Asymptotic analysis of boundary-value problems for the Laplace operator with frequently alternating type of boundary conditions

D. I. Borisov

Contemporary Mathematics. Fundamental Directions, Partial Differential Equations, Tome 67 (2021) no. 1, pp. 14-129.

Résumé

This work, which can be considered as a small monograph, is devoted to the study of two- and three-dimensional boundary-value problems for eigenvalues of the Laplace operator with frequently alternating type of boundary conditions. The main goal is to construct asymptotic expansions of the eigenvalues and eigenfunctions of the considered problems. Asymptotic expansions are constructed on the basis of original combinations of asymptotic analysis methods: the method of matching asymptotic expansions, the boundary layer method and the multi-scale method. We perform the analysis of the coefficients of the formally constructed asymptotic series. For strictly periodic and locally periodic alternation of the boundary conditions, the described approach allows one to construct complete asymptotic expansions of the eigenvalues and eigenfunctions. In the case of nonperiodic alternation and the averaged third boundary condition, sufficiently weak conditions on the alternation structure are obtained, under which it is possible to construct the first corrections in the asymptotics for the eigenvalues and eigenfunctions. These conditions include in consideration a wide class of different cases of nonperiodic alternation. With further, very serious weakening of the conditions on the structure of alternation, it is possible to obtain two-sided estimates for the rate of convergence of the eigenvalues of the perturbed problem. It is shown that these estimates are unimprovable in order. For the corresponding eigenfunctions, we also obtain unimprovable in order estimates for the rate of convergence.

Bibliographie

[1] V. M. Babich, N. Ya. Kirpichnikova, Boundary Layer Method, LGU, Leningrad, 1974 (in Russian)

[2] N. C. Bakhvalov, G. P. Panasenko, Averaging of Processes in Periodic Media, Nauka, M., 1984 (in Russian) | MR

[3] A. G. Belyaev, G. A. Chechkin, “Averaging of a mixed boundary-value problem for the Laplace operator in the case when the “limit” problem is unsolvable”, Math. Digest, 186:4 (1995), 47–60 (in Russian) | MR | Zbl

[4] A. Yu. Belyaev, G. A. Chechkin, “Averaging of operators with small-scale structure”, Math. Notes, 65:4 (1999), 496–510 (in Russian) | Zbl

[5] M. Sh. Birman, T. A. Suslina, “Second-order periodic differential operators. Threshold properties and averagings”, Algebra Anal., 15:5 (2003), 1–108 (in Russian)

[6] M. Sh. Birman, T. A. Suslina, “Threshold approximations of the resolvent of a factorized self-adjoint family with allowance for a corrector”, Algebra Anal., 17:5 (2005), 69–90 (in Russian)

[7] N. N. Bogolyubov, Yu. A. Mitropol'skiy, Asymptotic Methods in the Theory of Nonlinear Oscillations, Nauka, M., 1974 (in Russian) | MR

[8] D. I. Borisov, “On twoparameter asymptotics in a boundary-value problem for the Laplacian”, Math. Notes, 70:4 (2001), 520–534 (in Russian) | Zbl

[9] D. I. Borisov, “Two-parameter asymptotics of the Laplacian eigenvalues with frequent alternation of boundary conditions”, Bull. Young Sci. Ser. Appl. Math. Mech., 2002, no. 1, 32–52 (in Russian)

[10] D. I. Borisov, “On the Laplacian with frequently and non-periodically alternating boundary conditions”, Rep. Russ. Acad. Sci., 383:4 (2002), 443–445 (in Russian) | MR | Zbl

[11] D. I. Borisov, “On a boundaryvalue problem in a cylinder with frequently alternating type of boundary conditions”, Math. Digest, 193:7 (2002), 37–68 (in Russian) | MR | Zbl

[12] D. I. Borisov, “On a singularly perturbed boundary-value problem for the Laplacian in a cylinder”, Differ. Equ., 38:8 (2002), 1071–1078 (in Russian) | MR | Zbl

