Gaps in the spectrum of the Neumann Laplacian generated by a system of periodically distributed

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1 Gaps in the spectrum of the Neumann Laplacian generated by a system of periodically distributed trap Andrii Khrabustovsyi 3, Evgeni Khruslov arxiv:30.96v [math.sp] 4 Jan 03 Abstract. The article deals with a convergence of the spectrum of the Neumann Laplacian in a periodic unbounded domainω ε depending on a small parameterε>0. The domain has the formω ε =R n \ S ε, where S ε is anεz n -periodic family of trap-lie screens. We prove that for an arbitrarily large L the spectrum has just one gap in [0, L] whenεsmall enough, moreover whenε 0 this gap converges to some interval whose edges can be controlled by a suitable choice of geometry of the screens. An application to the theory of D-photonic crystals is discussed. Keywords: periodic domain, Neumann Laplacian, spectrum, gaps Introduction It is well-nown (see, e.g., [ 3]) that the spectrum of self-adjoint periodic differential operators has a band structure, i.e. it is a union of compact intervals called bands. The neighbouring bands may overlap, otherwise we have a gap in the spectrum (i.e. an open interval that does not belong to the spectrum but its ends belong to it). In general the existence of spectral gaps is not guaranteed. For applications it is interesting to construct the operators with non-void spectral gaps since their presence is important for the description of wave processes which are governed by differential operators under consideration. Namely, if the wave frequency belongs to a gap, then the corresponding wave cannot propagate in the medium without attenuation. This feature is a dominant requirement for so-called photonic crystals which are materials with periodic dielectric structure attracting much attention in recent years (see, e.g., [4 6]). In the present wor we derive the effect of opening of spectral gaps for the Laplace operator in R n (n ) perforated by a family of periodically distributed traps on which we pose the Neumann boundary conditions. The traps are made from infinitely thin screens (see Fig. ). In the case n= this operator describes the propagation of the H-polarized electro-magnetic waves in the dielectric medium containing a system of perfectly conducting trap-lie screens (see the remar in Section 3). We describe the problem and main result more precisely. Letε>0 be a small parameter. Let S ε = i Z n S ε i be a union of periodically distributed screens S ε i inr n (n ). Each screen S ε i is an (n )-dimensional surface obtained by removing of a small spherical hole from the boundary of a n-dimensional cube. It is supposed that the distance between the screens is equal toε, the length Department of Mathematics, Karlsruhe Institute of Technology, Germany Mathematical Division, B. Verin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Uraine 3 Correspondence to: Department of Mathematics, Karlsruhe Institute of Technology, Kaiserstrasse 89-93, Karlsruhe 7633, Germany andrii.hrabustovsyi@it.edu, hruslov@ilt.harov.ua

2 of their edges is equal to bε, while the radius of the holes is equal to dε n n if n> and e /dε if n=. Here d (0, ), b (0, ) are constants. S ε i ε Fig.. The system of screens S ε i ByA ε we denote the Neumann Laplacian in the domainr n \ S ε. Our goal is to describe the behaviour of its spectrumσ(a ε ) asε 0. The main result of this wor is as follows (see Theorem.). For an arbitrarily large L the spectrumσ(a ε ) has just one gap in [0, L] whenεis small enough. Whenε 0 this gap converges to some interval (σ,µ) depending in simple manner on the coefficients d and b. Moreover (see Corollary.) with a suitable choice of d and b this interval can be made equal to an arbitrary preassigned interval in (0, ). The possibility of opening of spectral gaps by means of a periodic perforation was also investigated in [7]. Here the authors studied the spectrum of the Neumann Laplacian inr perforated byz -periodic family of circular holes. It was proved that the gaps open up when the diameter d of holes is close enough to the distance between their centres (which is equal to ). However, the structure of the spectrum in [7] differs essentially from that one in the present paper. Namely, when d the spectrum converges (uniformly on compact intervals) to a sequence of points. Various examples of scalar periodic elliptic operators in the entire space with periodic coefficients were presented in [8 8]. In these wors spectral gaps are the result of high contrast in (some of) the coefficients of the operator. The outline of the paper is as follows. In Section we describe precisely the operatora ε and formulate the main result of the paper (Theorem.) describing the behaviour ofσ(a ε ) asε 0. Theorem. is proved in Section. Finally, in Section 3 on a formal level of rigour we discuss the applications to the theory of D photonic crystals.. Settingoftheproblemandthemainresult Let n N\{} and let ε > 0. We introduce the following sets: B={x=(x,..., x n ) R n : b/< x i < b/, i}, where b (0, ) is a constant. D ε = { x B : x x 0 <d ε}, where x 0 = (0, 0,..., 0, b/) and d ε is defined by the following formula: dε n, d ε ( n>, = ε exp ) (), n=. dε Here d>0is a constant. It is supposed thatεis small enough so that d ε < b/

