 1 3/2 C τ n,τ n, λn 1 + λn where the diagonal matrix D is D τn,τ n = δ nn δ ττ (1 + λ n) τ 3/2. Proof. The norms of the trace of the scattered electric and magnetic eld are γ(e sc ) 2 H 1/2 (div,) = τn (1 + λ n ) τ 3/2 e τn 2, and γ(h sc ) 2 H 1/2 (div,) = τn (1 + λ n ) τ 3/2 h τn 2 = τn (1 + λ n ) τ+3/2 h τn 2, or in short-hand matrix notation γ(e sc ) 2 H 1/2 (div,) = e De, γ(h sc ) 2 H 1/2 (div,) = h D 1 h, where e and h are the column vectors of the coecients e τn and h τn, respectively, and the matrix D is dened above. The Hermitian conjugate of these column vectors are denoted e and h. The norm of the exterior Calderón operator in H 1/2 (div, ) can then be formed, viz., C e 2 H 1/2 (div,) = sup e (Ce) D 1 (Ce) e De = sup e e D ( 1/2 D 1/2 C D 1 CD 1/2) D 1/2 e. e D 1/2 D 1/2 e This is a quadratic form and the largest eigenvalue of D 1/2 C D 1 CD 1/2 determines the norm.

16 15 ˆν B(a) a S a ˆν Figure 2: The spherical surface S a and the domain Ω. 4.1 Calculation of the exterior Calderón matrix The goal now is to nd an explicit representation of the exterior Calderón matrix C τn,τ n in terms of the geometry of the surface. A number of lemmata and propositions guide us. Denote by S r the sphere of radius r centered at the origin. The restriction of γ( Mf) to S r denes an operator A r : H 1/2 (div, ) H 1/2 (div, S r ). The explicit expression of the operator is, for f H 1/2 (div, ) f (A r f) (x) = ˆx g(k, x x )f(x ) ds, x S r, (4.6) where the radius 0 < r < R, R = min x x. Dene the radius a (0, R) such that the functions ψ l (ka) 0 and ψ l (ka) 0 for all l = 1, 2,..., where ψ l (z) are the Riccati-Bessel functions [21, 28]. This is always possible for small enough ka 0. Lemma 4.3. The compact operator A a : H 1/2 (div, ) H 1/2 (div, S a ), dened by (4.6), is injective with dense range. Proof. The kernel of the operator A a is continuous (analytic in the variable x) and hence A a is compact. The operator is injective if we can prove that (A a f) (x) = 0, x S a f = 0. To accomplish this, dene F (x) = g(k, x x )f(x ) ds, x R 3 \.

17 16 By assumption, γ(f ) = 0 on S a (the same limit from both sides). We proceed by proving that the only f that satises this condition is f = 0. Let B(a) denote the ball, centered at the origin, of radius a, see Figure 2. The function F (x) satises, see Theorem 2.3 ( F (x)) k 2 F (x) = 0, x R 3 \, therefore also in the ball B(a). Inside the ball B(a), the eld F (x) has an expansion in regular spherical vector waves v n (kx) [21]. Due to orthogonality of the vector spherical harmonics, and the choice of a such that ψ l (ka) 0 and ψ l (ka) 0 for all l = 1, 2,..., the expansion coecients of this expansion are all zero. Therefore, the interior boundary value problem has a unique solution F (x) = 0, x B(a). By analyticity, F (x) = 0 for all x Ω [30]. As a consequence, the traces γ(f ) = 0 and γ( F ) = 0. By Theorem 2.3, we also conclude that γ( F ) + = 0. As a function of x Ω e, F (x) satises the correct radiation conditions at innity and γ( F ) + = 0 on. Due to unique solvability of the exterior problem (Problem (E)), F (x) = 0 in Ω e. Since F = k 2 ( F ), F (x) = 0 in Ω e, and, consequently, γ(f ) + = 0. Finally, the jump condition on the trace of F shows, see Theorem = γ(f ) + γ(f ) = f. This proves the injectivity of the operator A a. To prove that the range is dense, and for this purpose, we dene the adjoint operator A a : H 1/2 (curl, S a ) H 1/2 (curl, ) of A a w.r.t. to the dual spaces (H 1/2 (div, ), H 1/2 (div, S a )). The explicit form of the adjoint operator is ( A a g ) (x) = ˆν(x) ˆν(x) g(k, x x ) [ˆx g(x ) ] ds S a = ˆν(x) (B(ˆx g)) (x), x, where (use g(k, x x ) = g(k, x x )) (Bg) (x) = ˆν(x) g(k, x x ) g(x ) ds S a = ˆν(x) g(k, x x )g(x ) ds, x. S a We now prove that A a is injective, i.e., B is injective, namely (Bg) (x) = 0, x g = 0. To this end assume that (Bg) (x) = 0, x, and similarly as above, dene the function F (x) = g(k, x x )g(x ) ds, x R 3 \ S a, S a