[13] D. I. Borisov, “Asymptotics and estimates of the Laplacian eigenelements with frequent non-periodic alternation of the boundary conditions”, Bull. Acad. Sci. USSR. Ser. Math., 67:6 (2003), 23–70 (in Russian) | MR | Zbl

[14] D. I. Borisov, “Asymptotics and estimates of the rate of convergence in a three-dimensional boundary value problem with frequent alternation of boundary conditions”, Siberian Math. J., 45:2 (2004), 274–294 (in Russian) | MR | Zbl

[15] D. I. Borisov, “On a problem with frequent non-periodic alternation of boundary conditions on rapidly oscillating sets”, J. Comput. Math. Math. Phys., 46:2 (2006), 284–294 (in Russian) | MR | Zbl

[16] D. I. Borisov, R. R. Gadyl'shin, “On the Laplacian spectrum with frequently alternating type of boundary conditions”, Theor. Math. Phys., 118:3 (1999), 347–353 (in Russian) | MR | Zbl

[17] D. I. Borisov, T. F. Sharapov, “On the resolvent of multidimensional operators with frequent alternation of boundary conditions in the case of the third averaged condition”, Probl. Math. Anal., 83 (2015), 3–40 (in Russian)

[18] M. Van Dyke, Perturbation Methods in Fluid Mechanics, Russian translation, Mir, M., 1967

[19] G. N. Watson, A Treatise on the Theory of Bessel Functions, Russian translation, v. 1, IL, M., 1949

[20] M. I. Vishik, L. A. Lyusternik, “Regular degeneration and boundary layer for linear differential equations with a small parameter”, Progr. Math. Sci., 12:5 (1957), 3–122 (in Russian) | Zbl

[21] R. R. Gadyl'shin, “The spectrum of elliptic boundary-value problems with a singular perturbation of the boundary conditions”, Asymptotic Properties of Solutions of Differential Equations, BNTs UrO AN SSSR, Ufa, 1988, 3–15 (in Russian)

[22] R. R. Gadyl'shin, “On the asymptotics of eigenvalues for a periodically fixed membrane”, Algebra Anal., 10:1 (1998), 3–19 (in Russian) | Zbl

[23] R. R. Gadyl'shin, “On a boundary-value problem for the Laplacian with rapidly oscillating boundary conditions”, Rep. Russ. Acad. Sci., 362:4 (1998), 456–459 (in Russian) | MR | Zbl

[24] R. R. Gadyl'shin, “Asymptotics of the eigenvalues of a boundary-value problem with rapidly oscillating boundary conditions”, Differ. Equ., 35:4 (1999), 540–551 (in Russian) | MR | Zbl

[25] R. R. Gadyl'shin, “Systems of resonators”, Bull. Russ. Acad. Sci. Ser. Math., 64:3 (2000), 51–96 (in Russian) | MR | Zbl

[26] R. R. Gadyl'shin, “Averaging and asymptotics in the problem of a densely fixed membrane”, J. Comput. Math. Math. Phys., 41:12 (2001), 1857–1869 (in Russian) | MR | Zbl

[27] R. R. Gadyl'shin, “On a model analogue of the Helmholtz resonator in averaging”, Proc. Math. Inst. Russ. Acad. Sci., 236, 2002, 79–86 (in Russian) | Zbl

[28] R. R. Gadyl'shin, “On analogs of the Helmholtz resonator in averaging theory”, Math. Digest, 193:11 (2002), 43–70 (in Russian) | MR | Zbl

[29] R. R. Gadyl'shin, G. A. Chechkin, “Boundary-value problem for the Laplacian with a rapidly alternating type of boundary conditions in a multidimensional domain”, Siberian Math. J., 40:2 (1999), 271–287 (in Russian) | MR | Zbl

[30] I. C. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products, Fizmatgiz, M., 1963 (in Russian) | MR

[31] E. I. Doronina, G. A. Chechkin, “On averaging of solutions of a second-order elliptic equation with nonperiodic rapidly alternating boundary conditions”, Bull. Moscow Univ. Ser. 1. Math. Mech., 2001, no. 1, 14–19 (in Russian) | MR | Zbl

