Homogenization of micro-resonances and localization of waves.

Transcription

1 Homogenization of micro-resonances and localization of waves. Valery Smyshlyaev University College London, UK July 13, 2012 (joint work with Ilia Kamotski UCL, and Shane Cooper Bath/ Cardiff) Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves.july 13, / 23

2 Outline: - Classical homogenization for waves ( = low frequencies) - Higher frequencies + higher contrasts ( degeneracies ) resonant homogenization - Effects: (frequency/wavenumber) band gaps, dispersion, negative materials, etc. - Partial degeneracies and resonances (more of effects; general theory) - Photonic Crystal Fibers as an example of partial degeneracies Band gaps in PCFs. (Cooper, Kamotski, V.S., 2012). Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves.july 13, / 23

3 Classical Periodic Homogenisation for waves (= low frequencies) ( anti-plane elastodynamics/ TM electrodynamics, for simplicity) Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves.july 13, / 23

4 Classical Periodic Homogenisation for waves (= low frequencies) ( anti-plane elastodynamics/ TM electrodynamics, for simplicity) ρ ε (x)u tt div (a ε (x) u ε )) = f (x, t) Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves.july 13, / 23

5 Classical Periodic Homogenisation for waves (= low frequencies) ( anti-plane elastodynamics/ TM electrodynamics, for simplicity) ρ ε (x)u tt div (a ε (x) u ε )) = f (x, t) f (x, t) 0, t 0; u(x, t) 0, t 0. a ε (x) = a(x/ε), ρ ε = ρ(x/ε) a stiffness, ρ density a(y), ρ(y) Q-periodic in y Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves.july 13, / 23

6 Classical Periodic Homogenisation for waves (= low frequencies) ( anti-plane elastodynamics/ TM electrodynamics, for simplicity) ρ ε (x)u tt div (a ε (x) u ε )) = f (x, t) f (x, t) 0, t 0; u(x, t) 0, t 0. a ε (x) = a(x/ε), ρ ε = ρ(x/ε) a stiffness, ρ density a(y), ρ(y) Q-periodic in y Asymptotic expansion u ε (x, t) u 0 (x, x/ε, t) + εu 1 (x, x/ε, t) +... u 0 (x, y, t), u 1 (x, y, t) Q-periodic in y Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves.july 13, / 23

7 Classical Periodic Homogenisation ε 2 : u 0 = u(x, t) ( long wavelengths low frequencies ) alery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves.july 13, / 23

8 Classical Periodic Homogenisation ε 2 : u 0 = u(x, t) ( long wavelengths low frequencies ) ε 1 : u 1 = j N j (y) u0 x j (x, t) N solutions of cell problem div y ( a(y)(e j + y N j ) ) = 0 Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves.july 13, / 23

9 Classical Periodic Homogenisation ε 2 : u 0 = u(x, t) ( long wavelengths low frequencies ) ε 1 : u 1 = j N j (y) u0 x j (x, t) ( N solutions of cell problem div y a(y)(e j + y N j ) ) = 0 ( ) ε 0 : ρ hom utt 0 div A hom u 0 ) = f (x), homogenized eqn ρ hom = ρ(y) y A hom = a(y) (I + y N) y homogenised (averaged) density homogenised stiffness tensor Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves.july 13, / 23

10 Classical Periodic Homogenisation ε 2 : u 0 = u(x, t) ( long wavelengths low frequencies ) ε 1 : u 1 = j N j (y) u0 x j (x, t) ( N solutions of cell problem div y a(y)(e j + y N j ) ) = 0 ( ) ε 0 : ρ hom utt 0 div A hom u 0 ) = f (x), homogenized eqn ρ hom = ρ(y) y A hom = a(y) (I + y N) y homogenised (averaged) density homogenised stiffness tensor 0 < νi A hom, ρ hom ν 1 I (uniform positivity inherited) Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves.july 13, / 23