3 S ε = B\D ε For i Z n we set: and (see Fig. ) S ε i=ε(s ε + i) () Ω ε =R n \ i Z n S ε i (3) Now we define precisely the Neumann Laplacian inω ε. We denote byη ε [u, v] the sesquilinear form in L (Ω ε ) which is defined by the formula η ε [u, v]= ( u, v) dx (4) Ω ε n and the definitional domain dom(η ε )=H (Ω ε u v ). Here ( u, v)=. The formη ε [u, v] is x = x densely defined closed and positive. Then (see, e.g., [9, Chapter 6, Theorem.]) there exists the unique self-adjoint and positive operatora ε associated with the formη ε, i.e. (A ε u, v) L (Ω ε )=η ε [u, v], u dom(a ε ), v dom(η ε ) (5) It follows from (5) thata ε u= u in the generalized sense. Using a standard regularity theory (see, e.g, [0, Chapter 5]) it is easy to show that each u dom(a) belongs to H loc (Ωε ), furthermore u n = 0 for any smoothγ Ω ε. Γ We denote byσ(a ε ) the spectrum ofa ε. To describe the behaviour ofσ(a ε ) asε 0 we need some additional notations. In the case n>we denote by cap(t) the capacity of the disc Recall (see, e.g, []) that it is defined by T={x=(x,..., x n ) R n : x <, x n = 0} cap(t)=inf w R n w dx where the infimum is taen over smooth and compactly supported inr n functions equal to on T. We set cap(t)d n, n>, σ= 4b n σ πd µ= (6) b, n=, b n It is clear thatσ<µ. The behaviour ofσ(a ε ) asε 0 is described by the following theorem. Theorem.. Let L be an arbitrary number satisfying L>µ. Then the spectrum of the operator A ε has the following structure in [0, L] whenεis small enough: σ(a ε ) [0, L]=[0, L]\(σ ε,µ ε ) (7) 3

4 4 where the interval (σ ε,µ ε ) satisfies limσ ε =σ, limµ ε =µ (8) ε 0 ε 0 Corollary.. For an arbitrary interval (σ,µ) (0, ) there is a family{ω ε } ε of periodic unbounded domains inr n such that for an arbitrary number L satisfying L>µ the spectrum of the corresponding Neumann LaplacianA ε has just one gap in [0, L] whenεis small enough and this gap converges to the interval (σ,µ) asε 0. Proof. It is easy to see that the map (d, b) (6) (σ,µ) is one-to-one and maps (0, ) (0, ) onto { (σ,µ) R : 0<σ<µ }. The inverse map is given by n 4σ( σµ )(cap(t)) d=, n>, σπ ( σµ ), n=, b= n σµ (9) Then the domainω ε considered in Theorem. with d and b being defined by formula (9) satisfies the requirements of the corollary. Remar.. It will be easily seen from the proof of Theorem. that the main result remains valid for an arbitrary open domain B which is compactly supported in the unit cube ( /, /) n and whose boundary contains an open flat subset on which we choose the point x 0. In this case the coefficientsσandµare defined again by formula (6) but with B instead of b n (here by we denote the volume of the domain). Remar.. One can guess that in order to open up m> gaps we have to place m screens S ε, S ε,...,sε m in the cube ( /, /) n. However the proof of this conjecture is more complicated comparing with the case m=. We prove it in our next wor. Below we only announce the result for the case m=. Let B and B be arbitrary open cuboids which are compactly supported in ( /, /) n. On B and B we choose the points x and x correspondingly. We suppose that these points do not belong to the edges of cuboids. Let D ε and Dε be open balls with the radii dε and dε and the centres at x and x correspondingly. Here d ε j, j=, are defined by formula () but with d j instead of d (d j > 0 are constants). For i Z n, j=, we set S i ε j =ε(sε j + i) and finally Ω ε =R n \ S ε i j i Z n, j=, ByA ε we denote the Neumann Laplacian inω ε. We introduce the numbersσ j, j =, by formula (6) with d j and B j instead of d and b n correspondingly. We suppose that d j are such that the inequalityσ <σ holds. Finally we define the numbersµ j by the formula µ j = ( ρ +ρ +σ +σ + ( ) j (ρ +ρ +σ +σ ) 4(ρ σ +ρ σ +σ σ ) ) whereρ j =σ j B j ( B j ). It is not hard to chec thatσ <µ <σ <µ. Now, let L be an arbitrary number satisfying L>µ. Thenσ(A ε ) has just two gaps in [0, L] whenεis small enough, moreover the edges of these gaps converge to the intervals (σ j,µ j ) as ε 0. Also it is easily to show that the pointsσ j,µ j can be controlled by a suitable choice of the numbers d j and the cuboids B j.