18 17 so that by assumption, γ( F ) ± = (Bg) (x) = 0 on (same limit from both sides). The function F (x) satises ( F (x)) k 2 F (x) = 0, x R 3 \ S a. Moreover, the function satises the appropriate radiation condition at innity and γ( F ) + = 0 on. The uniqueness of the exterior scattering problem (Problem (E)), F (x) = 0, x Ω e, and by analyticity, F = 0 also outside S a. As above, by Theorem 2.3, the curl of F has a continuous tangential component at S a. The interior problem is uniquely solvable, since ψ l (ka) 0 and ψ l (ka) 0 for all l = 1, 2,..., which implies that F (x) = 0, x B(a). The tangential components of F (x) has a jump discontinuity on S a, Theorem = ˆx F (x) ˆx F (x) = g(x), x S a. + This proves the injectivity of the operator B, and, consequently, that the operator A a has a dense range, since N(A a) = R(A a ) [7, p. 241]. Lemma 4.4. The expansion coecients e τn and h τn above are related by A τn,τ n h τ n = A τn,τ n e τ n, (4.7) τ n τ n where the dimensionless matrix A τn,τ n A τn,τ n = k is dened as u τn Y τ n ds. (4.8) The bar over the index τ denotes the dual index in τ (1 = 2 and 2 = 1). Proof. The extinction part of Theorem 2.1 reads g(k, x x )γ(h sc )(x ) ds = 1 ik g(k, x x )γ(e sc )(x ) ds }, x Ω. Introduce the Green dyadic for the electric eld in free space [21] G e (k, x x ) = (I 3 + 1k ) g(k, x x ) = (I k ) 2 g(k, x x ), where I 3 is the unit dyadic in R 3. Consequently, the extinction part is G e (k, x x ) γ(h sc )(x ) ds = 1 ik G e (k, x x ) γ(e sc )(x ) ds }, x Ω. (4.9)

19 18 The Green dyadic for the electric eld is [21, (7.24) on p. 370] G e (k, x x ) = ik τn = ik τn v τn(kx < )u τn (kx > ) u τn (kx > )v τn(kx < ), x x, (4.10) where x < (x > ) is the position vector with the smallest (largest) distance to the origin, i.e., if x < x then x < = x and x > = x. The denition of the spherical vector waves is given in Appendix B, and a star denotes complex conjugate. This expansion is uniformly convergent in compact (bounded and closed) domains, provided x x in the domain [19, 26]. Apply (4.10) to (4.9) for an x inside the inscribed sphere of and use the dual property of the spherical vector waves, i.e., v τn (kx) = kv τn (kx), u τn (kx) = ku τn (kx). We get ik 2 τn v τn(kx) u τn (kx ) γ(h sc )(x ) ds = k 2 τn v τn(kx) u τn (kx ) γ(e sc )(x ) ds, x Ω. Orthogonality of the vector spherical harmonics on the inscribed sphere implies u τn γ(h sc ) ds = i u τn γ(e sc ) ds, n, τ = 1, 2. (4.11) Insert the expansion of the eld in their Fourier series, (4.1) and (4.2), and we obtain u τn Y τ n ds = e τ n u τn Y τ n ds, n, τ = 1, 2, τ n τ n h τ n which is identical to the statement in the lemma. Corollary 4.1. The exterior Calderón operator satises the relation (C e ) 2 = I on H 1/2 (div, ). Proof. This corollary is a simple consequence of (4.11) in the proof of Lemma 4.4. We have for any m H 1/2 (div, ) u τn C e (C e (m)) ds = i u τn C e (m) ds = u τn m ds = π(u τn ) m ds, n, τ = 1, 2.