11 Classical Periodic Homogenisation ε 2 : u 0 = u(x, t) ( long wavelengths low frequencies ) ε 1 : u 1 = j N j (y) u0 x j (x, t) ( N solutions of cell problem div y a(y)(e j + y N j ) ) = 0 ( ) ε 0 : ρ hom utt 0 div A hom u 0 ) = f (x), homogenized eqn ρ hom = ρ(y) y A hom = a(y) (I + y N) y homogenised (averaged) density homogenised stiffness tensor 0 < νi A hom, ρ hom ν 1 I (uniform positivity inherited) The Homogenisation Theorem C > 0 independent of ε such that u ε (u 0 + εu 1 ) H 1 Cε 1/2 Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves.july 13, / 23

12 High-contrast (= micro-resonant ) homogenization and non-classical two-scale limits (Zhikov 2000, 2004) A ε u = div (a ε (x) u ε ) { a ε ε 2 on Ω (x) = ε 0 ( soft phase) 1 on Ω ε 1 ( stiff phase) Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves.july 13, / 23

13 High-contrast (= micro-resonant ) homogenization and non-classical two-scale limits (Zhikov 2000, 2004) A ε u = div (a ε (x) u ε ) { a ε ε 2 on Ω (x) = ε 0 ( soft phase) 1 on Ω ε 1 ( stiff phase) Contrast δ ε 2 is a critical scaling giving rise to non-classical effects (Khruslov 1980s; Arbogast, Douglas, Hornung 1990; Panasenko 1991; Allaire 1992; Sandrakov 1999; Brianne 2002; Bourget, Mikelic, Piatnitski 2003; Bouchitte & Felbaq 2004,...): elliptic, spectral, parabolic, hyperbolic, nonlinear, non-periodic/ random,.... Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves.july 13, / 23

14 High-contrast (= micro-resonant ) homogenization and non-classical two-scale limits (Zhikov 2000, 2004) A ε u = div (a ε (x) u ε ) { a ε ε 2 on Ω (x) = ε 0 ( soft phase) 1 on Ω ε 1 ( stiff phase) Contrast δ ε 2 is a critical scaling giving rise to non-classical effects (Khruslov 1980s; Arbogast, Douglas, Hornung 1990; Panasenko 1991; Allaire 1992; Sandrakov 1999; Brianne 2002; Bourget, Mikelic, Piatnitski 2003; Bouchitte & Felbaq 2004,...): elliptic, spectral, parabolic, hyperbolic, nonlinear, non-periodic/ random,.... WHY? Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves.july 13, / 23

15 Two-scale formal asymptotic expansion: div (a ε (x) u ε ) + ρω 2 u ε = 0 (timeharmonicwaves) A ε u ε = λu ε, λ = ρω 2 (spectral problem). Seek u ε (x) u 0 (x, x/ε) + εu 1 (x, x/ε) +... u j (x, y) Q periodic in y. Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves.july 13, / 23

16 Two-scale formal asymptotic expansion: div (a ε (x) u ε ) + ρω 2 u ε = 0 (timeharmonicwaves) A ε u ε = λu ε, λ = ρω 2 (spectral problem). Seek u ε (x) u 0 (x, x/ε) + εu 1 (x, x/ε) +... u j (x, y) Q periodic in y. THEN: Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves.july 13, / 23

17 Two-scale limit problem (Zhikov 2000, 2004) Then u 0 (x, y) = u 0 (x) in Q 1 (still low frequency) w(x,y) in Q 0 ( resonance frequency) alery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves.july 13, / 23

18 Two-scale limit problem (Zhikov 2000, 2004) Then u 0 (x, y) = u 0 (x) in Q 1 (still low frequency) w(x,y) in Q 0 ( resonance frequency) (u 0, w), w(x, y) := u 0 (x) + v(x, y), solves the two-scale limit spectral problem: div x (a hom x u(x)) = λu(x) + λ < v > (x) in Ω div y (a (0) y v(x, y)) = λ(u(x) + v(x, y)) in Q 0 v(x, y) = 0 on Q 0 alery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves.july 13, / 23

19 Two-scale limit problem (Zhikov 2000, 2004) Then u 0 (x, y) = u 0 (x) in Q 1 (still low frequency) w(x,y) in Q 0 ( resonance frequency) (u 0, w), w(x, y) := u 0 (x) + v(x, y), solves the two-scale limit spectral problem: div x (a hom x u(x)) = λu(x) + λ < v > (x) in Ω div y (a (0) y v(x, y)) = λ(u(x) + v(x, y)) in Q 0 v(x, y) = 0 on Q 0 alery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves.july 13, / 23