5 . Proof of Theorem. We present the proof of Theorem. for the case n 3 only. For the case n= the proof is repeated word-by-word with some small modifications. In what follows by C, C... we denote generic constants that do not depend onε. By u B we denote the mean value of the function v(x) over the domain B, i.e. u B = B B u(x)dx Recall that by B we denote the volume of the domain B... Preliminaries. We introduce the following sets (see Fig. ): Y={x R n : /< x i < /, i}. Y ε = Y\ S ε. F= Y\ B. Let A ε be the Neumann Laplacian inε Ω ε. It is clear that σ(a ε )=ε σ(a ε ) (0) A ε is anz n -periodic operator, i.e. A ε commutes with the translations u(x) u(x+i), i Z n. For us it is more convenient to deal with the operator A ε since the external boundary of its period cell is fixed (it coincides with Y). 5 D ε S ε B F Fig.. The period cell Y ε In view of the periodicity of A ε the analysis of the spectrumσ(a ε ) is reduced to the analysis of the spectrum of the Laplace operator on Y ε with the Neumann boundary conditions on S ε and so-calledθ-periodic boundary conditions on Y. Namely, let T n ={θ=(θ,...,θ n ) C n : θ =} Forθ T n we introduce the functional space H θ (Yε ) consisting of functions from H (Y ε ) that satisfy the following condition on Y: =, n : u(x+e )=θ u(x) for x=(x, x,..., /,..., x n ) -th place () where e = (0, 0,...,,..., 0).

6 6 Byη θ,ε we denote the sesquilenear form defined by formula (4) (with Y ε instead ofω) and the definitional domain H θ (Yε ). We define the operator A θ,ε as the operator acting in L (Y ε ) and associated with the formη θ,ε, i.e. (A ε,θ u, v) L (Y ε )=η ε,θ [u, v], u dom(a ε,θ ), v dom(η ε,θ ) The functions from dom(a θ,ε ) satisfy the Neumann boundary conditions on S ε, condition () on Y and the condition =, n : u u (x+e )=θ (x) for x=(x, x,..., /,..., x n ) () x x The operator A θ,ε has purely discrete spectrum. We denote by { λ θ,ε the sequence of eigenvalues of A θ,ε written in the increasing order and repeated according to their multiplicity. N The Floquet-Bloch theory (see, e.g., [ 3]) establishes the following relationship between the spectra of the operators A ε and A θ,ε : σ(a ε )= L, where L = = θ T n { } λ θ,ε The sets L are compact intervals. Also we need the Laplace operators on Y ε with the Neumann boundary conditions on S ε and either the Neumann or Dirichlet boundary conditions on Y= Y ε \ S ε. Namely, we denote by η N,ε (resp.η D,ε ) the sesquilinear form in L (Y ε ) defined by formula (4) (with Y ε instead ofω ε ) and the definitional domain H (Y ε ) (resp. Ĥ 0 (Yε )= { u H (Y ε ) : u=0on Y ε \ S ε} ). Then by A N,ε (resp. A D,ε ) we denote the operator associated with the formη N,ε (resp.η D,ε ), i.e. (A ε, u, v) L (Y ε )=η ε, [u, v], u dom(a ε, ), v dom(η ε, ) where is N (resp. D). The spectra of the operators A N,ε and A D,ε are purely discrete. We denote by { } λ N,ε { } (resp. N λ D,ε ) the sequence of eigenvalues of N AN,ε (resp. A D,ε ) written in the increasing order and repeated according to their multiplicity. From the min-max principle (see, e.g., [, Chapter XIII]) and the enclosure H (Y ε ) Hθ (Yε ) Ĥ 0 (Yε ) one can easily obtain the inequality N, θ T n : λ N,ε λ θ,ε } (3) λ D,ε (4) In this end of this subsection we introduce the operators which will be used in the description of the behaviour ofλ N,λD andλθ asε 0. By N F (resp. D F, θ F ) we denote the operator which acts in L (F) and is defined by the operation, the Neumann boundary conditions on B and the Neumann (resp. Dirichlet,θ-periodic) boundary conditions on Y. By B we denote the operator which acts in L (B) and is defined by the operation and the Neumann boundary conditions on B. Finally, we introduce the operators A N, A D, A θ which act in L (F) L (B) and are defined by the following formulae: ( ) ( ) ( ) A N N = F 0, A 0 D D = F 0, A B 0 θ θ = F 0 B 0 B

