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On the rate of convergence for perforated plates with a small interior Dirichlet zone. (English) Zbl 1264.74189
Summary: The aim of the paper is to compare the asymptotic behavior of solutions of two boundary value problems for an elliptic equation posed in a thin periodically perforated plate. In the first problem, we impose homogeneous Dirichlet boundary condition only at the exterior lateral boundary of the plate, while at the remaining part of the boundary Neumann condition is assigned. In the second problem, Dirichlet condition is also imposed at the surface of one of the holes. Although in these two cases, the homogenized problem is the same, the asymptotic behavior of solutions is rather different. In particular, the presence of perturbation in the boundary condition in the second problem results in logarithmic rate of convergence, while for non-perturbed problem the rate of convergence is of power-law type.

MSC:
74Q05 Homogenization in equilibrium problems of solid mechanics
74K20 Plates
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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[1] Ladyzhenskaya, O.A.: Boundary value problems of mathematical physics. Moscow: Nauka, 408 p. (1973) (English translation: The boundary value problems of mathematical physics. Translated from the Russian by Jack Lohwater. Applied Mathematical Sciences, 49. Springer, New York 1985) · Zbl 0588.35003
[2] Nazarov S.A.: Asymptotic Theory of Thin Plates and Rods Vol. 1. Dimension Reduction and Integral Estimates. Nauchnaya Kniga, Novosibirsk (2001)
[3] Nazarov S.A., Thäter G: Asymptotics at infinity of solutions to the Neumann problem in a sieve-type layer. C.R. Mecanique. 331, 85–90 (2003) · Zbl 1181.74106 · doi:10.1016/S1631-0721(02)00005-0
[4] Nazarov S.A., Thäter G.: Neumann problem in a perforated layer (sieve). Asymptot. Anal. 44(3,4), 259–298 (2005) · Zbl 1084.35017
[5] Il’in A.M.: A boundary value problem for the elliptic equation of second order in a domain with a narrow slit. 1. The two-dimensional case. Math. USSR Sbornik 28(4), 59–80 (1976)
[6] Il’in, A.M.: Matching of asymptotic expansions of solutions of boundary value problems. Translations of Mathematical Monographs, 102. AMS, Providence (1992)
[7] Maz’ya V.G., Poborchi S.V.: Differentiable Functions on Bad Domains. World Scientific Publishing Co., (1997) · Zbl 0918.46033
[8] Maz’ya, V.G., Nazarov, S.A., Plamenevskii, B.A.: On the asymptotic behavior of solutions of elliptic boundary value problems with irregular perturbations of the domain. Probl. Mat. Anal. (vol. 8, pp. 72–153). Leningrad: Leningrad University (1981, Russian)
[9] Maz’ya, V.G., Nazarov, S.A., Plamenevskij, B.A.: Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, Tbilisi, Tbilisi University (1981) (Russian) (German Trans.: 1. Berlin: Akademie-Verlag. 1991. 432 S.; English transl.: Operator Theory: Advances and Applications, 112, Birkhäuser Verlag, Basel) (2000)
[10] Gel’fand I.M., Shilov G.E.: Generalized functions. Vol. 3. Theory of differential equations. Academic Press, New York (1967)
[11] Kondratiev, V.A.: Boundary value problems for elliptic problems in domains with conical or corner points, Trudy Moskov. Matem. Obshch. 16, 209–292 (1967). (English transl. Trans. Moscow Math. Soc. 16, 227–313 (1967))
[12] Panasenko G., Reztsov M.: Asymptotic expansion of the solution of a system of equations in elasticity theory in an inhomogeneous thin layer. J. Soviet Math. 67(3), 3081–3084 (1993) · Zbl 0792.73006 · doi:10.1007/BF01098144
[13] Kohn R., Vogelius M.: A New Model for Thin Plates with Rapidly Varying Thickness. II. A Convergence Proof. Elsevier, Amsterdam (1985) · Zbl 0565.73046