20 19 Provided the set π(u τn ) on is a complete (closed), the result follows. In fact, since every vector u = π(u) + (u ˆν)ˆν, if u H 1/2 (curl, ), we have that u = π(u). By [4, Theorem 3], the whole set u τn is complete in T L 2 (; C 2 ), therefore any v T L 2 (; C 2 ), regardless of boundary behavior, has an expansion as v = τ,n b τnu τn. Apply the π operator on both sides to get π(v) = τ,n b τnπ(u τn ). If v H 1/2 (curl, ) this gives the required result. Integration by parts gives an alternative form of the matrix A τn,τ n, see (4.8) and use Denition 3.1. and A τn,1n = 1 λn A τn,2n = 1 λn Proposition 4.1. Dene a matrix (curl π(u τn )) Y n ds, n, τ = 1, 2, A τn,τ n = k (div π(u τn )) Y n ds, n, τ = 1, 2. u τn Y τ n ds. The mapping a τn τ n A τn,τ n a τ n, is injective. Proof. We prove the proposition by showing A τn,τ n a τ n = 0, n, τ = 1, 2, τ n implies that a τn = 0 for τ = 1, 2 and all n. Multiply this relation with v τn(kx), where x lies inside the inscribed sphere of the scatterer, and sum over τ and n. We obtain, see (4.10) 1 ik G e (k, x x ) a(x ) ds = 0, x inside the inscribed sphere, where a = τn a τn Y τn. Now consider the vector-valued function A(x) = G e (k, x x ) a(x ) ds, x R 3 \,

21 20 which is dened everywhere in R 3 \. This function is, by denition, zero inside the inscribed sphere of the scatterer. By analyticity, the function A(x) = 0 for all x Ω [30]. As a consequence, the traces γ(a) = 0 and γ( A) = 0. The vector eld A(x) satises ( A(x)) k 2 A(x) = 0, x R 3 \. Moreover, A(x) satises the correct radiation conditions at innity. Due to unique solvability of the exterior problem, A(x) = 0 in the entire exterior region Ω e. As a consequence, the traces γ(a) + = 0 and γ( A) + = 0. The curl of A(x) is F (x) = A(x) = g(k, x x ) a(x ) ds, x R 3 \, where g(k, x x ) = eik x x 4π x x. The trace of F (x) has a jump discontinuity on, see Theorem = γ( A) + γ( A) = γ(f ) + γ(f ) = a, x. and consequently, by orthogonality of the vector generalized harmonics, a τn = 0, which implies the injectivity of the mapping above. To simplify the analysis in the theorem below, we introduce a special notation for the matrix with dual τ indices. To this end, dene the matrix A τn,τ n = A τn,τ n, Theorem 4.2. The exterior Calderón matrix C can be approximated by Cτn,τ α n = (αi + A A) 1 τn,τ n A τ n,τ n A τ n,τ n, τ n τ n for adequately small α > 0, where denotes the Hermitian conjugated matrix. In shorthand matrix notation C α = (αi + A A) 1 A A. Proof. The expansion coecients e τn and h τn are related by, see (4.7) A τn,τ n h τ n = A τn,τ n e τ n, (4.12) τ n τ n This equation consists of a countable set of linear equations, the solution of which may be used to express h τn in terms of e τn, thus providing a matrix form representation of the exterior Calderón operator in terms of the chosen basis of generalized harmonics. Assuming the invertibility of the matrix A τn,τ n, we write the equation as h τn = τ n τ n A 1 τn,τ n A τ n,τ n e τ n,

22 21 so that C e admits the matrix representation C τn,τ n = A 1 τn,τ n A τ n,τ n, τ n τ n In shorthand matrix notation C = A 1 A, where C is the exterior Calderón matrix. However, by the denition of the matrix operator A and the connection of the spherical vector waves u τ,n with the Green dyadic for the electric eld, see left-hand side of (4.9) and (4.6), we see that A, and therefore also A, is related to a compact operator; hence A is not expected, in general, to be invertible and, even if it were, it would lead to an ill-posed problem which could not provide a well dened numerical scheme. We may, however, resort to a Tikhonov regularization approach of the solution of (4.12), which leads to a, well-suited for numerical approaches, approximation of the exterior Calderón operator. According to the theory of the Tikhonov regularization, see [20, Ch. 16], the regularized approximate solution of (4.12) is h α τn = τ n (αi + A A) 1 τn,τ n A τ n,τ n A τ n,τ n e τ n, α > 0, τ n or in shorthand matrix notation h α = (αi +A A) 1 A Ae, which leads to an approximation of C by C α, where C α := (αi + A A) 1 τn,τ n A τ n,τ n A τ n,τ n, α > 0, τ n τ n or in shorthand matrix notation C α = (αi + A A) 1 A A. The invertibility of the matrix αi + A A is easily obtained by the Lax-Milgram Lemma, since the regularization term αi introduces coercivity into the problem and the numerical inversion can be performed in terms of a variational approach related to the minimization problem min α z 2 + z l 1/2 l 1/2 (div) A A z, z l 1/2 (div). (div) Spherical geometry The only known geometry, where we can test the theory analytically, is the spherical geometry. In this subsection, we apply the results above to a sphere of radius r. The eigenvalues for the sphere is λ n = l(l + 1)/(kr) 2, and the vector spherical harmonics Y τn (x), see [21]. For the sphere, the matrix A is diagonal. Specically, A τn,τ n = δ nn δ ξ l (kr), τ = 1 ττ ξ l (kr), τ = 2,