[32] E. I. Doronina, G. A. Chechkin, “On eigen oscillations of a body with a large number of nonperiodically located concentrated masses”, Proc. Inst. Russ. Acad. Sci., 236, 2002, 158–166 (in Russian) | MR | Zbl

[33] W. Jager, O. A. Oleynik, A. S. Shamaev, “On the averaging problem for the Laplace equation in a partially perforated domain”, Rep. Russ. Acad. Sci., 333:4 (1993), 424–427 (in Russian)

[34] V. V. Zhikov, “On averaging of nonlinear variational problems in perforated domains”, Rep. Russ. Acad. Sci., 345:2 (1995), 156–160 (in Russian) | MR | Zbl

[35] V. V. Zhikov, “On one extension and application of the two-scale convergence method”, Math. Digest, 191:7 (2000), 31–72 (in Russian) | MR | Zbl

[36] V. V. Zhikov, “On the spectral method in homogenization theory”, Proc. Math. Inst. Russ. Acad. Sci., 250, 2005, 95–104 (in Russian) | Zbl

[37] V. V. Zhikov, “On operator estimates in homogenization theory”, Rep. Russ. Acad. Sci., 403:3 (2005), 305–308 (in Russian) | MR | Zbl

[38] V. V. Zhikov, S. M. Kozlov, O. A. Oleynik, Homogenization of Differential Operators, Fizmatlit, M., 1993 (in Russian)

[39] V. V. Zhikov, M. E. Rychago, “Averaging of nonlinear second-order elliptic equations in perforated domains”, Bull. Russ. Acad. Sci. Ser. Math., 61:1 (1997), 70–88 (in Russian)

[40] A. M. Il'in, “Boundary-value problem for a second-order elliptic equation in a domain with a narrow slit. I. Two-dimensional case”, Math. Digest, 99:4 (1976), 514–537 (in Russian) | MR | Zbl

[41] A. M. Il'in, “Boundary-value problem for a second-order elliptic equation in a domain with a narrow slit. II. Domain with a small hole”, Math. Digest, 103(145):2 (1977), 265–284 (in Russian) | MR | Zbl

[42] A. M. Il'in, Reconciliation of Asymptotic Expansions of Solutions of Boundary-Value Problems, Nauka, M., 1989 (in Russian)

[43] T. Kato, Perturbation Theory for Linear Operators, Russian translation, Mir, M., 1972

[44] S. M. Kozlov, A. L. Pyatnitskiy, “Averaging under vanishing viscosity”, Math. Digest, 181:6 (1990), 813–832 (in Russian) | Zbl

[45] J. Cole, Perturbation Methods in Applied Mathematics, Russian translation, Mir, M., 1979

[46] O. A. Ladyzhenskaya, N. N. Ural'tseva, Linear and Quasilinear Equations of Elliptic Type, Nauka, M., 1973 (in Russian) | MR

[47] V. G. Maz'ya, S. A. Nazarov, B. A. Plamenevskiy, Asymptotics of Solutions of Elliptic Boundary-Value Problems under Singular Perturbations of the Domain, Tbilisi Univ., Tbilisi, 1981 (in Russian)

[48] V. G. Maz'ya, S. A. Nazarov, B. A. Plamenevskiy, “Asymptotic expansions of eigenvalues of boundaryvalue problems in domains with small holes”, Bull. Acad. Sci. USSR. Ser. Math., 48:2 (1984), 347–371 (in Russian) | MR

[49] V. A. Marchenko, E. Ya. Khruslov, Boundary-Value Problems in Domains with a Fine-Grained Boundary, Naukova Dumka, Kiev, 1974 (in Russian) | MR

[50] T. A. Mel'nik, S. A. Nazarov, “Asymptotic behavior of the solution to the Neumann spectral problem in a domain of the “dense ridge” type”, Proc. Petrovskii Semin., 19, 1996, 138–174 (in Russian)

[51] V. P. Mikhaylov, Partial Differential Equations, Nauka, M., 1976 (in Russian)