7 We denote by { } λ N (resp. { } λ D N,{ } λ θ N ) the sequence of eigenvalues of N AN (resp. A D, A θ ) written in the increasing order and repeated according to their multiplicity. It is clear that λ N =λn = 0,λN 3 > 0 (5) λ D = 0,λD > 0 (6) λ θ =λθ = 0,λθ 3 > 0 ifθ=(,,..., ) (7) λ θ = 0,λθ > 0 ifθ (,,..., ) (8).. Asymptotic behaviour of Dirichlet eigenvalues. We start from the description of the asymptotic behaviour of the eigenvalues of the operator A D,ε asε 0. Lemma.. For each None has Furthermore whereσis defined by formula (6). limλ D,ε ε 0 =λ D (9) λ D,ε σε asε 0 (0) Proof. We start from the proof of (9). It is based on the following abstract theorem. Theorem. (Iosifyan et al. []) LetH ε,h 0 be separable Hilbert spaces, letl ε :H ε H ε,l 0 : H 0 H 0 be linear continuous operators, iml 0 V H 0, wherevis a subspace inh 0. Suppose that the following conditions C C 4 hold: C. The linear bounded operators R ε :H 0 H ε exist such that R ε f f H ε 0 for any ε H 0 f V. Here >0is a constant. C. OperatorsL ε,l 0 are positive, compact and self-adjoint. The norms L ε L(H ε ) are bounded uniformly inε. C 3. For any f V: L ε R ε f R ε L 0 f H ε ε 0 0. C 4. For any sequence f ε H ε such that sup ε exist such that L ε f ε R ε w H ε ε=ε 0 0. Then for any N f ε H ε< the subsequenceε ε and w V µ ε ε 0 µ where{µ ε } = and{µ } = are the eigenvalues of the operatorslε andl 0, which are renumbered in the increasing order and with account of their multiplicity. Let us apply this theorem. We seth ε = L (Y ε ),H 0 = L (F) L (B),L ε = (A D,ε + I), L 0 = (A D + I) (here I is the identity operator),v=h 0, the operator R ε :H 0 H ε is defined by the formula [R ε f F (y), y F, f ](y)= f= ( f F, f B ) H 0 () f B (y), y B, Obviously conditions C (with =) and C hold (namely, L ε L(H ε ) ). Let us verify condition C 3. Let f= ( f F, f B ) H 0. We set f ε = R ε f, v ε =L ε f ε. By v ε F and vε B we denote the restrictions of v ε onto F and B correspondingly. It is clear that v ε F H (F) + vε B H (B) = vε H (Y ε ) fε L (Y ε ) = f H 0 () 7

8 8 We denote Ĥ 0 (F)={ w H (F) : w Y = 0 } Since Ĥ 0 (F) H (B) is compactly embedded intoh 0 then due to estimate () there is a subsequenceε εand v F Ĥ 0 (F), v B H (B) such that v ε F v ε B v ε=ε F wealy in H (F) and strongly in L (F) 0 v ε=ε B wealy in H (B) and strongly in L (B) 0 One has the integral equality [ ( v ε, w ε) ] + v ε w ε f ε w ε dy=0, w ε Ĥ 0 (Yε ) (4) (3) Y ε We introduce the set { W= (w F, w B ) C (F) C (B) : w F Y = 0, ( supp(wf ) supp(w B ) ) } {x 0 }= (here as usual by supp( f ) we denote the closure of the set{x : f (x) 0}). Let w=(w F, w B ) be an arbitrary function from W. We set w ε = R ε w. It follows from the definition of W that supp(w ε ) D ε = whenεis small enough and therefore w ε C (Y ε ) Ĥ 0 (Yε ). We substitute w ε into (4) and taing into account (3) we pass to the limit in (4) asε=ε 0. As a result we obtain F [ ( vf, w F ) + vf w F f F w F ] dy+ B [ ( vb, w B ) + vb w B f B w B ] dy=0 (5) Since W is a dense subspace of Ĥ 0 (F) H (B) then (5) is valid for an arbitrary (w F, w B ) Ĥ 0 (F) H (B). It follows from (5) that D F v F+ v F = f F and B v B + v B = f B and consequently v=l 0 f, where v=(v F, v B ) (6) We remar that v do not depend on the subsequenceε and therefore the whole sequence (v ε F, vε B ) converges to (v F, v B ) asε 0. Condition C 3 follows directly from (), (3) and (6). Obviously condition C 4 was proved during the proof of C 3. Thus the eigenvaluesµ ε of the operatorlε converges to the eigenvaluesµ of the operatorl 0 as ε 0. Butλ D,ε = (µ ε ),λ = (µ ) that implies (9). Now we focus on the proof of (0). Let v D,ε be the eigenfunction of A D,ε that corresponds to the eigenvalueλ D,ε and satisfies One has the following inequalities L (Y ε )= (7) v D,ε B 0 (8) L (F) C vd,ε L (F) (9) v D,ε B L (B) C vd,ε L (B) (30) B v D,ε B L (B) (3)