23 22 and ξ l (kr) C τn,τ n = δ nn δ ξ ττ l (kr), τ = 1 ξ l (kr) ξ l (kr), τ = 2, where ξ l (z) is the Riccati-Bessel (Hankel) function [21, 28]. Lemma 4.1, i.e., C 1 τn,τ n = C τn,τ n. Moreover, 2 (1 + λ n ) ξ l (kr) P τn,τ n = δ nn δ ξ ττ l (kr), τ = 1 (1 + λ n ) 1 ξ l (kr) 2 ξ l (kr), τ = 2, Notice the result of which is, apart from a dierent normalization, in agreement with [22], see Figure 3. The static limit of the exterior Calderón operator for a spherical geometry is of interest. We have l + 1, τ = 1 lim P τn,τ n = δ nn δ l ττ kr 0 l l + 1, τ = 2, and consequently lim kr 0 C e H 1/2 (div, B r) = 2. We can also check the validity of Lemma 4.2. ξ l(kr) ( 1) τ C τn,τ n = δ nn δ ξ ττ l (kr), τ = 1 ξ l (kr) ξ l (kr), τ = 2. Therefore, 1 } ( 1) τ C τn,τ 2i n ( 1)τ Cτ n,τn = δ nn δ ττ ξ l(kr)ψ l (kr) ξ l (kr)ψ l(kr) = iδ nn δ ττ ξ l (kr) 2 ξ l (kr)ψ l(kr) ξ l (kr)ψ l (kr) ξ l (kr) 2 Im ξ l(kr) ξ l (kr) Im ξ l (kr) ξ l (kr) 1 ξ, τ = 1 l = δ nn δ (kr) 2 ττ 1 ξ l (kr) 2, τ = 2, by the use of ξ l (kr) = 2ψ l(kr) ξ l (kr) and the Wronskian for the Riccati-Bessel functions ψ l (z)ξ l (z) ψ l (z)ξ l(z) = i [21]. Obviously, this matrix is positive denite.

24 23 C e H 1/2 (div, B x) l = 1 l = 2 l = κ = kx Figure 3: The norm of the exterior Calderón operator C e H 1/2 (div, B x) for a sphere of radius x is depicted. The dashed blue lines depict the function P 1l,1l for l = 1, 2, The nite dimensional problem This section contains a generalization of the result presented in [17] for a spherical surface to a general surface. Denote } N S N = f H 1/2 (div, ) : f = a τn Y τn, a τn C. We dene the orthogonal projection P N : H 1/2 (div, ) S N in the H 1/2 (div, ) inner product. The following proposition holds: Proposition 4.2. and holds for any s 1/2, where P N f f in H 1/2 (div, ) as N, (I P N )f H 1/2 (div,) λ (s+1/2)/2 N f H s (div,), τn f 2 H s (div,) = τn (1 + λ n ) s+τ 1 a τn 2. Proof. The convergence P N f f in H 1/2 (div, ) as N,

25 24 is a consequence of the generalized Fourier transform properties. We estimate for every s 1/2 (I P N )f 2 H 1/2 (div,) = n>n τ=1,2 (1 + λ n ) τ 3/2 a τn 2 = (1 + λ n ) s 1/2 (1 + λ n ) s+τ 1 a τn 2 n>n τ=1,2 (1 + λ N ) s 1/2 (1 + λ n ) s+τ 1 a τn 2 λ s 1/2 N f 2 H (div,). s n>n τ=1,2 Remark 4.2. The analysis can be extended for the case of non simply connected surfaces, by extending the proposed orthonormal basis with the nite-dimensional basis of the kernel of the Laplace-Beltrami operator on, see [27, p. 206]. 5 Connection to the transition matrix for a PEC obstacle Scattering by a perfectly conducting obstacle (PEC) with bounding surface is related to the exterior Calderón operator C e. This section develops and claries this connection. The transition matrix (T-matrix), T τn,τ n, connects the expansion coecients of the incident eld E inc, with sources in Ω e and the scattering E sc in terms of the regular spherical vector waves, v τn (kx), and the radiating spherical vector waves, u τn (kx), respectively. The denition of the spherical vector waves is given in Appendix B. Specically, E inc (x) = τn a τn v τn (kx), E sc (x) = τn f τn u τn (kx), where the expansion coecients f τn and a τn are related as f τn = τ n T τn,τ n a τ n. The expansion of the incident eld is absolutely convergent, at least, inside the inscribed sphere of the PEC obstacle, 5 and the expansion of the scattered eld converges, at least, outside the circumscribed sphere of the PEC obstacle. The transition matrix completely characterizes the scattering process. The following theorem shows that when the exterior Calderón operator is known, the transition matrix for a PEC obstacle is obtained by some simple operations: 5 More precisely, the convergence is guaranteed inside the largest inscribable ball not including the sources of the incident eld.

