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February 1, 2008 21:3 WSPC/INSTRUCTION FILE H-measures 5 0 0 2 H-measures and system of Maxwell’s n a J HassanTAHA 9 Université d’Orléans ] MAPMO-UMR 6628, BP 6759 P 45067 ORLEANSCEDEX-France A [email protected] . h t a Weare interested inthe homogenization of energy likequantities inelectromagnetism. m WeproveageneralpropagationTheoremforH-measuresassociatedtoMaxwell’ssystem, in the full space Ω = R3, without boundary conditions. We shall distinguish between [ twocases:constant coefficient case,andnoncoefficient-scalar case.Inthetwocaseswe 1 givethebehaviour oftheH-measuresassociatedtothissystem. v Keywords:Electromagnetism,homogenizationofenergy,H-measures,Maxwell’ssystem. 8 2 MathematicsSubjectClassification2000:35BXX,35B27 1 1 0 1. Introduction 5 Herein, we are interested in the homogenization of energy like quantities in elec- 0 / tromagnetism, and more particularly in Maxwell’s equations, without boundary h conditions. We use the notion of H-measures, introduced by Gérard and Tartar t a [5], [23]. We prove a general propagation Theorem for H-measures associated to m Maxwell’s system. This result, combined with the localisation property, is then : v used to obtain more precise results on the behaviour of H-measures associated to i X this system. As known, an H-measure is a (possibly matrix of) Radon measures onthe product r a spaceΩ Sn−1,whereΩ RnisanopendomainandSn−1istheunitsphereinRn. × ⊆ InordertoapplyFouriertransform,functionsdefinedonthewholeofRn shouldbe consideredandthiscanbeachievedbyextendingthembyzerooutsidethedomain. For this reason, we consider Maxwell’s system in the full space R3, which means without boundary conditions. Let us mention that similar works already exist, see in particuliar [2]. However,in [2], computations are far from being complete. Ifoneis interestedincouplingwithboundaryconditions,the usualpseudodifferen- tial calculus behind the notion of H-measures is not sufficient, and one should use much more technical tools. In the case of semi-classical measures, this is now well known, see for instance [6], [8], [12]. In the context of H-measures, and in particuliar without a typical scale, then one should use tools similar to those developped in recents works, see for instance [1]. However,theresultspresentedherewillbeimportantforthefullMaxwellproblem, with suitable boundary conditions. 