20 Two-scale limit problem (Zhikov 2000, 2004) Then u 0 (x, y) = u 0 (x) in Q 1 (still low frequency) w(x,y) in Q 0 ( resonance frequency) (u 0, w), w(x, y) := u 0 (x) + v(x, y), solves the two-scale limit spectral problem: div x (a hom x u(x)) = λu(x) + λ < v > (x) in Ω div y (a (0) y v(x, y)) = λ(u(x) + v(x, y)) in Q 0 v(x, y) = 0 on Q 0 Decouple it alery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves.july 13, / 23

21 Two-scale limit spectral problem Decouple by choosing v(x, y) = λu(x)b(y) div y (a (0) y b(y)) λu = 1 in Q 0 b(y) = 0 on Q 0 div x (a hom u(x)) = β(λ)u(x), in Ω, where β(λ) = λ + λ 2 φ j 2 y λ j λ, (λ j, φ j ) Dirichlet eigenvalues/functions of inclusion Q 0 ( = micro-resonances : β < 0 negative denstity/magnetism etc) j=1 Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves.july 13, / 23

22 Rigorous analysis: Two-scale Convergence Definition 1. Let u ε (x) be a bounded sequence in L 2 (Ω). We say (u ε ) weakly two-scale converges to u 0 (x, y) L 2 2 (Ω Q), denoted by u ε u 0, if for all φ C0 (Ω), ψ C # (Q) ( x ) u ε (x)φ(x)ψ dx u 0 (x, y)φ(x)ψ(y) dxdy Ω ε Ω Q as ε We say (u ε ) strongly two-scale converges to u 0 L 2 (Ω Q), denoted 2 2 by u ε u 0, if for all v ε v 0 (x, y), u ε (x)v ε (x)dx u 0 (x, y)v 0 (x, y) dxdy Ω as ε 0. (implies convergence of norms upon sufficient regularity) Ω Q Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves.july 13, / 23

23 Rigorous analysis (Zhikov 2000, 2004): 1. Two-scale resolvent convergence: α > 0, A ε u ε + αu ε = f ε L 2 (Ω); u ε H0 1(Ω). If f ε 2 f 0 (x, y) then u ε 2 u 0 (x, y). (If f ε 2 f 0 (x, y) then u ε 2 u 0 (x, y).) Here u 0 solves two-scale limit problem A 0 u 0 + αu 0 = f 0. Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23

24 Rigorous analysis (Zhikov 2000, 2004): 1. Two-scale resolvent convergence: α > 0, A ε u ε + αu ε = f ε L 2 (Ω); u ε H0 1(Ω). If f ε 2 f 0 (x, y) then u ε 2 u 0 (x, y). (If f ε 2 f 0 (x, y) then u ε 2 u 0 (x, y).) Here u 0 solves two-scale limit problem A 0 u 0 + αu 0 = f Spectral band gaps: (Let Ω = R d.) A 0 self-adjoint in H L 2 (R Q 0 ), with a band-gap spectrum σ(a 0 ). σ(a ε ) σ(a 0 ) in the sense of Hausdorff. (Hence a Band-gap effect: For small enough ε waves of certain frequencies do not propagate, in any direction). The proof is based on a two-scale spectral compactness. Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23

25 Rigorous analysis (Zhikov 2000, 2004): 1. Two-scale resolvent convergence: α > 0, A ε u ε + αu ε = f ε L 2 (Ω); u ε H0 1(Ω). If f ε 2 f 0 (x, y) then u ε 2 u 0 (x, y). (If f ε 2 f 0 (x, y) then u ε 2 u 0 (x, y).) Here u 0 solves two-scale limit problem A 0 u 0 + αu 0 = f Spectral band gaps: (Let Ω = R d.) A 0 self-adjoint in H L 2 (R Q 0 ), with a band-gap spectrum σ(a 0 ). σ(a ε ) σ(a 0 ) in the sense of Hausdorff. (Hence a Band-gap effect: For small enough ε waves of certain frequencies do not propagate, in any direction). The proof is based on a two-scale spectral compactness. Limit band gaps ={λ : β(λ) < 0}. Interpretation : β(λ) = µ(ω) < 0 negative magnetism/density (Bouchitte & Felbacq, 2004). Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23