26 25 Theorem 5.1. The transition matrix for a PEC obstacle, T τn,τ n, with bounding surface and the corresponding exterior Calderón matrix, C τn,τ n, is: T τn,τ n = i } W τn,τ n V τ n,τ n + V τ n,τ n, τ n τ n C τ n,τ n W τn,τ n where the dimensionless matrices W τn,τ n and V τn,τ n are W τn,τ n = k v τn Y τ n ds, V τn,τ n = k γ(v τn ) Y τ n ds. Notice that W τn,τ n and V τn,τ n are related, i.e.,. V τn,τ n = ( 1)τ+1 W τn,τ n. Proof. For a given incident eld E inc, the boundary condition on the surface is γ(e inc + E sc ) = 0, which implies γ(e sc ) = γ(e inc ). The trace of the scattered magnetic eld on is γ(h) = γ(h inc + H sc ) = γ(h inc ) C e (γ(e inc )). The expansion coecients of the scattered eld for a PEC surface, f τn, are [21, (9.3) on p. 481] f τn = k 2 v τn γ(h) ds = k 2 v τn γ(h inc ) C e (γ(e inc ))} ds. Inserting the expansions of the incident elds, we obtain an explicit form of the transition matrix, viz., T τn,τ n = k2 v τn iγ(v τ n ) + Ce (γ(v τ n ))} ds, where we also used the explicit form of the trace of the incident magnetic eld H inc (x) = i τn a τn v τn (kx). The regular spherical vector wave γ(v τn ) has a Fourier series expansion in Y τn. kγ(v τn ) = V τn,τ n Y τ n, V τn,τ n = k γ(v τn ) Y ds, τ n τ n and also (4.5) C e (Y τn ) = i τ n C τ n,τny τ n.

27 26 Combine these expansions kc e (γ(v τn )) = i V τn,τ n C τ n,τ n Y τ n = i V τn,τ n C τ n,τ n Y τ n. τ n,τ n τ n,τ n These expressions lead to T τn,τ n = ik τ n } v τn V τ n,τ n Y τ n + V τ n,τ n ds. τ n C τ n,τ n Y τ n If we denote W τn,τ n = k v τn Y τ n ds, we get in matrix notation T τn,τ n = i τ n W τn,τ n V τ n,τ n + V τ n,τ n which proves the theorem. τ n C τ n,τ n W τn,τ n }, 5.1 Spherical geometry We illustrate the result in Theorem 5.1 with a sphere of radius r. The vector generalized harmonics are Y τn (x) dened in Appendix B. From above, we have ξ l (kr) C τn,τ n = δ nn δ ξ ττ l (kr), τ = 1 ξ l (kr) ξ l (kr), τ = 2. Moreover, we have and V τn,τ n = δ nn δ ψ l (kr), τ = 1 ττ ψ l (kr), τ = 2, W τn,τ n = δ nn δ ψ l (kr), τ = 1 ττ ψ l (kr), τ = 2. The transition matrix becomes ψ l (kr)ψ T τn,τ n = iδ nn δ l (kr) + ψ l(kr)ψ l (kr) ξ l (kr) ξ l (kr), τ = 1 ττ ψ l (kr)ψ l (kr) ψ l (kr)ψ l (kr)ξ l(kr) τ = 2, (kr), ξ l