[14] Kohn R., Vogelius M.: A new model for thin plates with rapidly varying thickness. III. Comparison of different scalings. Quart. Appl. Math. 44(1), 35–48 (1986) · Zbl 0605.73048
[15] Caillerie D.: Thin elastic and periodic plates. Math. Methods Appl. Sci. 6, 159–191 (1984) · Zbl 0543.73073 · doi:10.1002/mma.1670060112
[16] Ansini N., Braides A.: Homogenization of oscillating boundaries and applications to thin films. J. Anal. Math. 83(1), 151–182 (2001) · Zbl 0983.49008 · doi:10.1007/BF02790260
[17] Bouchitte G., Fragala I.: Homogenization of thin structures by two-scale method with respect to measures. SIAM J. Math. Anal. 32(6), 1198–1226 (2001) · Zbl 0986.35015 · doi:10.1137/S0036141000370260
[18] Zhikov V.V.: On an extension and an application of the two-scale convergence method. Sb. Math. 191(7–8), 973–1014 (2000) · Zbl 0969.35048 · doi:10.1070/SM2000v191n07ABEH000491
[19] Akimova, E.A., Nazarov, S.A., Chechkin, G.A.: Asymptotics of the solution of the problem of the deformation of an arbitrary locally periodic thin plate. Transations on Moscow Mathematical Society, pp. 1–29 (2004) · Zbl 1167.74480
[20] Pazy A.: Asymptotic expansions of ordinary differential equations in Hilbert space. Arch. Rational Mech. Anal. 24, 193–218 (1967) · Zbl 0147.12303 · doi:10.1007/BF00281343
[21] Agalaryan, O.B., Nazarov, S.A.: On the variation of the intensity factor due to soldering of a longitudinal crack in a prismatic rod. Dokl. Akad. Nauk. Armenian SSR, 72(1), 18–21 (1981, Russian) · Zbl 0514.73112
[22] Nazarov, S.A., Romashev, Yu.A.: Variation of the intensity factor under rupture of the ligament between two collinear cracks. Izv. Akad. Nauk Armenian SSR. Mekh. (4), 30–40 (1982, Russian) · Zbl 0538.73126
[23] Maz’ya, V.G., Nazarov, S.A., Plamenevskii, B.A.: Evaluation of the asymptotic form of the ”intensity coefficients” on approaching corner or conical points. Zh. Vychisl. Mat. i Mat. Fiz. 23(2), 333–346 (1983). (English transl.: USSR Comput. Math. Math. Phys. 23(2), 50–58 (1983))
[24] Maz’ya, V.G., Nazarov, S.A., Plamenevskii, B.A.: Asymptotic expansions of the eigenvalues of boundary value problems for the Laplace operator in domains with small holes. Izv. Akad. Nauk SSSR. Ser. Mat. 48(2), 347–371 (1984). (English transl.: Math. USSR Izvestiya. 24, 321–345 (1985))
[25] Campbell A., Nazarov S.A.: An asymptotic study of a plate problem by a rearrangement method. Application to the mechanical impedance. RAIRO Model. Math. Anal. Numer. 32(5), 579–610 (1998) · Zbl 0905.73029
[26] Campbell A., Nazarov S.A.: Asymptotics of eigenvalues of a plate with small clamped zone. Positivity 5(3), 275–295 (2001) · Zbl 1029.74031 · doi:10.1023/A:1011469822255
[27] Nazarov S.A., Sokolowski J.: Self-adjoint extensions for the Neumann Laplacian and applications. Acta Math. Sin. 22(3), 879–906 (2006) · Zbl 1284.49048 · doi:10.1007/s10114-005-0652-z
[28] Nazarov, S.A.: Asymptotic behavior of eigenvalues of the Neumann problem for systems with masses concentrated on a thin toroidal set. Vestnik St.-Petersburg Univ. (3), 61–71 (2006). (English transl.: Vestnik St.-Petersburg Univ. Math. 39(3), 149–157 (2006))
[29] Movchan N.V.: Oscillations of elastic bodies with small holes. Vestnik Leningrad Univ. Math. 22(1), 50–55 (1989) · Zbl 0681.73041
[30] Movchan A.B., Movchan N.V.: Mathematical Modelling of Solids with Nonregular Boundaries. CRC Mathematical Modelling Series. CRC Press, London (1995) · Zbl 0868.73005
[31] Acerbi E., Chiado Piat V., Dal Maso G., Percivale D.: An extension theorem from connected sets, and homogenization in general periodic domains. Nonlinear Anal Theory Methods Appl 18(5), 481–496 (1992) · Zbl 0779.35011 · doi:10.1016/0362-546X(92)90015-7