9 Here the first one is the Friedrichs inequality, the second one is the Poincaré inequality and the third one is the Cauchy inequality. Furthermore one has ( v D,ε L (Y ε ) =λd,ε v D,ε L (F) + vd,ε v D,ε B L (B) + B vd,ε B ) (3) Below we will prove (see inequality (56)) that Then it follows from (7), (9)-(33) that The last inequality can be specified. Namely, one has λ D,ε Cε (33) v D,ε L (Y ε ) Cε (34) L (F) Cε (35) v D,ε B L (B) Cε (36) = L (F) + vd,ε v D,ε B L (B) + B vd,ε B (37) Then in view of (35)-(37) v D,ε B B Cε Finally, taing into account (8) we get: lim ε 0 B / L (B) = 0 (38) Now we construct a convenient approximation v D,ε for the eigenfunction v D,ε. We consider the following problem ψ=0inr n \ T (39) ψ=in T (40) ψ(x)=o() as x (4) Recall that T={x R n : x <, x n = 0}, obviously T= T. It is well-nown that this problem has the unique solutionψ(x) satisfying ψ dx<. Moreover it has the following properties: The first two properties imply: R n \T ψ C (R n \ T) (4) ψ(x, x,..., x n, x n )=ψ(x, x,..., x n, x n ) (43) cap(t)= ψ dx (44) R n \T ψ = 0 in{x R n : x n = 0}\T (45) x n ψ + ψ = 0 in T (46) x n x n xn =+0 xn = 0 Furthermore the functionψ(x) satisfies the estimate (see, e.g, [3, Lemma.4]): D α ψ(x) C x n α for x >, α =0,, (47) 9

10 0 We define the function v D,ε by the formula ( ) ( ) x x 0 x x 0 v D,ε (x)= B ψ ϕ, x F d ε l ( ) ( ) x x 0 x x 0 (48) B B ψ ϕ, x B D ε d ε l whereϕ :R R is a twice-continuously differentiable function such that l is an arbitrary constant satisfying ϕ(ρ)=asρ / andϕ(ρ)=0 asρ, (49) 0<l< min{ b, b} (50) 4 Here we also suppose thatεis small enough so that d ε < l/. It is easy to see that the constructed function v D,ε (x) belongs to dom(a D,ε ) in view (40), (4), (45), (46), (49), (50). Taing into account (44), (47) we obtain: v D,ε L (Y ε ) 4 cap(t)d n B ε =σε (ε 0) (5) v D,ε L (Y ε ) Cε4 (5) Since v D,ε = 0 on Y and [ v D,ε B /] int= 0 on B\ { x : x x 0 l } (here [... ] int means the value of the function when we approach B from inside of B) we have the following Friedrichs inequalities L (F) C vd,ε L (F) (53) B / L (B) C vd,ε L (B) (54) It follows from (5), (53), (54) that L (Y ε ) (ε 0) (55) Using the min-max principle (see, e.g., [, Chapter XIII]) and taing into account (5), (55) we get λ D,ε = v D,ε L (Y ε ) vd,ε Now let us estimate the difference L (Y ε ) L (Y ε ) cap(t)d n B ε =σε (ε 0) (56) w ε = v D,ε v D,ε (57) One has w ε L (Y ε ) ( ) ( ) L (F) + vd,ε L (F) + v D,ε B / L (B ε ) + B / v D,ε L (B ε ) and thus in view of (35), (38), (5), (53), (54) we get w ε L (Y ε ) Cε (58) Substituting the equality v D,ε = v D,ε + w ε into (56) and integrating by parts we obtain w ε L (Y ε ) ( vd,ε, w ε ) L (Y ε )+ v D,ε L (Y ε ) v D,ε L (Y ε ) L (Y ε )

11 and in view of (5), (5), (55), (58) we conclude that Finally using (5), (59) we obtain The lemma is proved. limε w ε L ε 0 (Y ε ) = 0 (59) λ D,ε v D,ε L (Y ε ) σε (ε 0) (60).3. Asymptotic behaviour of Neumann eigenvalues. In this subsection we study the behaviour of the eigenvalues of the operator A N,ε asε 0. Lemma.. For each None has Furthermore whereµis defined by formula (6). limλ N,ε ε 0 =λ N (6) λ N,ε µε asε 0 (6) Proof. The proof of (6) is similar to the proof of (9), so we focus on the proof of (6). Let v N,ε be the eigenfunction of A N,ε that corresponds toλ N,ε and satisfies v N,ε L (Y ε )= (63) v N,ε B 0 (64) Since the eigenspace which corresponds toλ N,ε = 0 consists of constants then (v N,ε, ) L (Y ε )= 0 (65) and in view of the min-max principle we have λ N,ε = v N,ε L (Y ε ) = min v L (Y ε ) 0 v H# (Yε ) v L (Y ε ) where H # (Yε )= { v H (Y ε ) : (v ε, ) L (Y ε )= 0 } Below we will prove (see inequality (76)) that λ N,ε Cε Then in the same way as we obtain the estimates for v D,ε in Lemma. we derive the estimates for v N,ε : v N,ε L (Y ε ) Cε (67) v N,ε v N,ε F L (F) + vn,ε v N,ε B L (B) Cε (68) Using (67), (68) we obtain from (63) and (65) that v N,ε F F + v N,ε B B (ε 0) v N,ε F F + v N,ε B B =0 and therefore taing into account (64) we get (66) v N,ε F B / F, v N,ε B F / B (ε 0) (69)