1 February 1, 2008 21:3 WSPC/INSTRUCTION FILE H-measures 2 Let Ω be an open set of R3. We will consider Maxwell’s system in the material Ω with electric permeability¨ǫ, conductivity σ¨ and magnetic susceptibility η¨given by ε ε ε ε i) ∂ D (x,t)+J (x,t)= rotH (x,t)+F , t  ε ε ε ii) ∂ B (x,t) = rotE (x,t)+G (x,t), iii) ditvBε(x,t) = − 0, (1.1) ε ε where x Ω iavn)d t d(i0v,DT),(xE,εt), Hε,=Dε, Jε an̺d(Bx,εt)a,re the electric, magnetic, ∈ ∈ induced electric, current density and induced magnetic fields, respectively. ε ε ε Morever, ρ (the charge density), F and G are given, and we have the three constitutive relations ε ε 1)D (x,t) = ǫ¨(x)E (x,t), 2) Jε(x,t) = σ¨(x)Eε(x,t), (1.2) 3) Bε(x,t) =η¨(x)Hε(x,t), where ǫ¨, σ¨ and η¨ are 3 3 matrix valued functions and ε is a typical lenght going × to 0. WeshallconsiderthissysteminthefullspaceΩ=R3,withoutboundaryconditions. Since we are not taking into account the initial data, we will also assume that the time variable t belongs to R. We shall use the notion of H-measure to compute for instance energy quantities in the following cases : i) Constant coefficient case: here, we assume that the electric permittivity ¨ǫ, conductivity σ¨ and magnetic susceptibility ηäre 3 3 identity matrices, i.e. × ¨ǫ=σ¨ =η¨=(Id) . (1.3) 3×3 ii) Non constant coefficient-scalar case: in this case, we consider that the matrix¨ǫ, σ¨, ηäre scalar 3 3 matrix valued smooth functions, i.e. × ¨ǫ=ǫ(Id) , σ¨ =σ(Id) , η¨=η(Id) (1.4) 3×3 3×3 3×3 where ǫ, σ, η are smooth functions, given in C1(R3), bounded from below. b We will also assume that fε =(Fε,Gε)t, ̺ε ⇀(0,0,0) in [L2(R R3)6] L2(R R3) weakly (1.5) × × × and uε (Eε,Hε)t ⇀0 in L2(R R3)6 weakly. (1.6) ≡ × After some prerequisitesonH-measures,see [6] and[23],presentedin SectionI,we use this notion in Section II to prove February 1, 2008 21:3 WSPC/INSTRUCTION FILE H-measures 3 Theorem 1.1. Constant coefficient case Assume (1.2), (1.5), (1.6) and (1.3). Then, up to a suitable extraction, the H- measure b = b(t,x,ζ), ζ = (ζ ,ζ′), ζ′ = (ζ ,ζ ,ζ ), associated to (uε), can be 0 1 2 3 expressed as follows ′ ′ ′ ′ ζ ζ a(t,x,ζ) ζ ζ c(t,x,ζ) b= ⊗ ⊗ . (1.7)  ′ ′ ′ ′  ζ ζ d(t,x,ζ) ζ ζ b(t,x,ζ) ⊗ ⊗   Here a(t,x,ζ) and b(t,x,ζ) are positive measures, while c(t,x,ζ) and d(t,x,ζ) are complex measures such that c=d¯, all supported in [ ζ =0 ζ′ =0 ] ζ ζ ζ = 0 1 2 3 { }∪{ } ∩{ 0 . } They satisfy the transport system (propagation property) 3 |ζ′ |2 (−∂∂at −2a)− ∂l′.[Tr(ζ′ ⊗ζ′∂l′E)c]=2Re Trµuf11,  l′=1  |ζlX′′=3|21∂(−l′.∂∂[Tbtr)(+ζ′ ⊗3ζX∂′∂l′l.′[ET)ra(]ζ−′ ⊗|ζζ′′∂|2l′∂E∂ct)d=]=2R2eRTerTµruµfu12f,22, (1.8) l′=1 Above,a−dlXe′=3ri1v∂alt′i.v[eTrw(ζit′h⊗aXζn′∂ul′pEpe)rb]+(re|sζp′.|2lo(w∂∂edtr)−i2ndd)ex=d2eRneotTesrµaufd2e1r.ivative wrt. variable ζ′ (resp. x). E=E(ζ′) is the constant 3 3 matrix whose action is given × by E.