Rustem R. Gadyl'shin

Comptes Rendus Mécanique (CRM)

ition ond havingns) withourne courbedéduisons leune petiteicalcutthetedC. R. Mecanique 331 (2003) 595–600On resonances in a homogenization problemRustem R. Gadyl’shinBashkir State Pedagogical University, 3a October revolution str., 450000 Ufa, RussiaReceived and accepted 27 May 2003Presented by Évariste Sanchez-PalenciaAbstractWe consider a two-dimensional boundary value problem for the Helmholtz equation with Neumann boundary conda set of arcs. This set is obtained from a closed curve by cutting out small holes situated closely each to other anlocally periodic structure. We construct asymptotics of scattering frequencies (poles of analytic continuation of solutiosmall imaginary parts and show that these scattering frequencies imply resonances.To cite this article: R.R. Gadyl’shin, C. R.Mecanique 331 (2003). 2003 Académie des sciences. Published by Éditions scientifiques et médicales Elsevier SAS. All rights reserved.RésuméSur les résonances dans un problème d’homogénéisation. On considère un problème aux limites bidimensionnel pl’équation de Helmholtz avec des conditions de Neumann sur un ensemble d’arcs. Cet ensemble s’obtient à partir d’ufermée en supprimant des petites parties très proches les unes des autres et disposées de façon périodique. Nouscomportement asymptotique des fréquences de diffusion (pôles des prolongements analytiques des solutions) avecpartie imaginaire et nous montrons qu’elles impliquent l’existence des résonances.Pour citer cet article : R.R. Gadyl’shin,C. R. Mecanique 331 (2003). 2003 Académie des sciences. Published by Éditions scientifiques et médicales Elsevier SAS. All rights reserved.Keywords: Vibrations; Asymptotics; Small parameter; HomogenizationMots-clés : Vibrations ; Asymptotique ; Un petit paramètre ; Homogénéisation1. IntroductionIt is known that the scattering ofH -polarized electromagnetic field with an ideally conducting cylindrsurface the cross-section of which is the curveΓδ , and vibrations of a membrane which is not fixed on theΓδ , are described by the solution of the following boundary value problem inΩδ = R2 \ Γδ:( + k2)uδ = f, x ∈ Ωδ, ∂uδ∂ν= 0, x ∈ Γδ, ∂uδ∂r− ikuδ = o(r−1/2), r → ∞ (1)wherex = (x1, x2), r = |x|, ν is the normal toΓδ, k is a positive number. Hereinafter we assume thatfunctionf belongs toL2(R2) and has bounded support,Γ0 ∈ C∞ is a boundary of a bounded simply-connecE-mail address: gadylshin@bspu.ru (R.R. Gadyl’shin).1631-0721/$ – see front matter 2003 Académie des sciences. Published by Éditions scientifiques et médicales Elsevier SAS. All rightsreserved.doi:10.1016/S1631-0721(03)00139-6596 R.R. Gadyl’shin / C. R. Mecanique 331 (2003) 595–600gseeneblemat thethee peakFig. 