28 27 which by the use of the Wronskian for the Riccati-Bessel functions ψ l (z)ξ l(z) ψ l(z)ξ l (z) = i, simplies to ψ l (kr) T τn,τ n = δ nn δ ξ l (kr), τ = 1 ττ ψ l (kr), τ = 2, (kr) in agreement with the result of Mie scattering [21]. ξ l 6 Conclusions This paper deals with a novel approach to compute the exterior Calderón operator, and, in particular, the computation of its norm in the space H 1/2 (div, ). This operator is instrumental in the understanding of the scattering problem. The approach is constructive, and employs the eigenfunctions of the Beltrami-Laplace operator of the surface. These functions are intrinsic to the surface, and constitute the natural orthonormal set for a matrix representation of the operator. The norm of the operator is explicitly given as the largest eigenvalue of a quadratic form that contains this representation of the exterior Calderón operator. The paper is closed by an investigation of the connection between the exterior Calderón operator and the transition matrix of the same perfectly conducting surface. In a future paper, the numerical behavior of the suggested algorithm is intended to be conducted. Acknowledgements To pursue this work, the Royal Physiographic Society of Lund, Sweden, has nancially supported the stay for one of the authors (GK) in Athens. This support is gratefully acknowledged. Appendix A Function spaces In this appendix, we list the various function spaces used in this paper. Let Ω be an open, bounded domain in R 3 with simply connected Lipschitz boundary Ω, see [3]. The space C(Ω) is the space of continuous functions in Ω. We also use C 0 (Ω) which consists of all uniformly continuous functions, which are zero at the boundary. The space C (Ω) is the space of innitely continuously dierentiable functions in Ω, and C 0 (Ω) are the functions in this space with compact support in Ω, which we also denote D(Ω).

29 28 Several function spaces with square integrable functions are used in this paper. The basic space is given by functions u (x u(x)) dened on Ω R 3 C } L 2 (Ω) def = u Lebesgue integrable in Ω, u 2 dv <, Ω with scalar product and norm u, v L 2 (Ω) = uv dv, u L 2 (Ω) = u 2 dv} 1/2, Ω Ω where bar denotes the complex conjugate. Similarly for vector-valued spaces we have the scalar product u, v L 2 (Ω;C 3 ) = u v dv, Ω and the norm u L 2 (Ω;C 3 ) = u 2 dv} 1/2, where and denotes the Euclidean scalar product and norm in C 3, respectively. We also dene the function spaces H(div, Ω) def = u L 2 (Ω; C 3 ) : u L 2 (Ω) } Ω H(curl, Ω) def = u L 2 (Ω; C 3 ) : u L 2 (Ω; C 3 ) }, which are Hilbert spaces with norms ( ) 1/2 u H(div,Ω) = u 2 L 2 (Ω;C 3 ) + u 2 L 2 (Ω) ( 1/2 u H(curl,Ω) = u 2 L 2 (Ω;C 3 ) + u 2 L 2 (Ω;C )). 3 The curl and the divergence are dened in the weak sense as u, φ L 2 (Ω;C 3 ) = u, φ L 2 (Ω;C 3 ), φ D(Ω; C 3 ) u, φ L 2 (Ω) = u, φ L 2 (Ω;C 3 ), φ D(Ω). In the exterior region, we dene spaces of locally integrable functions as Hloc (div, Ω e ) def = u D (Ω e ; C 3 ) : ξu H(div, Ω e ), ξ D(R 3 ) } H loc (curl, Ω e ) def = u D (Ω e ; C 3 ) : ξu H(curl, Ω e ), ξ D(R 3 ) }, where Ω e = R 3 \ Ω and D (Ω e ) is the space of distributions with nite support in Ω e.

30 29 A.1 Surface spaces, trace and lifting operators On the boundary, we have the L 2 spaces } L 2 () = u : u 2 ds <, where ds denotes the surface measure of. For the vector-valued functions, we have } L 2 (; C 3 ) = u : u 2 ds < } T L 2 (; C 3 ) = u : u ˆν = 0 and u 2 ds <, where ˆν is the outward pointing unit normal to. The scalar products and norms are 1/2 u, v L 2 () = uv dv, u L 2 () = u ds} 2, and u, v L 2 (;C 3 ) = u v dv, u L 2 (;C 3 ) = u 2 ds} 1/2, With our assumptions on Ω and, there exists a unique linear continuous map ϑ : H 1 (Ω) L 2 (), such that for any u H 1 (Ω) C(Ω) one has ϑ(u) = u. The function ϑ(u) is called the trace of u on. Note that ϑ is not onto L 2 (). Now, dene H 1/2 () := ϑ(h 1 (Ω)). This is a Banach space for the norm dened by and u H 1/2 () = u(x) 2 ds x + u(x) u(y) 2 In view of the above, one has the following denitions x y 4 ds x ds y. H 1 0(Ω) = u : u H 1 (Ω) and ϑ(u) = 0}, H 1 (Ω) = (H 1 0(Ω)), H 1/2 () = (H 1/2 ()), where by prime is denoted the dual space. For important properties related to the above spaces, we refer to [11]. We just note here that