α = ζ′ α, for all α R3. Finally, µ is the 6 6 matrix correlating the uf ε∧ ε ∈ × sequences u and f , written with blocks of size 3 3. In particuliar, it is zero if ε × at least f is strongly convergent to 0. Finally, one has the following constraint ζ2∂ a+ζ2∂ a+ζ2∂ a=2Re Trµ 1 x1 2 x2 3 x3 u̺˜11 and similarly for b, c and d, where µ is the 6 6 matrix correlating the sequence u̺˜ uε with the sequence ̺˜ε (̺ε,0,0,0,0,0)t. × ≡ Theorem 1.2. Non constant coefficient-scalar case Assume (1.2), (1.5), (1.6) and (1.4). Let the dispersion matrix (see formula (3.35) 3 below) be L(x,ζ) = A−1(x)ζ Aj, which has three eigenvalues, each with fixed 0 j j=1 X multiplicity two, for ζ′ =0 and given by 6 ′ ′ ω =0 ,ω =v ζ ,ω = v ζ . 0 + − | | − | | Then the matrix P(x,ζ)=A (ζ Id+L(x,ζ)) has also the following three eigenval- 0 0 ues ′ ′ ω =ζ ,ω =ζ +v ζ ,ω =ζ v ζ , 0 0 + 0 − 0 | | − | | February 1, 2008 21:3 WSPC/INSTRUCTION FILE H-measures 4 1 each with fixed multiplicity two, where v(x)= is the propagation speed. ǫ(x)η(x) Using the propagation basis and the eigenvector basis introduced in (3.51) and p (3.52), it follows that the H-measure b associated to a suitable subsequence of ε u can be expressed as: b b b= 11 12 b b (cid:18) 21 22(cid:19) where b are 3 3 matrix valued measures. Furthermore, one has ij × 1 1 1 1 1 b = [(ζˆ′ ζˆ′)a + (z1 z1)a + (z2 z2)b + (z1 z1)a + (z2 z2)b ] 11 ǫ ⊗ 0 2 ⊗ + 2 ⊗ + 2 ⊗ − 2 ⊗ −   bb1221 == vv22[[((zz12⊗⊗zz21))aa++−−((zz12⊗⊗zz21))bb++−−((zz21⊗⊗zz21))aa−−++((zz21⊗⊗zz12))bb−+]] usibng22no=taµ1ti[o(ζnˆ′s⊗giζˆv′e)nb0b+y21((3z.521⊗).zA2)bao+ve+a210(,zb10⊗, az±1)ba+n+d b21±(z2ar⊗e za2ll)as−ca+lar21(pzo1s⊗itivze1)b−]. measures supported in the set [ ζ =0 ζ = v ζ′ ] ζ ζ ζ =0 . Finally, 0 0 1 2 3 { }∪{ ± | |} ∩{ } one has the following propagation type system 3 3 −ε(x)∂tb11+ζ0 ∂l′ε(x)∂l′b11−2σb11− ∂l′E.∂l′b12 =2Reµuf11  l′=1 l′=1 X X −ε(x)∂−tηb(x2)1b+1ζ20+3ζ0∂lX′l=3′ε1(∂xl)′∂η(l′xb)∂1l1′b−122σ+blX′2=311−∂l′E3∂∂l′lb′E1.1∂l=′b22R2e=µu2fR12eµuf21 l′=1 l′=1 where w−eηa(rxe)ubsi2n2g+saζXm0leX′=3n1o∂tal′tηi(oxn)s∂al′sbin22T+helX′o=3r1em∂l′1E.X1∂l′fbor2t1he=r2igRhetµhuafn22d side. 2. Some basic facts on H-measures In this Section, we recall some results from the H-measures theory, taking the presentationofTartar[23].However,thisisalsosimilartotheexpositionofGérard [5,7], relying upon Hormander [10], [11]. Definition 1. Let Ω be an open set of Rn and let uε be a sequence of functions defined in Rn with values in Rp. We assume that uε converges weakly to zero in (L2(Rn))p. Then after extracting a subsequence (still denoted by ε), there exists a family of complex-valued Radon measures (b (x,ζ)) on Rn Sn−1, ij 1≤i,j≤p × such that for every functions φ ,φ in C (Rn), the space of continuous functions 1 2 0 February 1, 2008 21:3 WSPC/INSTRUCTION FILE H-measures 5 converging to zero at infinity, and for every function ψ in C(Sn−1), the space of continuous functions on the unit sphere Sn−1 in Rn, one has <b ,φ φ¯ ψ >= φ φ¯ψ(ζ/ζ )b (x,ζ)dxdζ  ij 1 2⊗ ZRnZSn−1 1 2 | | ij (2.1)  =ε lim 0 Rn[F(φ1uεi)(ζ)][F(φ2uεj)(ζ)]ψ(ζ/|ζ|)dζ. −→ Z Above,F denotes the Fourier transform operator defined in L2(Rn), for an inte- grablefunctionf as[F(f)](ζ)= f(x)e−2πix.