[52] A. B. Movchan, S. A. Nazarov, “Influence of small surface irregularities on the stress state of the body and the energy balance during crack growth”, Appl. Math. Mech., 55:5 (1991), 819–828 (in Russian) | MR | Zbl

[53] A. H. Nayfeh, Perturbation Methods, Mir, M., 1986 (in Russian)

[54] S. A. Nazarov, B. A. Plamenevskiy, Elliptic Problems in Domains with Piecewise Smooth Boundary, Nauka, M., 1991 (in Russian)

[55] O. A. Oleynik, G. A. Iosif'yan, A. S. Shamaev, Mathematical Problems in the Theory of Strongly Inhomogeneous Elastic Media, MGU, M., 1990 (in Russian) | MR

[56] O. A. Oleynik, G. A. Chechkin, “Boundary-value problems for elliptic equations with rapidly alternating type of boundary conditions”, Progr. Math. Sci., 48:6 (294) (1993), 163–165 (in Russian) | MR

[57] O. A. Oleynik, G. A. Chechkin, “On one problem of boundary averaging for a system of elasticity theory”, Progr. Math. Sci., 49:4 (1994), 114 (in Russian)

[58] O. A. Oleynik, A. S. Shamaev, “On averaging of solutions of a boundary-value problem for the Laplace equation in a partially perforated domain with Dirichlet conditions on the boundary of cavities”, Rep. Russ. Acad. Sci., 337:2 (1994), 168–171 (in Russian) | Zbl

[59] S. E. Pastukhova, “On the nature of the distribution of the temperature field in a perforated body with its given value at the outer boundary under conditions of heat transfer at the boundary of the cavities according to Newton's law”, Math. Digest, 187:6 (1996), 85–96 (in Russian) | MR | Zbl

[60] M. E. Peres, G. A. Chechkin, E. I. Yablokova, “On eigne vibrations of the body with “light” concentrated masses on the surface”, Progr. Math. Sci., 57:6 (2002), 195–196 (in Russian) | MR

[61] M. Yu. Planida, “On convergence of solutions of singularly perturbed boundary-value problems for the Laplacian”, Math. Notes, 71:6 (2002), 867–877 (in Russian) | MR | Zbl

[62] E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, Russian translation, Mir, M., 1984 | MR

[63] I. V. Skrypnik, Research Methods for Nonlinear Elliptic Boundary-Value Problems, Nauka, M., 1990 (in Russian) | MR

[64] I. V. Skrypnik, “Asymptotics of solutions of nonlinear elliptic boundary-value problems in perforated domains”, Math. Digest, 184:10 (1993), 67–90 (in Russian) | Zbl

[65] G. A. Chechkin, “On boundary-value problems for a second-order elliptic equation with oscillating boundary conditions”, Nonclassical Partial Differential Equations, IM SOAN SSSR, Novosibirsk, 1988, 95–104 (in Russian)

[66] G. A. Chechkin, “Averaging of boundary-value problems with singular perturbation of boundary conditions”, Math. Digest, 184:6 (1993), 99–150 (in Russian) | Zbl

[67] G. A. Chechkin, “Complete asymptotic expansion of a solution to a boundary value problem with rapidly changing boundary conditions in a layer”, Progr. Math. Sci., 48:4 (1993), 218–219 (in Russian)

[68] G. A. Chechkin, “Asymptotic expansion of the solution to a boundary-value problem with a rapidly alternating type of boundary conditions”, Proc. Petrovskii Semin., 19, 1996, 323–337 (in Russian) | Zbl

[69] T. A. Shaposhnikova, “Averaging of a boundary-value problem for a biharmonic equation in a domain containing thin channels of small length”, Math. Digest, 192:10 (2001), 131–160 (in Russian) | Zbl

[70] T. F. Sharapov, “On the resolvent of multidimensional operators with frequent alternation of boundary conditions in the case of the averaged Dirichlet condition”, Math. Digest, 205:10 (2014), 125–160 (in Russian) | Zbl