12 We construct an approximation v N,ε for the eigenfunction v N,ε by the following formula: ( ) ( ) B v N,ε (x)= F + x x 0 x x 0 B F ψ ϕ, x F d ε l ( ) ( ) (70) F B x x 0 x x 0 B F ψ ϕ, x B D ε d ε l Recall that ψ is a solution of (39)-(4), ϕ : R R is a twice-continuously differentiable function satisfying (49), l is a constant satisfying (50). One can easily show (taing into account the equality B + F =) that v N,ε dom(a N,ε ). In the same way as we obtain the estimates for v D,ε in Lemma. we get the estimates for v N,ε : v N,ε L (Y ε ) 4 F B cap(t)d n ε =µε (ε 0) (7) v D,ε L (Y ε ) Cε4 (7) v N,ε + B / F + v N,ε L (F) F / B L (B) Cε (73) v N,ε Y ε 0 (ε 0) (74) v N,ε (ε 0) (75) Since v N,ε v N,ε H # (Yε ) then it follows from (66), (7), (74), (75) that L (Y ε ) v N,ε v N,ε Y ε L (Y ε ) λ N,ε = v N,ε L (Y ε ) v N,ε Now let us estimate the difference µε (ε 0) (76) w ε = v N,ε v N,ε (77) One has ( w ε L (Y ε ) v N,ε + B / F + B / F v N,ε ) L (F) + L (F) ( + v N,ε F / B + F / B v N,ε ) L (B ε ) and thus in view of (68), (69), (73) we get L (B ε ) lim w ε L ε 0 (Y ε ) = 0 (78) Substituting the equality v N,ε = v N,ε + w ε into (76) and integrating by parts we get w ε L (Y ε ) ( vn,ε, wε ) L (Y ε )+ v N,ε L (Y ε ) v N,ε v N,ε Y ε L (Y ε ) v N,ε L (Y ε ) and then in view of (7), (7), (74), (75), (78) we conclude that limε w ε L ε 0 (Y ε ) = 0 (79) Finally using (7), (79) we obtain λ N,ε v N,ε L (Y ε ) µε (ε 0) (80) The lemma is proved.

13 .4. Asymptotic behaviour of θ-periodic eigenvalues. To complete the proof of Theorem. we have study the behaviour of the spectrum of the operator A θ,ε. Lemma.3. For eachθ T n and None has 3 Furthermore limλ θ,ε ε 0 =λ θ (8) λ θ,ε µε ifθ=(,,..., ) (8) λ θ,ε σε ifθ (,,..., ) (83) Proof. The proof of (8) is similar to the proof of (9). The proof of (8) is similar to the proof of (6). Namely, we approximate the eigenfunction v θ,ε of A θ,ε that corresponds toλ θ,ε and satisfies v θ,ε L (Y ε )=, v θ,ε B 0 by the function v N,ε (70) (since v N,ε is constant in the vicinity of Y then it satisfies ()-() withθ=(,..., )). The justification of the asymptotic equality λ θ,ε v N,ε L (Y ε ),θ=(,..., ) is completely similar to the proof of (80). Now, we focus on the proof of (83). Let v θ,ε be the eigenfunction of A θ,ε that corresponds toλ θ,ε and satisfies v θ,ε L (Y ε )=, v θ,ε B 0. Sinceθ (,..., ) then there exists l {,..., n} such that u(x+e l )=θ l u(x) for x=(x, x,..., /,..., x n ), whereθ l (84) l-th place We denote by S ± l the faces of Y which are orthogonal to the axis x l that is { S ± l = x Y : x l =± } We need an additional estimate Lemma.4. For any v H (F) the following inequality holds: v S ± l v F C v L (F) (85) (here v S ± l = S ± l S ± l v(x)ds x, where ds x is the volume (area) form on S ± l, S± l = S ± l ds x ). Proof. Let v be an arbitrary function from C (F). One has α v(x) v(x+αe l )= 0 dv dt (x+e lt)dt (86) where x S l, 0<α<( b)/ (and therefore x+αe l F). We denote F={x F : /< x n < b/}. Integrating equality (86) by x, x,..., x l, x l+,..., x n from / to / and byαfrom 0 to ( b)/,