26 Rigorous analysis (Zhikov 2000, 2004): 1. Two-scale resolvent convergence: α > 0, A ε u ε + αu ε = f ε L 2 (Ω); u ε H0 1(Ω). If f ε 2 f 0 (x, y) then u ε 2 u 0 (x, y). (If f ε 2 f 0 (x, y) then u ε 2 u 0 (x, y).) Here u 0 solves two-scale limit problem A 0 u 0 + αu 0 = f Spectral band gaps: (Let Ω = R d.) A 0 self-adjoint in H L 2 (R Q 0 ), with a band-gap spectrum σ(a 0 ). σ(a ε ) σ(a 0 ) in the sense of Hausdorff. (Hence a Band-gap effect: For small enough ε waves of certain frequencies do not propagate, in any direction). The proof is based on a two-scale spectral compactness. Limit band gaps ={λ : β(λ) < 0}. Interpretation : β(λ) = µ(ω) < 0 negative magnetism/density (Bouchitte & Felbacq, 2004). β(λ) > 0 - propagation with (high) dispersion (λ λ j ), due to coupled resonances (V.S. & P. Kuchment, 2007). Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23

27 Frequency vs directional gaps and partial degeneracies Cherednichenko, V.S., Zhikov (2006): spatial nonlocality for homogenised limit with highly anisotropic fibers. 1 in Q 1 (matrix) a ε (x) = ε 2 in Q 0 across fibers 1 in Q 0 along fibers alery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23

28 Frequency vs directional gaps and partial degeneracies Cherednichenko, V.S., Zhikov (2006): spatial nonlocality for homogenised limit with highly anisotropic fibers. 1 in Q 1 (matrix) a ε (x) = ε 2 in Q 0 across fibers 1 in Q 0 along fibers V.S. (2009): a directional localization (via formal asymptotic expansions): for certain frequencies waves can propagate in some directions (e.g. along the fibers above) but cannot in others (e.g. orthogonal to the fibers). Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23

29 Frequency vs directional gaps and partial degeneracies Cherednichenko, V.S., Zhikov (2006): spatial nonlocality for homogenised limit with highly anisotropic fibers. 1 in Q 1 (matrix) a ε (x) = ε 2 in Q 0 across fibers 1 in Q 0 along fibers V.S. (2009): a directional localization (via formal asymptotic expansions): for certain frequencies waves can propagate in some directions (e.g. along the fibers above) but cannot in others (e.g. orthogonal to the fibers). Notice, in the above fibres, a ε (x) = a (1) (x/ε) + ε 2 a (0) (x/ε), where a (1) = 0 0 0, i.e. is partially degenerate Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23

30 Photonic crystal fibers: A partially degenerate problem (Cooper, I. Kamotski, V.S. 2012); cf scalar prototype problem I.Kamotski V.S. 2006; M. Cherdantsev 2009 ( full contrast) Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23

31 Photonic crystal fibers: Problem Formulation E = iωµh, H = iωɛe ɛ = ɛ 0 χ 0 (x/ε) + ɛ 1 χ 1 (x/ε) ɛ 0 > ɛ 1, µ constant E=exp(ikx 3 )E(x 1, x 2 ), H=exp(ikx 3 )H(x 1, x 2 ) Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23

32 Photonic crystal fibers: Problem Formulation E = iωµh, H = iωɛe ɛ = ɛ 0 χ 0 (x/ε) + ɛ 1 χ 1 (x/ε) ɛ 0 > ɛ 1, µ constant E=exp(ikx 3 )E(x 1, x 2 ), H=exp(ikx 3 )H(x 1, x 2 ) In each phase E 3 and H 3 satisfy the following equations E 3 + a ε E 3 = 0, H 3 + a ε H 3 = 0 where a ε = ω 2 µɛ(x/ε) - k 2. E 3 and H 3 coupled across interface Γ ε : [ ɛ ] [ ] [ ] 1 1 [ µ ] ω a ε E 3 n = k a ε H 3 n, k a ε E 3 n = ω a ε H 3 n Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23