ζdx,whileF¯ istheinverseFourier Rn Z transform defined as [F(f)](x) = f(ζ) e2πix.ζdζ. z¯denotes the complex conju- Rn Z gate of the complex number z. The matrix valued measure b = (b ) is called the H-measure associated ij 1≤i≤p ε with the extracted subsequence u . Remark 1. H-measures are hermitian and non-negative matrices in the following sense b =b and ij ji  p (2.2)  bijφiφj ≥0 for all φ1 φ2 ...........φn ∈C0(Rn), i,j=1 X and it is clear that the H-measure for a strongly convergent sequence is zero. Although we consider the scalar case for all properties of H-measures, all the following facts are easily extended to the vectorial case. Definition 2. Let a C(Sn−1), b C (Rn). We associate with a the linear 0 ∈ ∈ continuous operator A on L2(Rn) defined by F(Au)(ζ)=a(ζ/ζ )F(u)(ζ) a.e. ζ Rn (2.3) | | ∈ and with b we associate the operator Bu(x)=b(x)u(x) a.e. x Rn . (2.4) ∈ A continuous function P on Rn Sn−1 with values in R is called an admissible × symbol if it can be written as +∞ +∞ P(x,ζ)= b (x) a (ζ)= b (x)a (ζ) (2.5) n n n n ⊗ n=1 n=1 X X where a are continuous functions on Sn−1 and b are continuous bounded func- n n tions converging to zero at infinity on Rn with +∞ max a (ζ) max b (x) < . (2.6) ζ | n | x | n | ∞ n=1 X February 1, 2008 21:3 WSPC/INSTRUCTION FILE H-measures 6 An operator L with symbol P is defined by: 1) L is linear continuous on L2(Rn). 2) P is an admissible symbol with a decomposition (2.5), satisfying (2.6). 3) L can written as the following form n L= A B +compact operator n n n=1 X where A ,B are the operators associated with a ,b as in (2.3) and (2.4). n n n n Withnotationsasin(2.3)and(2.4),onecanshowthattheoperatorC :=AB BA − is a compact operator from L2(Rn) into itself. Denote by Xm(Rn) the space of functions v with derivatives up to order m belonging to the image by the Fourier transform of the space L1(Rn) i.e. (F(L1(Rn))), equipped with the norm v Xm = (1+ 2πζ m)F(v)(ζ) dζ . || ||| Rn | | | | Z Then,ifAandB areoperatorswithsymbolsaandbasin(2.3)and(2.4),satisfying one the following conditions 1) a C1(Sn−1) and b X1(Rn), ∈ ∈ 2) a X1 (Rn 0 ) and b C1(Rn), ∈ loc \{ } ∈ 0 it follows that the operator C = AB BA is a continuous operator from L2(Rn) − into H1(Rn) and extending a to be homogeneous of degree zero on Rn, then ∂ C = (AB BA) has the symbol ∇ ∂x − i n ∂a ∂b ( ζa. xb)ζ =ζi . (2.7) ∇ ∇ ∂ζ ∂x k k k=1 X The main results of H-measures theory are given by the next two results ε Theorem 2.1. Localisation property Let u be a sequence converging weakly to zero in (L2(Rn))p and let b be the H-measure associated to uε. Assume that one has the balance relation n ∂ (Akuε) 0 (H−1(Ω))p strongly, ∂x −→ loc