[71] T. F. Sharapov, “On the resolvent of multidimensional operators with frequent alternation of boundary conditions: critical case”, Ufa Math. J., 8:2 (2016), 66–96 (in Russian) | Zbl

[72] G. I. Barenbaltt, J. B. Bell, W. Y. Crutchfiled, “The thermal explosion revisited”, Proc. Natl. Acad. Sci. USA, 95:23 (1998), 13384–13386 | DOI | MR

[73] A. Bensoussan, J. L. Lions, G. Papanicolau, Asymptotic analysis for periodic structures, North Holland, Amsterdam-New-York-Oxford, 1978 | MR | Zbl

[74] D. I. Borisov, “The asymptotics for the eigenelements of the Laplacian in a cylinder with frequently alternating boundary conditions”, C. R. Acad. Sci. Paris. Sér. IIb, 329:10 (2001), 717–721

[75] D. I. Borisov, “On a model boundary value problem for Laplacian with frequently alternating type of boundary condition”, Asymptot. Anal., 35:1 (2003), 1–26 | MR | Zbl

[76] D. Borisov, R. Bunoiu, G. Cardone, “On a waveguide with frequently alternating boundary conditions: homogenized Neumann condition”, Ann. Henri Poincare, 11:8 (2010), 1591–1627 | DOI | MR | Zbl

[77] D. Borisov, R. Bunoiu, G. Cardone, “Homogenization and asymptotics for a waveguide with an infinite number of closely located small windows”, J. Math. Sci. (N.Y.), 176:6 (2011), 774–785 | DOI | MR | Zbl

[78] D. Borisov, R. Bunoiu, G. Cardone, “Waveguide with non-periodically alternating Dirichlet and Robin conditions: homogenization and asymptotics”, Z. Angew. Math. Phys., 64:3 (2013), 439–472 | DOI | MR | Zbl

[79] D. Borisov, G. Cardone, “Homogenization of the planar waveguide with frequently alternating boundary conditions”, J. Phys. A: Math. Gen., 42:36 (2009), 365205 | DOI | MR | Zbl

[80] D. Borisov, G. Cardone, T. Durante, “Homogenization and uniform resolvent convergence for elliptic operators in a strip perforated along a curve”, Proc. Roy Soc. Edinburgh Sect. A, 146:6 (2016), 1115–1158 | DOI | MR | Zbl

[81] D. Borisov, G. Cardone, L. Faella, C. Perugia, “Uniform resolvent convergence for a strip with fast oscillating boundary”, J. Differ. Equ., 255:12 (2013), 4378–4402 | DOI | MR | Zbl

[82] A. Brillard, M. Lobo, M. E. Pérez, “Homogénéisation de frontières par épi-convergence en élasticité linéaire”, Mod el. Math. Anal. Numér., 24:1 (1990), 5–26 | DOI | MR | Zbl

[83] G. A. Chechkin, E. I. Doronina, “On asymptotics of spetrum of boundary value problem with nonperiodic rapidly alternating boundary conditions”, Funct. Differ. Equ., 8:1-2 (2001), 111–122 | MR | Zbl

[84] A. Damlamian, “Le problème de la passoire de Neumann”, Rend. Semin. Mat. Univ. Politec. Torino, 43 (1985), 427–450 | MR | Zbl

[85] A. Damlamian, T. Li (D. Li), “Boundary homogenization for ellpitic problems”, J. Math. Pures Appl., 66:4 (1987), 351–361 | MR | Zbl

[86] A. Damlamian, T. Li (D. Li), “Homogénéisation sur le bord pour les problèmes elliptiques”, C. R. Acad. Sci. Paris. Ser. I. Math., 299:17 (1984), 859–862 | MR | Zbl

[87] J. Davila, “A nonlinear elliptic equation with rapidly oscillating boundary conditions”, Asymptot. Anal., 28:3-4 (2001), 279–307 | MR | Zbl

[88] J. Filo, “A note on asymptotic expansion for a periodic boundary condition”, Arch. Math. (Brno), 34:1 (1998), 83–92 | MR | Zbl