33 Photonic crystal fibers: A partially degenerate problem Consider the problem in its weak form: ( ωɛ ) ( ωɛ ) ( ) ( ) k k 1 a ε E 3,1 + 2 a ε E 3,2 + 1 a ε H 3,2 2 a ε H 3,1 = ωɛe 3 ( ) ( ) k k ( ωµ ) ( ωµ ) 1 a ε E 3,2 2 a ε E 3,1 1 a ε H 3,1 2 a ε H 3,2 = ωµh 3. alery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23

34 Photonic crystal fibers: A partially degenerate problem Consider the problem in its weak form: ( ωɛ ) ( ωɛ ) ( ) ( ) k k 1 a ε E 3,1 + 2 a ε E 3,2 + 1 a ε H 3,2 2 a ε H 3,1 = ωɛe 3 ( ) ( ) k k ( ωµ ) ( ωµ ) 1 a ε E 3,2 2 a ε E 3,1 1 a ε H 3,1 2 a ε H 3,2 = ωµh 3. For u = (E 3, H 3 ), find u such that ω ( ) k ({ } { }) ɛ u1 R 2 a ε φ 1 + µ u 2 φ 2 + φ1 a ε, u 2 + u1, φ 2 dx = ωρ(x/ε)u φ φ C0 (R 2 ) R 2 ( ) ɛ(y) 0 {u, v} := u,1 v,2 v,1 u,2 (Poisson bracket); ρ(y) = 0 µ Form positive if k 2 < ω 2 µɛ 1. Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23

35 Photonic crystal fibers: A partially degenerate problem Consider the problem in its weak form: ( ωɛ ) ( ωɛ ) ( ) ( ) k k 1 a ε E 3,1 + 2 a ε E 3,2 + 1 a ε H 3,2 2 a ε H 3,1 = ωɛe 3 ( ) ( ) k k ( ωµ ) ( ωµ ) 1 a ε E 3,2 2 a ε E 3,1 1 a ε H 3,1 2 a ε H 3,2 = ωµh 3. For u = (E 3, H 3 ), find u such that ω ( ) k ({ } { }) ɛ u1 R 2 a ε φ 1 + µ u 2 φ 2 + φ1 a ε, u 2 + u1, φ 2 dx = ωρ(x/ε)u φ φ C0 (R 2 ) R 2 ( ) ɛ(y) 0 {u, v} := u,1 v,2 v,1 u,2 (Poisson bracket); ρ(y) = 0 µ Form positive if k 2 < ω 2 µɛ 1. Consider a near critical k: k 2 = ω 2 µ(ɛ 1 ε 2 ), ω 2 µ = λ Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23

36 Photonic crystal fibers: a partially degenerate problem An emergent high contrast: A ε (x) u φ dx = λ R 2 ρ(x/ε)u φdx R 2 φ C0 (R 2 ) Here A ε (x) = A(x/ε) where A(y) = a (1) (y) + ε 2 a (0) (y) + O(ε 4 ). a (1) 0 BUT a (1) (y) + a (0) (y) > νi, ν > 0 a (1) (y) u u = χ 1 (y) ( u 1,1 + u 2,2 2 + u 1,2 u 2,1 2) (partially) DEGENERATES for u s.t. RHS zero. i.e. satisfies Cauchy-Riemann type equations in matrix phase. Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23

37 Photonic crystal fibers: a partially degenerate problem An emergent high contrast: A ε (x) u φ dx = λ R 2 ρ(x/ε)u φdx R 2 φ C0 (R 2 ) Here A ε (x) = A(x/ε) where A(y) = a (1) (y) + ε 2 a (0) (y) + O(ε 4 ). a (1) 0 BUT a (1) (y) + a (0) (y) > νi, ν > 0 a (1) (y) u u = χ 1 (y) ( u 1,1 + u 2,2 2 + u 1,2 u 2,1 2) (partially) DEGENERATES for u s.t. RHS zero. i.e. satisfies Cauchy-Riemann type equations in matrix phase. Moral: This appears a particular case of homogenization for partially degenerating PDE systems Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23