31 30 (i) H 1/2 () L 2 () H 1/2 (), where the injections are compact. (ii) For u H(div, Ω) it holds that ˆν u H 1/2 () and the map u ˆν u is linear and continuous. Using (ii) (which is an important result, due to Jacques-Louis Lions and Enrico Magenes, of the late 1960's), we note that even though, as mentioned above, elements u of L 2 (Ω) do not necessarily have a trace on the boundary, nevertheless ˆν u makes sense, if, additionally, u is also in L 2 (Ω). The appropriate trace spaces which we use in this paper are H 1/2 (div, ) and H 1/2 (curl, ) dened by H 1/2 (div, ) def = u H 1/2 (; C 3 ), ˆν u = 0, div u H 1/2 () } H 1/2 (curl, ) def = u H 1/2 (; C 3 ), ˆν u = 0, curl u H 1/2 () }, where the surface divergence, div, and the surface curl, curl, are dened by duality and restriction, see [10, 27, 31] div u, φ L 2 () curl u def = ˆν ( u), def = u, grad φ L 2 ( Ω;C 3 ), φ D() (A.1) and the surface gradient, grad, is dened by the orthogonal projection of on the surface, i.e., grad φ = π( φ), where π is dened in Theorem A.1 below. Notice that the surface curl operator, curl, provides a scalar quantity. We can dene a vector valued curl operator acting on scalars, curl u def = grad u ˆν, alternatively by duality (A.2) curl u, v T L 2 () def = u, curl v L 2 (), v D 2 (). The space H 1/2 (div, ) is dened as the completion of the tangential elds in H 1 () w.r.t. the norm m 2 = m 2 H 1/2 () + m 2 H 1/2 (), With the assumptions made on the boundary, the space H 1/2 (curl, ) is the dual of H 1/2 (div, ), i.e., ( H 1/2 (div, ) ) = H 1/2 (curl, ). In [10, Lemma 4, p. 34], we have the following result Lemma A.1. For any u H(curl, Ω) holds curl π(u) = div (ˆν u ) = div (γ(u)). It follows that, see [10, Corollary 2, p. 38], for v H 1/2 (curl, ) we have curl v = div (ˆν v),

32 31 which implies that if curl π(u) H 1/2 () then div (γ(u)) H 1/2 () as well, or in other words, π(u) H 1/2 (curl,) = γ(u) H 1/2 (div,), and if div π(u) H 1/2 () then curl (γ(u)) H 1/2 () and π(u) H 1/2 (div,) = γ(u) H 1/2 (curl,). The following theorem is proved in [27]: Theorem A The trace mapping π : H(curl, Ω) H 1/2 (curl, ), that assigns to any u H(curl, Ω) its tangential component ˆν (u ˆν), is continuous and surjective from H(curl, Ω) onto H 1/2 (curl, ). That is π(u) H 1/2 (curl,) C π u H(curl,Ω), u H(curl, Ω). 2. The trace mapping γ : H(curl, Ω) H 1/2 (div, ), that takes u H(curl, Ω) to its (rotated) tangential component ˆν u, is continuous and surjective from H(curl, Ω) onto H 1/2 (div, ). That is γ(u) H 1/2 (div,) C γ u H(curl,Ω), u H(curl, Ω). 3. In both cases, a continuous lifting with zero divergence for these trace operators in H(curl, Ω) exists. More precisely, there exists an operator R : H 1/2 (div, ) H(curl, Ω) such that for every m H 1/2 (div, ) there exists a u H(curl, Ω) satisfying γ(u) = m, and R(m) H(curl,Ω) C m H 1/2 (div,), m H 1/2 (div, ), and similarly from H 1/2 (curl, ) to H(curl, Ω), corresponding to the π-trace. 4. For any u, v H(curl, Ω), the following Stokes' formula holds: u, v L 2 (Ω;C 3 ) u, v L 2 (Ω;C 3 ) = γ(u), π(v) L 2 (;C 3 ). A.2 Function spaces expressed in Fourier coecients We dene the pertinent function spaces used frequently in this paper in terms of the orthogonal basis Y n and Y τn. The generalized Fourier series of a function f is (we assume the surface is simply connected) f = n a n Y n, a n = f, Y n L 2 (), where convergence is in the L 2 () norm (dened below). The space L 2 () is dened as L 2 () = f D () : } a n 2 <, n