14 4 dividing by ( b)/ and squaring we get v S l v F = / / ( b)/ α = dv dt (x+e lt)dt dαdx... dx l dx l+... dx n / / 0 0 C v L (F) (87) The fulfilment of inequality (87) for v H (F) follows from the standard embedding and trace theorems. It is well-nown (see, e.g, [3, Chapter 4]) that the operatorπ : H (F) H (Y) exists such that for an arbitrary v H (F) one has Πv F = v, Πv H (Y) C v H (F) (88) We have the estimate which follows directly from [8, Lemma 3.] v F v F C Πv L (Y) (89) Then inequality (85) (with S l ) follows from (87)-(89). The proof of (85) with S+ l Lemma.4 is proved. Now using Lemma.4 and (84) we get v θ,ε = θ l v θ,ε F θ l v θ,ε F F ( v θ l θ,ε F v θ,ε S + +θ l It follows from (90) and the Poincaré inequality that v θ,ε L (F) = vθ,ε v θ,ε F L (F) + v θ,ε F v θ,ε S v θ,ε F is similar. ) C v θ,ε L (F) (90) θ,ε F C v L (F) (9) Thus similarly to the Dirichlet eigenfunction v D,ε (see (9)) the function v θ,ε satisfies the Friedrichs inequality in F (although v θ,ε 0 on Y!). As for the rest the proof of (83) repeats word-by-word the proof of (0): we approximate the eigenfunction v θ,ε by the function v D,ε (48) (since v D,ε = 0 in the vicinity of Y then it satisfies ()-() with an arbitraryθ) and then prove the asymptotic equality λ θ,ε v D,ε L (Y ε ),θ (,..., ) (9) The proof of (9) (taing into account (9)) is completely similar to the proof of (60). Lemma.3 is proved..5. End of the proof of Theorem.. It follows from (0) and (3) that σ(a ε )= [a (ε), a+ (ε)] (93) where the compact intervals [a (ε), a+ (ε)] are defined by [a (ε), a+ (ε)]= { } ε λ θ,ε = θ T n (94)

15 We denoteθ = (,,..., ),θ = θ. It follows from (4) and (94) that ε λ N,ε a (ε) ε λ θ,ε (95) ε λ θ,ε a + (ε) ε λ D,ε (96) Obviously if = then the left and right-hand-sides of (95) are equal to zero. It follows from (6), (8) that in the case = they both converge toµasε 0, while if 3 they converge to infinity in view of (5), (7), (6), (8). Thus a (ε)=0, lim ε 0 a (ε)=µ, lim ε 0 a (ε)=, =3, 4, 5... (97) Similarly in view of (6), (8), (9), (0), (8), (83) one has lim a + (ε)=σ, lim ε 0 ε 0 a+ (ε)=, =, 3, 4... (98) Then (7)-(8) follow directly from (93), (97)-(98). Theorem. is proved. 3. Application to the theory of D photonic crystals In this section we apply the results obtained above to the theory of D photonic crystals. Photonic crystal is a dielectric medium with periodic structure whose main property is that the electromagnetic waves of a certain frequency cannot propagate in it without attenuation. From the mathematical point of view it means that the corresponding Maxwell operator has gaps in its spectrum. We refer to [4 6] for more details. It is nown that if the crystal is periodic in two directions and homogeneous with respect to the third one (so-called D photonic crystals) then the analysis of the Maxwell operator reduces to the analysis of scalar elliptic operators. In the case when dielectric medium occupies the entire space the rigorous justification of this reduction was carried out in [0] (in this wor spectral gaps open up due to a high contrast electric permittivity). In the current wor we consider the dielectric medium with a periodic family of perfectly conducting trap-lie screens embedded into it. It means that we should supplement the Maxwell equations by suitable boundary conditions on these screens. In this case the analysis of the Maxwell operator reduces to the analysis of the Neumann or Dirichlet Laplacians in a -dimensional domain which is a cross-section of the crystal along periodicity plane. In this wor we derive this reduction on a formal level of rigour. Then using Theorem. and Lemma 3. below we open up gaps in the spectrum of the Maxwell operator. Let us introduce the following sets inr 3 : S ε i= { (x, x, z) : x=(x, x ) S ε i, z R},Ω ε ={(x, x, z) : x=(x, x ) Ω ε, z R} where S ε i andω ε belong tor and are defined by () and (3) correspondingly. We suppose that Ω ε is occupied by a dielectric medium while the union of the screens S ε i is occupied by a perfectly conducting material. It is supposed that the electric permittivity and the magnetic permeability of the material occupyingω ε are equal to. Then the propagation of electro-magnetic waves inω ε is described by the Maxwell equations curle= H t, curlh= E t, divh=dive=0 supplemented by the following boundary conditions on S ε i : i Z E τ = 0, H ν = 0 5