[32] Braides A., Fonseca I., Francfort G: 3D–2D asymptotic analysis for inhomogeneous thin films. Indiana Univ. Math. J. 49(4), 1367–1404 (2000) · Zbl 0987.35020 · doi:10.1512/iumj.2000.49.1822
[33] Nazarov, S.A.: Asymptotic expansions at infinity of solutions to the elasticity theory problem in a layer. Trudy Moskov. Mat. Obschch. 60, 3–97 (1998). (English transl.: Trans. Moscow Math. Soc. 60, 1–85 (1999))
[34] Nazarov S.A., Pileckas K.I.: On the solvability of the Stokes and Navier–Stokes problem in the domain that are layer-like at infinity. J. Math. Fluid Mech. 1(1), 78–116 (1999) · Zbl 0941.35062 · doi:10.1007/s000210050005
[35] Nazarov S.A., Pileckas K.: The asymptotic properties of the solution to the Stokes problem in domains that are layer-like at infinity. J. Math. Fluid Mech. 1(2), 131–167 (1999) · Zbl 0940.35155 · doi:10.1007/s000210050007
[36] Nazarov, S.A.: Elastic capacity and polarization of a defect in an elastic layer. Mekhanika tverd. tela. (5), 57–65 (1990, Russian)
[37] Nazarov, S.A.: Asymptotics of the solution of the Dirichlet problem for an equation with rapidly oscillating coefficients in a rectangle. Mat. sbornik. 182(5), 692–722 (1991). (English transl.: Math. USSR Sbornik. 73(1), 79–110 (1992)) · Zbl 0782.35005
[38] Blanc F., Nazarov S.A.: Asymptotics of solutions to the Poisson problem in a perforated domain with corners. J. Math. Pures. Appl. 76(10), 893–911 (1997) · Zbl 0906.35011
[39] Nazarov, S.A., Slutskii, A.S.: Asymptotic behavior of solutions of boundary-value problems for equations with rapidly oscillating coefficients in a domain with a small cavity. Mat. sbornik. 189(9), 107–142 (1998). (English transl.: Sb. Math. 189(9), 1385–1422 (1998))
[40] Cardone, G., Corbo Esposito, A., Nazarov, S.A.: Homogenization of the mixed boundary value problem for a formally self-adjoint system in a periodically perforated domain. St. Petersburg Math. J. (to appear) · Zbl 1200.35100
[41] Lions J.L., Magenes E.: Non-Homogeneous Boundary Value Problems and Applications. Springer, New York (1972) · Zbl 0223.35039
[42] Kondratiev V.A.: Singularities of the solution of the Dirichlet problem for a second order elliptic equation in the neighborhood of an edge. Diff Equ 13(11), 1411–1415 (1978)
[43] Nazarov, S.A., Plamenevsky, B.A.: Elliptic problems in domains with piecewise smooth boundaries, Nauka, Moscow (1991) (English transl.: Elliptic problems in domains with piecewise smooth boundaries. Berlin: Walter de Gruyter (1994)) · Zbl 0806.35001
[44] Nazarov S.A.: Asymptotics at infinity of the solution to the Dirichlet problem for a system of equations with periodic coefficients in an angular domain. Russ. J. Math. Phys. 3(3), 297–326 (1995) · Zbl 0907.35032
[45] Polya G., Szegö G.: Isoperimetric Inequalties in Mathematical Physics. Annals of Mathematics Studies, 27. Princeton University Press, Princeton (1951)
[46] Landkof N.S.: Foundations of Modern Potential Theory Die. Grundlehren der mathematischen Wissenschaften, Band 180. Springer, New York (1972) · Zbl 0253.31001
[47] Mazja, V.G., Plamenevskii, B.A.: On coefficients in asymptotics of solutions of elliptic boundary value problems in a domain with conical points, Math. Nachr. Bd. 76(S), 29–60 (1977). (English translation in Am. Math. Soc. Transl. 123, 57–89 (1984))
[48] Agmon S., Douglis A., Nirenberg L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I. Comm. Pure Appl. Math. 12, 623–727 (1959) · Zbl 0093.10401 · doi:10.1002/cpa.3160120405
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