1. The perturbed problem.domainΩ , and for δ = ε > 0 the curvesΓε is obtained fromΓ0 by cutting out a great number of openinof small diameter located closely to each other. Namely (see Fig. 1), letω be a unit circle with center at thorigin. Suppose thatγ0 = ∂ω, N  1 is an integer number,ε = 2N−1, γε = {(r, θ): r = 1, ε(a(ε) + mπ) < θ <ε(π(m + 1) − a(ε)), m = 1, . . . ,N}, whereθ is a polar angle andP is a diffeomorphismR2 into R2. We denoteΩ =P(ω), Γδ =P(γδ). Forδ = ε we will call the problem (1) the perturbed problem. SinceΩ0 = Ω ∪ (R2 \ Ω),it follows that forδ = 0 the problem (1) decomposes into two Neumann problems, inΩ and inR2 \ Ω . We willcall them the limit internal and the limit external problems, respectively. In what follows we assume that:a(ε) = exp(− 1εA(ε)), A(ε) > 0, limε→0A(ε) = 0, ε → 0 (2)For this case it is known [1] (see also [2]) that ifλ = k2 is not the eigenvalue of the limit internal problem tha solution of the perturbed problem converges to solutions of the limit problems inΩ a dR2 \ Ω . Assume thatk0 > 0 andλ0 = k20 is an eigenvalue of the Neumann problem for the operator− in Ω . In [3] one can find that theanalytic continuation (with respect tok) of the solution to a boundary value problem (1) has a poleτ ε with smallimaginary part (a scattering frequency), converging tok0 asε → 0. This pole lies in the lower complex half-planImk < 0, and the residue of the analytic continuation of the solution is a solution of the boundary value pro( + (τ ε)2)ψε = 0, x ∈ Ωε, ∂ψε∂ν= 0, x ∈ Γε (3)Let us emphasize that for fixedε the functionψε increases exponentially asr → ∞. At the same time, it beingsolution to the homogeneous boundary value problem (3), we will call it a quasi-eigenfunction. Letλ0 = k20 > 0be a simple eigenvalue of the limit problem, andψ0 be the associated eigenfunction normalized inL2(Ω). For thiscase in [3] one can find that fork, close tok0, solutions of the problem (1) and their analytic continuation meerepresentationuε(x, k) = ψε(x)k2 − (τ ε)2∫R2ψε(y) f (y)dy + ũε(x, k) (4)ψε →ε→0ψ0 in H1(Ω), ψε →ε→00, ũε →ε→0u in H1loc(R2 \ Ω ), ‖ũε‖H1(Ω)  C‖f ‖L2(R2) (5)whereu(x; k) is a solution (and its analytic continuation) of the boundary value problem( + k2)u = f, x ∈ R2 \ Ω, ∂u∂ν= 0, x ∈ Γ0, ∂u∂r− iku = o(r−1/2), r → ∞ (6)From (4) it follows that for the real frequenciesk the first term in the right-hand side causes most affect incasek = k(ε) = Reτ ε + O(Imτ ε) (7)The main goal of this paper is to construct the leading terms of the asymptotics for Imτ ε andψε in R2 \ Ωallowing to employ formula (4) to establish the resonance nature of the first term in this formula for thfrequencies (7).R.R. Gadyl’shin / C. R. Mecanique 331 (2003) 595–600 597ics ofn ordertveerboundary2. Asymptotics of quasi-eigenelementsThe leading terms of the asymptotics for the poleτ ε and associated quasi-eigenfunctionψε outside aneighborhood ofΓ0 are constructed in the formτ ε = τ0(A(ε)) + · · · , τ0 →A→0k0 (8)ψε(x) = Ψ+0(x;A(ε)) + · · · , x ∈ Ω, Ψ+0 →A→0ψ0 (9)ψε(x) = A(ε)Ψ−0(x; τ ε;A(ε)) + · · · , x ∈ R2 \ Ω (10)whereΨ−0 (x; k;A) is a solution (and its analytic continuation) of the equation and radiation condition( + k2)Ψ−0 = 0, x ∈ R2 \ Ω, ∂Ψ−0∂r− ikΨ−0 = o(r−1/2), r → ∞ (11)DenoteΛ0(A) = τ20 (A). Substituting (8), (9) in (3), we obtain the following equation forΨ+0 :( +Λ0)Ψ+0 = 0, x ∈ Ω (12)In a small neighborhood ofΓ0 (moreover, outside smaller neighborhood of the openings) the asymptotψε are constructed in another form, employing the method of matching asymptotic expansions [4–6]. Ito do so, we introduce local coordinates(s, t) in a neighborhood ofΓ0. Here t is the distance from the pointo Γ0 measured in the direction of the internal normal, containing this point, ands is an arc length of thecurveΓ0. SinceΓ0 = P(γ0), it is obvious that the angleθ parameterized the curveγ0, can be expressed bys:θ = θ(s), θ ′(s) > 0, θ(0) = 0. We denoteφ+0 = Ψ +0∣∣Γ0, φ+1 =∂Ψ+0∂ν∣∣∣∣Γ0, φ−0 = Ψ−0∣∣Γ0,k=τ0, φ−1 =∂Ψ−0∂ν∣∣∣∣Γ0,k=τ0(13)ξ = (ξ1, ξ2) = (θ(s)ε−1, tθ ′(s)ε−1). Hereafter we considerν as an outer normal toΩ . By the matching conditionof the series (9), (10) with new series in theξ variables in a neighborhood ofΓ0 we deduce that these series hathe formψε(x) = V ±0 (ξ; s;A)+ εV ±1 (ξ; s;A)+ · · · , ±ξ2 > 0 (14)V +0 (ξ; s;A) ∼ φ+0 (s;A), V +1 (ξ; s;A) ∼ −φ+1 (s;A)θ ′(s)ξ2, ξ2 → +∞V −0 (ξ; s;A) ∼ Aφ−0 (s;A), V −1 (ξ; s;A) ∼ −Aφ−1 (s;A)θ ′(s)ξ2, ξ2 → −∞(15)Heres is a ‘slow’ variable. All the openings expressed inξ being exponentially small, we will introduce anothexpansion at each opening. Therefore, the coefficients of the series (14) should obey the homogeneouscondition∂V ±j∂ξ2= 0, ξ2 = 0, ξ1 = πm, m ∈ Z (16)Substituting (14) and (8) in (3) and passing toξ , we obtain the equation forV ±j and, for instance:ξV±j = 0, ±ξ2 > 0 (17)for j = 0. It is clear that the functionsV +0 (ξ; s;A) ≡ φ+0 (s;A), V −0 (ξ; s;A) ≡ Aφ−0 (s;A) (18)598 R.R. Gadyl’shin / C. R. Mecanique 331 (2003) 595–600ner.iablesofoticedFig. 2. The half-strip,Π . Fig. 3. The boundary value problem.satisfy (17) and have the asymptotics (15). Keeping in mind (18) we conclude that the equations forV ±1 have theform (17) wherej = 1. DenoteX(ξ) = Re lnsinz + ln2, wherez = ξ1 + iξ2 is a complex variable. By definitiothe functionX is harmonic in a half-stripΠ = {ξ : −π/2< ξ1 < π/2, ξ2 > 0} (see Fig. 2), can be continued ovthe periodπ on the half-planeξ2 > 0 and satisfies the condition∂X∂ξ2= 0, ξ2 = 0, ξ1 = πm (19)Let us extend this function on the semi-planeξ2 < 0 in a even way and keep the same notationX for this extensionAs a result, we conclude that the functionX has the asymptoticsX(ξ) = ±ξ2 + O(exp{∓2ξ2}), ξ2 → ±∞, X(ξ) = lnρ + ln2+ O(ρ2), ρ = |ξ | → 0 (20)Using (19) and (20), we obtain that the functionsV +1 (ξ; s;A) = −φ+1 (s;A)θ ′(s)X(ξ), V −1 (ξ; s;A) = Aφ−1 (s;A)θ ′(s)X(ξ) (21)are the solutions of Eqs. (17) satisfying the boundary condition (16) and having the asymptotics (15).The coefficients ofV ±j have the jump on the openings, that is why in a neighborhood of them-th opening theasymptotics of the quasi-eigenfunctionψε is sought as a new series whose coefficients depend on the varζ (m) = (ζ (m)1 , ζ2) = ((ξ1 − πm)a−1, ξ2a−1), wherea(ε) is defined by formula (2). Rewriting the asymptoticsthe series (14) as(ξ1 − πm,ξ2) → 0 in the variablesζ (m) and according to the method of matching asymptexpansions, by (18), (20) and (21) we obtain that in the neighborhood of them-t opening the asymptotics of thfunctionψε has the formψε(x) = W(m)0(ζ (m); s;A) + εW(m)1 (ζ (m); s;A) + · · · (22)whereW(m)j satisfies the asymptotics as|ζ (m)| → ∞W0(ζ ; s;A)∼ φ+0 (s;A)+ A−1φ+1 (s;A)θ ′, W1(ζ ; s;A)∼ −φ+1 (s;A)θ ′(ln |ζ | + ln2), ζ2 > 0W0(ζ ; s;A)∼ Aφ−0 (s;A) −φ−1 (s;A)θ ′ , W1(ζ ; s;A)∼ Aφ−1 (s;A)θ ′(ln |ζ | + ln2), ζ2 < 0(23)Hereinafter for the sake of brevity we omit the indexesm of W(m)j andζ(m). Substituting (22) and (8) in (3) anpassing toζ , we obtain the boundary value problems forWj , and, in particular:ζW0 = 0, ζ ∈ R2 \ Γ 0, ∂W0∂ζ2= 0, ζ ∈ Γ 0 (24)whereΓ 0 is an axisOζ1 without the segment[−1,1] (see Fig. 3). Clearly, under the conditionφ+0 (s;A)+(Aθ ′(s))−1φ+1 (s;A) = Aφ−0 (s;A)−(θ ′(s))−1φ−1 (s;A) (25)R.R. Gadyl’shin / C. R. Mecanique 331 (2003) 595–600 599thequasi-the solution to the boundary value problem (24), having the asymptotics (23), has the formW0(ζ ; s;A)≡ φ+0 (s;A)+φ+1 (s;A)Aθ ′(s)(26)and it is independent ofζ . By (26) the boundary value problem forW1 isζW1 = 0, ζ ∈ R2 \ Γ 0, ∂W1∂ζ2= 0, ζ ∈ Γ 0 (27)DenoteY (ζ ) = Re ln(w + √w2 − 1), wherew = ζ1 + iζ2 is a complex variable. It is easily checked thatharmonic inR2 \ Γ 0 functionY ∈ H 1loc(R2 \ Γ 0) ∩C∞(R2 \ Γ 0) satisfies the boundary condition∂Y∂ζ2= 0, ζ ∈ Γ 0 (28)and has the following asymptotic at infinity:Y = ±(ln |ζ | + ln2) + O(|ζ |−2), ±ζ2 > 0 (29)By (28), (29) the functionW1(ζ ; s;A)= −φ+1 (s;A)θ ′(s)Y (ζ )is a solution to the boundary value problem (27) and meets the asymptotics (23) ifφ+1 (s;A)θ ′= Aφ−1 (s;A)θ ′(30)From (30), (25) and (13) for Eqs. (11), (12) we obtain the boundary conditions of the conjugation type:∂∂ν(Ψ+0 (x;A)+ AΨ−0(x; τ0(A);A)) + Aθ ′(s)(Ψ+0 (x;A)− AΨ−0 (x; τ0(A);A)) = 0∂∂ν(Ψ+0 (x;A)− AΨ−0(x; τ0(A);A)) = 0, x ∈ Γ0(31)From the solvability conditions of (11) and (12), the boundary conditions (31) and the conditionsτ0 → k0 andΨ+0 → ψ0 asA → 0 by (8) and (9) we determineτ0(A), Ψ±0 and, in particular, we get thatτ0(A) = k0 +Aτ0,1 + A2τ0,2 + O(A3), Ψ−0 (x; k;A) →A→0ψ−0 (x; k) (32)whereψ−0 (x; k) is a solution of the boundary value problem( + k2)ψ−0 = 0, x ∈ R2 \ Ω, ∂ψ−0∂ν= −θ′ψ02, x ∈ Γ0, ∂ψ−0∂r− ikψ−0 = o(r−1/2), r → ∞ (33)if k > 0 and its analytic continuation in the complex plane with the cut along the imaginary axis,τ0,1 = 14k0∫Γ0ψ20(x(s))dθ(s) > 0, Im τ0,2 = −σ2< 0, σ = limR→∞∫|x|=R∣∣ψ−0 (x; k0)∣∣2 ds > 0 (34)Thus from (8)–(10) and (32) it follows that the pole of the analytic continuation and the associatedeigenfunction have the asymptoticsτ ε ∼ k0 + Aτ0,1 +A2τ0,2, ψε(x) ∼ ψ0(x), x ∈ Ω, ψε(x) ∼ Aψ−0 (x; k0), x ∈ R2 \ Ω (35)whereτ0,j is defined by (34),ψ0 is a normalized inL2(Ω) eigenfunction of the Neumann problem for− in Ω ,associated with the eigenvalueλ0 = k20 > 0, andψ−0 is the solution of (33).600 R.R. Gadyl’shin / C. R. Mecanique 331 (2003) 595–6004),tionennnersitiesversity974 (inar Partialp. 