38 General Partial Degeneracies (I. Kamotski and V.S. 2012) Consider a resolvent problem: Ω R d, α > 0, - div (a ε (x) u ε ) + αρ ε u ε = f ε L 2 (Ω), u ε ( H 1 0 (Ω)) n, n 1. A general degeneracy a ε (x) = a (1) ( ) x ε + ε 2 a (0) ( ) x ε, ( ) n d n d a (l) L # (Q), a (1) 0, a (1) + a (0) > 0, Weak formulation: Ω [ ( a (1) x ) ( u φ(x) + ε 2 a (0) x ) ] u φ(x) + α ρ ε (x) u φ(x) dx ε ε = f ε (x) φ(x) dx, φ ( H0 1 (Ω) ) d. Ω alery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23

39 General Partial Degeneracies (I. Kamotski and V.S. 2012) Consider a resolvent problem: Ω R d, α > 0, - div (a ε (x) u ε ) + αρ ε u ε = f ε L 2 (Ω), u ε ( H 1 0 (Ω)) n, n 1. A general degeneracy a ε (x) = a (1) ( ) x ε + ε 2 a (0) ( ) x ε, ( ) n d n d a (l) L # (Q), a (1) 0, a (1) + a (0) > 0, Weak formulation: Ω [ ( a (1) x ) ( u φ(x) + ε 2 a (0) x ) ] u φ(x) + α ρ ε (x) u φ(x) dx ε ε = f ε (x) φ(x) dx, φ ( H0 1 (Ω) ) d. A priori estimates: Ω alery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23

40 Weak two-scale limits. Key assumption on the degeneracy Introduce V := { v ( ) } n H# 1 (Q) a (1) (y) y v = 0. (subspace { of microscopic oscillations ), and ) n d ( W:= ψ (L 2 (a # (Q) divy (1) (y) ) ) 1/2 ψ(y) = 0 in ( microscopic fluxes ) Then, up to a subsequence, u ε 2 u 0 (x, y) L 2 (Ω; V ) ε u ε 2 y u 0 (x, y) ξ ε (x) := ( a (1) (x/ε) ) 1/2 u ε 2 ξ0 (x, y) L 2 (Ω; W ). ( H 1 # (Q) ) n } Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23

41 Weak two-scale limits. Key assumption on the degeneracy Introduce V := { v ( ) } n H# 1 (Q) a (1) (y) y v = 0. (subspace { of microscopic oscillations ), and ) n d ( W:= ψ (L 2 (a # (Q) divy (1) (y) ) ) 1/2 ψ(y) = 0 in ( microscopic fluxes ) Then, up to a subsequence, u ε 2 u 0 (x, y) L 2 (Ω; V ) ε u ε 2 y u 0 (x, y) ξ ε (x) := ( a (1) (x/ε) ) 1/2 u ε 2 ξ0 (x, y) L 2 (Ω; W ). ( H 1 # (Q) ) n } Key assumption ) n There exists a constant C > 0 such that for all v (H# 1 (Q) P V v (H 1 # (Q)) n C a (1) (y) y v L 2 Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23

42 The two-scale Limit Operator Let Ω be bounded Lipschitz, or Ω = R d. Introduce U L 2 (Ω; V ): { U := u(x, y) L 2 (Ω; V ) ξ(x, y) L2 (Ω; W ) s.t., Ψ(x, y) C ( ( ) ) 1/2 ξ(x, y) Ψ(x, y)dxdy = u(x, y) x a (1) (y) Ψ(x, y) Ω Q Define T : U L 2 by Tu = ξ. Strong two-scale resolvent convergence: Let f ε 2 f 0 (x, y). Then u ε 2 u 0 (x, y) solving: Find u 0 U such that φ U Ω { Tu 0 (x, y) T φ 0 (x, y) + a (0) (y) y u 0 (x, y) y φ 0 (x, y) + Ω Q } + α ρ(y)u 0 (x, y) φ 0 (x, y) dy dx = f 0 (x, y) φ 0 (x, y) dy dx. Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23 Q Ω Q