k k=1 X where Ak are continuous matrix valued functions on Ω Rn. Then, on Ω Sn−1, ⊂ × one has n P(x,ζ)b ( ζ Ak(x))b=0. (2.8) k ≡ k=1 X This result shows that the support of the H-measure b is contained in the (char- acteristic) set (x,ζ) Ω Sn−1 , detP(x,ζ)=0 . { ∈ × } February 1, 2008 21:3 WSPC/INSTRUCTION FILE H-measures 7 Theorem 2.2. Propagation property for symmetric systems Let be given matrix valued functions Ak in the class C1(Ω). Assume that the pair of sequences 0 ε ε (u ,f ) satisfies the symmetric system n ε Ak∂u +Buε =fε (2.9) ∂x k k=1 X and that both sequences (uε), (fε) converge weakly to zero in L2(Ω)p. Then the ε ε H-measure µ associated to the sequence (u ,f ) and given under the form µ11 µ12 µ= (2.10)   µ21 µ22   satisfies the equation n <µ11, P,ψ +ψ ∂kAk 2ψS >=<2 (Trµ12),ψ > (2.11) { } − ℜ k=1 X for all smooth functions ψ(x,ζ). Here S := 1/2(B+B∗) is the hermitian part of the matrix B and P,ψ is the Poisson bracket of P and ψ, i.e. { } n ∂P ∂ψ ∂ψ ∂P P,ψ =∂lP∂ ψ ∂lψ∂ P ( ). (2.12) l l { } − ≡ ∂ζ ∂x − ∂ζ ∂x l l l l l=1 X 3. Applications to Maxwell’s system Thissectionis devotedto theproofsofourmainresultsstatedintheIntroduction. 3.1. Proof of Theorem 1.1: Constant coefficient case This case corresponds to the assumption (1.3), that is all the matrices ǫ¨,η¨,andσ¨ are the identity matrix, i.e 1 00 ¨ǫ=σ¨ =η¨= 0 10 =(Id) . (3.1) 3×3   0 01   In this case, system (1.1) can be rewritten as ε ∂D ε ε ε i) (x,t)+E (x,t) = rotH (x,t)+F (x,t), ∂t  ε iiiii)) d∂iv∂HHt ε((xx,,tt)) == −rotEε(x,t0),+Gε(x,t), (3.2) ε ε iv) divE (x,t) = ρ (x,t). February 1, 2008 21:3 WSPC/INSTRUCTION FILE H-measures 8 Recalling the notation of the Introduction, it follows that Maxwell’s system (3.2) can be written as 3 ε Ai∂u +Cuε =fε (3.3) ∂x i i=0 X and 3 ε Bi∂u =̺˜ε. (3.4) ∂x i i=1 X Here ǫ¨0 Id 0 A0 = = (3.5) 0 η¨ 0 Id (cid:18) (cid:19) (cid:18) (cid:19) and t t t A1 = 0 Q1 ,A2 = 0 Q2 ,A3 = 0 Q3 . (3.6) Q1 0 Q2 0 Q3 0 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) The constant antisymmetric matrices Qk ,1 k 3 are given by ≤ ≤ 00 0 0 01 0 10 − Q1 = 00 1 ,Q2 = 0 00 ,Q3 = 1 0 0 (3.7)  −      01 0 100 0 0 0 −       ε the matrix C and f by ε σ¨ 0 Id0 ε F C = = ,f = ε . (3.8) 0 0 0 0 G (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) Matrices Bi, i=1,2,3, are given by βi 0 Bi = 0 βi (cid:18) (cid:19) where the 3 3 matrices βi are such that × βi =0 except for βi =1 kl ii Finally we have denoted ̺˜ε (̺ε,0,0,0,0,0)t. ≡ ε Denote the H-measure corresponding to (a subsequence of) the sequence u by νe νem b= . (3.9)   νme νm   The measure b is a 2 2 block matrix measure, each block being of size 3 3. × × In the following, let x = t, x˜ = (x ,x), x = (x ,x ,x ). We let ζ denote the dual 0 0 1 2 3 variable to x˜, with ζ =(ζ ,ζ′), ζ′ =(ζ ,ζ ,ζ ). 