33 32 equipped with the norm f 2 L 2 () = n a n 2, and the space H s () is dened as H s () = f D () : n (1 + λ n ) s a n 2 < }, equipped with the norm f 2 H s () = n (1 + λ n ) s a n 2. Similarly, the generalized Fourier series of a tangential vector function f is f = τn a τn Y τn, a τn = f, Y τn T L 2 (;C 2 ), where convergence is in the T L 2 (; C 2 ) norm. The space T L 2 (; C 2 ) is dened as T L 2 (; C 2 ) = f D (; C 2 ) : } a τn 2 <, τn equipped with the norm f 2 T L 2 (;C 2 ) = τn a τn 2, and the space T H s (; C 2 ) is dened as T H s (; C 2 ) = f D (; C 2 ) : τn (1 + λ n ) s a τn 2 < }, equipped with the norm f 2 T H s (;C 2 ) = τn (1 + λ n ) s a τn 2. (A.3) Remark A.1. In [27] is this norm dened as f 2 T H s (;C 2 ) = τn (λ n ) s a τn 2. which is equivalent with (A.3) as long as the smallest λ n = λ min is strictly positive. The operations of curl and div imply, using Lemma 3.2, curl f = a τn curl Y τn = k λn a 1n Y n, τn n and λn a 2n Y n. div f = a τn div Y τn = k τn n Note that only one of Y 1n and Y 2n survive the respective dierentiation. motivates the following denition. This

34 33 Denition A.1. We dene H 1/2 (div, ) and H 1/2 (curl, ) as equipped with the norm H 1/2 (div, ) = f T H 1/2 (; C 2 ), div f H 1/2 () }, f 2 H 1/2 (div,) = τn (1 + λ n ) τ 3/2 a τn 2, and equipped with the norm H 1/2 (curl, ) = f T H 1/2 (; C 2 ), curl f H 1/2 () }, f 2 H 1/2 (curl,) = τn (1 + λ n ) τ 3/2 a τn 2. We also have use for the weighted spaces l 1/2 (div) dened by l 1/2 (div) = a τn C : } (1 + λ n ) τ 3/2 a τn 2 <. τn We notice that the space l 1/2 (div) and H 1/2 (div, ) are equivalent in the sense that f H 1/2 (div, ) if and only if its Fourier coecients a τn l 1/2 (div). We have the following Parseval type of identity Lemma A.2. Let u, v L 2 () with expansions u = e τn Y τn τn v = τn h τn Y τn, then u, v L 2 () = τn e τn h τn. Proof. The proof follows the proof of the orthogonality of the vector generalized harmonics in Lemma 3.2. Remark A.2. Let u H 1/2 (div, ) and v H 1/2 (curl, ). The two norms are explicitly given as u 2 H 1/2 (div,) = 1 e 1n λn e 2n 2, n 1 + λn n and v 2 H 1/2 (curl,) = n 1 + λn h 1n 2 + n λn h 2n 2, respectively. A duality pairing between the spaces H 1/2 (div, ) and H 1/2 (curl, ) yields u, v H 1/2 (div,),h 1/2 (curl,) = u, v L 2 ()

35 34 Lemma A.3. Let u H 1/2 (div, ) and v H 1/2 (curl, ) with expansions u = e τn Y τn τn v = τn h τn Y τn, then and u 2 H 1/2 (div,) = ˆν u 2 H 1/2 (curl,) = τn v 2 H 1/2 (curl,) = ˆν v 2 H 1/2 (div,) = τn (1 + λ n ) τ 3/2 e τn 2, (1 + λ n ) τ 3/2 h τn 2. Proof. The proof follows from the construction of the vector generalized harmonics in Denition 3.1 and Lemma 3.2. Appendix B Spherical vector waves The spherical harmonics Y n (x) are dened as Y n (x) = 1 2l + 1 (l m)! x 4π (l + m)! P l m (cos θ)e imφ, in terms of the spherical angles θ (polar angle) and φ (azimuthal angle) of the unit vector ˆx. The associated Legendre function is denoted Pl m (cos θ). The index n is a multi-index for the integer indices l = 0, 1, 2, 3,..., m = l, l+1,..., 1, 0, 1,..., l. Note, the extra factor 1/x in the denition of the spherical harmonics, which makes the spherical harmonics orthonormal on the sphere of radius x. The vector spherical harmonics are dened as, cf. [6, 21] Y 1n (x) = SY n (x) ˆx l(l + 1) Y 2n (x) = SY n (x) l(l + 1), where S is the nabla-operator on the unit sphere. The radiating solutions to the Maxwell equations in vacuum are dened as (outgoing spherical vector waves) u 1n (kx) = ξ l(kx) Y 1n (ˆx) k u 2n (kx) = 1 ( ) k ξl (kx) Y 1n (ˆx). k