[89] J. Filo, S. Luckhaus, “Asymptotic expansion for a periodic boundary condition”, J. Differ. Equ., 120:1 (1995), 133–173 | DOI | MR | Zbl

[90] J. Filo, S. Luckhaus, “Homogenization of a boundary condition for the heat equation”, J. Eur. Math. Soc., 2:3 (2000), 217–258 | DOI | MR | Zbl

[91] A. Friedman, Ch. Huang, J. Yong, “Effective permeability of the boundary of a domain”, Commun. Part. Differ. Equ., 20:1–2 (1995), 59–102 | DOI | MR | Zbl

[92] R. R. Gadyl'shin, “Asymptotics of minimum eigenvalue for a circle with fast oscillating boundary conditions”, C. R. Acad. Sci. Paris. Ser. I. Math., 323:3 (1996), 319–323 | MR | Zbl

[93] R. R. Gadyl'shin, “On an analog of the Helmholtz resonator in the averaging theory”, C. R. Acad. Sci. Paris. Ser. I. Math., 329:12 (1999), 1121–1126 | DOI | MR | Zbl

[94] R. R. Gadyl'shin, “Eigenvalues and scattering frequencies for domain with narrow appendixes and tubes”, C. R. Acad. Sci. Paris. Ser. IIb, 329:10 (2001), 723–726

[95] G. Griso, “Error estimate and unfolding for periodic homogenization”, Asymptot. Anal., 40:3–4 (2004), 269–286 | MR | Zbl

[96] G. Griso, “Interior error estimate for periodic homogenization”, Anal. Appl., 4:1 (2006), 61–79 | DOI | MR | Zbl

[97] W. Jäger, O. A. Oleinik, A. S. Shamaev, “Asymptotics of solutions of the boundary value problem for the Laplace equation in a partially perforated domain with boundary conditions of the third kind on the boundaries of the cavities”, Trans. Moscow Math. Soc., 1997, 163–196 | MR | Zbl

[98] A. Kratzer, W. Franz, Transzendte funcktionen, Akademische Verlagsgesellschaft, Leipzig, 1960 | MR

[99] C. E. Kenig, F. Lin, Z. Shen, “Convergence rates in $L_2$ for elliptic homogenization problems”, Arch. Ration. Mech. Anal., 203:3 (2012), 1009–1036 | DOI | MR | Zbl

[100] M. Lobo, M. E. Perez, “Asymptotic behaviour of an elastic body with a surface having small stuck regions”, RAIRO Model. Math. Anal. Numer., 22:4 (1988), 609–624 | DOI | MR | Zbl

[101] M. Lobo, M. E. Perez, “Boundary homogenization of certain elliptic problems for cylindrical bodies”, Bull. Sci. Math. Ser. 2, 116 (1992), 399–426 | MR | Zbl

[102] M. Lobo, M. E. Perez, “On vibrations of a body with many concentrated masses near the boundary”, Math. Models Methods Appl. Sci., 3:2 (1993), 249–273 | DOI | MR | Zbl

[103] M. Lobo, M. E. Perez, “Vibrations of a membrane with many concentrated masses near the boundary”, Math. Models Methods Appl. Sci., 5:5 (1995), 565–585 | DOI | MR | Zbl

[104] O. A. Oleinik, G. A. Chechkin, “Solutions and eigenvalues of the boundary value problems with rapidly alternating boundary conditions for the system of elasticity”, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 7:1 (1996), 5–15 | MR | Zbl

[105] A. I. Ovseevich, A. I. Pyatnitskii, A. S. Shamaev, “Asymptotic behavior of solutions to a boundary value problem with small parameter”, Russ. J. Math. Phys., 4:4 (1996), 487–498 | MR | Zbl

[106] V. Rybalko, “Vibrations of elastic system with a large number of tiny heavy inclusions”, Asymptot. Anal., 32:1 (2002), 27–62 | MR | Zbl

[107] V. V. Zhikov, S. E. Pastukhova, “On operator estimates for some problems in homogenization theory”, Russ. J. Math. Phys., 12:4 (2005), 515–524 | MR | Zbl