0 1 2 3 To state the localisation property (2.1), we need first to express the symbol of the differential operator appeating in (3.3), for which one has February 1, 2008 21:3 WSPC/INSTRUCTION FILE H-measures 9 3 P(x,ζ) ζ Aj(x) j ≡ j=0 X and thus P(x,ζ)=ζ A0(x)+ζ A1(x)+ζ A2(x)+ζ A3(x)= 0 1 2 3 t t Id 0 0 Q 0 Q 1 2 =ζ +ζ +ζ 0 0 Id 1 Q1 0 2 Q2 0 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) t +ζ3(cid:18)Q03 Q03(cid:19)=(cid:18)ζ0EIdζ−0IEd(cid:19) where 0 ζ ζ 3 2 − E ζ 0 ζ . (3.10) 3 1 ≡ −  ζ ζ 0 2 1 −   t Clearly E is antisymmetric (i.e. E = E), so that P is a symmetric matrix. − Using the localisation property, it follows ζ0Id E νe νem Pb= − =0 (3.11)    E ζ0Id νme νm    and thus t 1) ζ0Id.νe+E .νme = 0, t  2) ζ0Id.νem+E .νm = 0, (3.12)  3) E.νe+ζ0Id.νem = 0, 4) E.νem+ζ0Id.νm = 0. First note that from (3.11), since (see also next subsection) P has ζ , ζ′ as 0 ± | | eigenvalues, that b is supported in ζ =0 ζ′ =0 . Then, one has 0 { }∪{ } Lemma 1. The H-measure b can be written under the form ′ ′ ′ ′ ζ ζ a(t,x,ζ) ζ ζ c(t,x,ζ) b= ⊗ ⊗ (3.13)  ′ ′ ′ ′  ζ ζ d(t,x,ζ) ζ ζ b(t,x,ζ) ⊗ ⊗   where a, b are scalar positive measures, c and d are scalar complex measures such that c¯=d, all supported in ζ =0 ζ′ =0 . 0 { }∪{ } Proof of Lemma 1 February 1, 2008 21:3 WSPC/INSTRUCTION FILE H-measures 10 t Multiplying (3.12-1) par ζ , (3.12-3) par E and substracting the results, one has 0 (ζ02Id+E2).νe =0. (3.14) We discuss the following distinct cases i) case ζ = 0. Then note that ζ′ = 0 since ζ belongs to the unit sphere of R4. 0 6 From (3.12), we have E.νe =0. Then, we use the following Lemma Lemma 2. If E.A=0, then the matrix A has the form A=ζ′ a, for some vector a R3. ⊗ ∈ Proof of Lemma 2 We denote the columns of the matrix A by the vectors A=[~a ~a ~a ]. (3.15) 1 1 1 But as E.A=0, we get that [E~a E~a E~a ]=0 (3.16) 1 1 1 or E~a =0 ,i=1,2,3. (3.17) i v 1 For i = 1, and similarly for the other cases, a = v R3. Then from (3.10), 1 2   ∈ v 3   (3.17), we get that 0 ζ ζ v v ζ +v ζ 3 2 1 2 3 3 2 − − ′ ζ 0 ζ v = v ζ v ζ =ζ ~a =0 (3.18) 3 1 2 1 3 3 1 1  −    −  ⊗ ζ ζ 0 v v ζ +v ζ 2 1 3 1 2 2 1 − −      which implies that ζ′//~a and thus~a =c ζ′ , for some constant c R. 1 1 1 1 ∈ Thus all in all, and for i = 1,2,3, all the columns of the matrix A are parallel to ′ ′ the vector ζ = (ζ ,ζ ,ζ ), so we can write that~a = c ζ , and by arranging these 1 2 3 i i numbers c as components of the vector a, we get that i ′ A=~a ζ . (3.19) ⊗ End of the proof of Lemma 1 ′ ′ ′ Using Lemma 2, we can conclude that νe =ζ ζ a(t,x,ζ ) and thus the blocks of ⊗ the matrix H-measure, which satisfy system (3.12), are such that ′ ′ νe =ζ ζ a(t,x,ζ), ⊗  νm =ζ′ ζ′b(t,x,ζ),  νem =ζ′⊗⊗ζ′c(t,x,ζ), (3.20) ′ ′ ′ ′  νme =ζ ⊗ζ d(t,x,ζ)=ν¯em =ζ ⊗ζ c¯(t,x,ζ)