309–tion: Sb.dence,Differ-. i Mat.3. Resonances of solutions to problem (1)Let t be an arbitrary real number. From (7), (35), (34) it follows that the peak frequencies have the formk(ε) = k0 + A(ε)τ0,1 + A2(ε)t + o(A2(ε))(36)DenoteT (t) = 2k0(t − τ0,2). Suppose that suppf ⊂ Ω (the radiation problem). Then substituting (36) in (using (34), (35) and (5) we obtain that for realk satisfying (36) the leading terms of the asymptotics of the soluto the boundary value problem (1) have the formuε(x; k) ∼ ψ0(x)A2(ε)T (t)∫Ωψ0f dy, x ∈ Ω, uε(x; k)∼ ψ−0 (x; k0)A(ε)T (t)∫Ωψ0f dy, x ∈ R2 \ Ω (37)Let suppf ⊂ R2 \ Ω (the scattering problem) andu(x; k) be a solution of the boundary value problem (6). Thfrom the same formulae and due to (6) and (33) it follows thatuε(x; k) ∼ − ψ0(x)2A(ε)T (t)∫Γ0ψ0(x(s))u(x(s); k0)dθ(s), x ∈ Ωuε(x, k) ∼ −ψ−0 (x; k0)2T (t)∫Γ0ψ0(x(s))u(x(s); k0)dθ(s)+ u(x; k0), x ∈ R2 \ ΩFrom (37) it follows that in the case of the radiation problem the solution increases unboundedly inR2 on thepeak frequencies. In the case of the scattering problem on the peak frequencies the solution increases iΩ only,while in R2 \ Ω it differs from the solution of the limit external problem (6) up to O(1). Exactly this phenomenowas discovered by Rayleigh for the classic acoustic resonator with one opening in [7].AcknowledgementsThe work is supported by grants from RFBR (02-01-00693), Scientific Schools (1446.2003.1) and Univof Russia (UR.04.01.010). The author is also grateful for the hospitality extended to him at Narvik UniCollege where this work was done.References[1] V.A. Marchenko, E.Ya. Khruslov, Boundary Value Problems in Domains with Fine-Grained Boundaries, Naukova Dumka, Kyiv, 1Russian).[2] É. Sanchez-Palencia, Boundary value problems in domains containing perforated walls, in: H. Brezis, J.-L. Lions (Eds.), NonlineDifferential Equation and Their Applications College de France Seminar III, in: Res. Notes Math., Vol. 70, Pitman, London, 1982, p325.[3] R.R. Gadyl’shin, On analogs of Helmholtz resonator in averaging theory, Mat. Sb. 193 (11) (2002) 43–70. English translaMath. 193 (11) (2002) 1611–1638.[4] A.M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, American Mathematical Society, ProviRI, 1992.[5] R.R. Gadyl’shin, Asymptotics of the eigenvalues of a boundary value problem with rapidly oscillating boundary conditions,entsial’nye Uravneniya 35 (4) (1999) 540–551. English translation: Differential Equations 35 (4) (1999) 540–551.[6] R.R. Gadyl’shin, Homogenization and asymptotics for a membrane with closely spaced clamping points, Zh. Vychisl. MatFiz. 41 (12) (2001) 1857–1869. English translation: Comput. Math. Math. Phys. 41 (12) (2001) 1765–1776.[7] Lord Rayleigh, The theory of Helmholtz resonator, Proc. Roy. Soc. London Ser. A 92 (1916) 265–275.

On resonances in a homogenization problem

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On resonances in a homogenization problem

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