43 Back to Photonic Crystals { V θ := v ( } Hθ 1(Q)) n a(1) (y) y v = 0. v V θ iff v is θ-quasi-periodic, and v 1,1 + v 2,2 = 0, v 1,2 v 2,1 = 0 in Q 1. Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23

44 Back to Photonic Crystals { V θ := v ( } Hθ 1(Q)) n a(1) (y) y v = 0. v V θ iff v is θ-quasi-periodic, and v 1,1 + v 2,2 = 0, v 1,2 v 2,1 = 0 in Q 1. Theorem 1 (Key assumption holds) There exists a constant c > 0 such that for any u H 1 θ (Q) P V θ u H 1 (Q) c ( u 1,1 + u 2,2 L 2 (Q 0 ) + u 1,2 u 2,1 L 2 (Q 0 )). Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23

45 Back to Photonic Crystals { V θ := v ( } Hθ 1(Q)) n a(1) (y) y v = 0. v V θ iff v is θ-quasi-periodic, and v 1,1 + v 2,2 = 0, v 1,2 v 2,1 = 0 in Q 1. Theorem 1 (Key assumption holds) There exists a constant c > 0 such that for any u H 1 θ (Q) P V θ u H 1 (Q) c ( u 1,1 + u 2,2 L 2 (Q 0 ) + u 1,2 u 2,1 L 2 (Q 0 )). Second Result The generalised flux ξ =: Tu is zero. i.e. Tu = 0 u U. (Two-scale limit solution u 0 determined by miscroscopic behaviour only) Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23

46 Limit Spectral problem: Find u V θ such that a (0) (y) y u(y) y (φ(y)) dy = λ Q Q ρ(y)u(y) φ(y) dy φ V θ. alery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23

47 Limit Spectral problem: Find u V θ such that a (0) (y) y u(y) y (φ(y)) dy = λ Q Q ρ(y)u(y) φ(y) dy φ V θ. Theorem 2 (Spectral compactness: not trivial) Let λ ε σ(a ε ) (Bloch spectrum). Let u ε be associated normalized Bloch s waves: A ε u ε = λ ε u ε, u ε (x) = e iθε x/ε v ε (x/ε), v ε (y) L 2 (Q) = 1, θ ε ( π, π] d. Let λ λ 0 and θ ε θ 0. Then λ 0 σ 0 (A 0, θ 0 ) (spectrum of the Limit operator), and u ε u 0 (y), eigenfunction of A 0 (θ). Hence the spectra converge. Implication: If the limit problem displays a band-gap, the original problem must also have a gap for small enough ε. alery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23

48 Example: Band gaps in ARROW fibres An extreme problem: Q 0 a circle of small radius δ Theorem: There exist constants c 1, c 2 > 0 independent of δ, such that, for all quasimomenta θ 0, λ 2 (θ 0 ) c 1 δ 2 ln δ, λ 3(θ 0 ) c 2 δ 2. This implies that, for small enough δ, there is a wide spectral gap in the limit spectrum, and therefore also for small enough ε for the original problem by the spectral compactness. + Higher gaps: λ λ D j (Q δ 0 ) δ 2 (re micro-resonances ) Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23

49 The band gaps in a Photonic Crystal Fiber: Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23

50 Summary: - Homogenization for a critical high contrast scaling δ ε 2 gives rise to numerous non-classical effects described by two-scale limit problems due to micro-resonances. - Partial degeneracies often happen in physical problems, and give rise to more of such effects (e.g. Band Gaps in Photonic Crystal Fibers). These however have to be analysed in a new way. - A general two-scale homogenization theory can be constructed for such partial degeneracies, under a generically held decomposition condition. Resulting limit (homogenized) operator is generically two-scale (and non-local ). - Associated two-scale operator and spectral convergence and compactness are held generally or for particular physical examples. - In principle, some of this applies also to nonlinear (cf Cherednichenko & Cherdantsev 2011), discrete-to-continuous, as well as non-periodic problems. Valery Smyshlyaev (University College London, Homogenization UK) of micro-resonances and localization of waves. July 13, / 23