16 6 Here E and H are the electric and magnetic fields, E τ and H ν are the tangential and normal components of E and H correspondingly. Looing for monochromatic waves E(x, t)=e iωt E(x), H(x; t)=e iωt H(x) we obtain M ε U=ωU, divh=dive=0 inω ε, E τ = 0, H ν = 0 on S ε i (99) i Z n where U=(E, H) T,M ε is the Maxwell operator: ( ) 0 curl M ε = i curl 0 (the subscriptεemphasizes that this operator acts on functions defined inω ε ). For more precise definition of the Maxwell operator we refer to [4] (the theory developed in this paper covers, in particular, domains with a screen-lie boundary). We are interested on the waves propagated along the plane z=0, i.e. when (E, H) depends on x=(x, x ) only: E=(E (x, x ), E (x, x ), E 3 (x, x )), H=(H (x, x ), H (x, x ), H 3 (x, x )) (00) Also we suppose that U is a Bloch wave that is θ=(θ,...,θ n ) T n : U(x+i)=θ i U(x), i Z n (0) (hereθ i = n (θ ) i ). We call the set ofωfor which there is U=(E, H) 0 satisfying (99)-(0) a = (Bloch) spectrum of the operatorm ε. We denote it byσ(m ε ). Using (00) we can easily rewrite the equalitym ε U=ωU as i H 3 =ωe, i H ( 3 H =ωe, i H ) =ωe 3 (0) x x x x i E 3 =ωh, i E ( 3 E =ωh, i E ) =ωh 3 (03) x x x x Let us show that ω σ(m ε ) ω σ(a ε 0 ) σ(aε ) (04) wherea ε 0 andaε are, correspondingly, the Dirichlet and Neumann Laplacians inω ε. Suppose that ω σ(m ε ). Ifω=0 thenω σ(a ε ) otherwise we express H and H (resp. E and E ) from the first two equalities in (03) (resp. (0)) and plug them into the third equality in (0) (resp. (03)). As a result we get the following equalities onω ε : E 3 =ω E 3 (resp. H 3 =ω H 3 ) (05) Let n=(n, n, 0) be the unit normal to S ε i. Since E τ=0 then E n and therefore E (0, 0, ) and E ( n, n, 0). Taing this and the first two equalities in (0) into account we obtain the following boundary conditions for E 3 and H 3 on i Z n S ε i : E 3 = 0, H 3 n = H 3 x n + H 3 x n = iω ( E n +λ θ,ε µε ifθ=(,,..., ))=0 (06) Furthermore it follows from (0)-(03) that U 0 E 3 0 or H 3 0 (07)

17 It follows from (05)-(07) and (0) thatω belongs to the spectrum of either the Dirichlet or Neumann Laplacian inω ε. The converse implication in (04) is proved similarly. Remar 3.. Above we have dealt with the space J= { (E, H) : dive=divh=0onω ε, E τ = H µ = 0 on S ε i and (00) (0) holds } We introduce the following subspaces J E ={(E, H) J : E = E = H 3 = 0}, J H ={(E, H) J : H = H = E 3 = 0} Their elements are usually called E- and H-polarized waves. These subspaces are L -orthogonal and it is easy to see that each U J can be represented in unique way as U=U E + U H, where U E J E, U H J H Moreover J E and J H are invariant subspaces ofm ε. Thusσ(M ε ) is a union ofσ(m ε JE ) (Esubspectrum) andσ(m ε JH ) (H-subspectrum). We have just shown that the set of squares of points from E-subspectrum is just the spectrum of the Dirichlet Laplacian, while the the set of squares of points from H-subspectrum is the spectrum of the Neumann Laplacian. The spectrum ofa ε has been studied above (Theorem.). Now, we describe the spectrum of the Dirichlet Laplacian inω ε. We define it via a sesquilinear formη ε 0 which is defined by formula (4) and the definitional domain dom(η ε 0 )=H 0 (Ωε ) (here H 0 (Ωε ) is a closure in H (Ω ε ) of the set of compactly supported inω ε functions). The Diriclet Laplacian inω ε (we denote ita ε 0 ) is the operator which is generated by this form, i.e. (5) holds (witha ε 0,ηε 0 instead ofaε,η ε ). It turns out that its spectrum goes to infinity asε 0, namely the following lemma holds true. Lemma 3.. One has: min{λ :λ σ(a ε 0 )} ε 0 (08) Proof. For i Z n we denote B ε i=ε(b+i), F ε i=ε(f+ i). Let u doma ε 0. Since u=0 on Sε i one has the following Friedrichs inequalities: u L (B ε i ) Cε u L (B ε i ), u L (F ε i ) Cε u L (F ε i ) Here C is independent of u, i andε. Summing up these equalities by i Z n and then integrating by parts we get: (A ε 0 u, u) L (Ω ε ) C ε u L (Ω ε ) (09) It follows from (09) (see, e.g., [5, Chapter 6]) that the ray (, C ε ) belongs to the resolvent set ofa ε 0 that imply (08). The lemma is proved. Thus using Theorem., Lemma 3. and (04) we conclude that the spectrum of the Maxwell operatorm ε has just two gaps in large finite intervals whenεis small enough; asε 0 these gaps converge to the intervals ( σ, µ) and ( σ, µ). References [] Reed M, Simon B. Methods of Modern Mathematical Physics. IV: Analysis of Operators. Academic Press: New- Yor, 978. [] Kuchment P. Floquet Theory For Partial Differential Equations. Birhauser Verlag: Basel, 993. [3] Brown BM, Hoang V, Plum M, Wood IG. Floquet-Bloch theory for elliptic problems with discontinuous coefficients. Spectral theory and analysis, Oper. Theory Adv. Appl., 4, Birhuser/Springer Basel AG, Basel, 0. DOI: 0.007/

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