On the asymptotic behaviour of eigenvalues of a~boundary-value problem

R. R. Gadyl'shin ; A. L. Piatnitski ; G. A. Chechkin

Izvestiya. Mathematics , Tome 82 (2018) no. 6, pp. 1108-1135.

Résumé

We consider a two-dimensional spectral problem of Steklov type for the Laplace operator in a domain divided into two parts by a perforated partition with a periodic microstructure. The Steklov boundary condition is imposed on the lateral sides of the perforation, the Neumann condition on the remaining part of the boundary, and the Dirichlet and Neumann conditions on the outer boundary of the domain. We construct and justify two-term asymptotic expressions for the eigenvalues of this problem. We also construct a two-term asymptotic formula for the corresponding eigenfunctions.

Détail

Keywords: asymptotic behaviour of eigenvalues, spectral problem, Steklov problem, homogenization of spectral problems.

Bibliographie

[1] M. Vanninathan, “Homogenization of eigenvalue problems in perforated domains”, Proc. Indian Acad. Sci. Math. Sci., 90:3 (1981), 239–271 | DOI | MR | Zbl

[2] S. Kesavan, “Homogenization of elliptic eigenvalue problems: Part 1”, Appl. Math. Optim., 5:2 (1979), 153–167 | DOI | MR | Zbl

[3] A. G. Chechkina, “Convergence of solutions and eigenelements of Steklov type boundary value problems with boundary conditions of rapidly varying type”, J. Math. Sci. (N. Y.), 162:3 (2009), 443–458 | DOI | MR

[4] A. G. Chechkina, C. D'Apice, U. De Maio, “Rate of convergence of eigenvalues to singularly perturbed Steklov-type problem for elasticity system”, Appl. Anal. (to appear) , Publ. online: 21 Dec. 2017 | DOI

[5] G. A. Chechkin, T. A. Mel'nyk, “Spatial-skin effect for eigenvibrations of a thick cascade junction with ‘heavy’ concentrated masses”, Math. Methods Appl. Sci., 37:1 (2014), 56–74 | DOI | MR | Zbl

[6] G. Griso, A. Migunova, J. Orlik, “Asymptotic analysis for domains separated by a thin layer made of periodic vertical beams”, J. Elasticity, 128:2 (2017), 291–331 | DOI | MR | Zbl

[7] G. A. Chechkin, T. P. Chechkina, “Plane scalar analog of linear degenerate hydrodynamic problem with nonperiodic microstructure of free surface”, J. Math. Sci. (N. Y.), 207:2 (2015), 336–349 | DOI | MR | Zbl

[8] V. V. Zhikov, “On an extension of the method of two-scale convergence and its applications”, Sb. Math., 191:7 (2000), 973–1014 | DOI | DOI | MR | Zbl

[9] Yu. D. Golovaty, D. Gómez, M. Lobo, E. Pérez, “On vibrating membranes with very heavy thin inclusions”, Math. Models Methods Appl. Sci., 14:7 (2004), 987–1034 | DOI | MR | Zbl

[10] N. Babych, Yu. Golovaty, “Low and high frequency approximations to eigenvibrations of string with double contrasts”, J. Comput. Appl. Math., 234:6 (2010), 1860–1867 | DOI | MR | Zbl

[11] O. A. Oleĭnik, A. S. Shamaev, G. A. Yosifian, Mathematical problems in elasticity and homogenization, Stud. Math. Appl., 26, North-Holland Publishing Co., Amsterdam, 1992, xiv+398 pp. | MR | MR | Zbl | Zbl

[12] S. A. Nazarov, “Two-term asymptotics of solutions of spectral problems with singular perturbations”, Math. USSR-Sb., 69:2 (1991), 307–340 | DOI | MR | Zbl

[13] 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 | MR | Zbl

[14] D. I. Borisov, “Dvuparametricheskie asimptotiki sobstvennykh chisel laplasiana s chastym cheredovaniem granichnykh uslovii”, Vestnik molodykh uchenykh, Ser. Prikl. matem. i mekh., 2002:1 (2002), 36–52

[15] Y. Amirat, G. A. Chechkin, R. R. Gadyl'shin, “Asymptotics for eigenelements of Laplacian in domain with oscillating boundary: multiple eigenvalues”, Appl. Anal., 86:7 (2007), 873–897 | DOI | MR | Zbl

[16] R. R. Gadyl'shin, D. V. Kozhevnikov, G. A. Chechkin, “Spectral problem in a domain perforated along the boundary. Perturbation of a multiple eigenvalue”, J. Math. Sci. (N. Y.), 196:3 (2014), 276–292 | DOI | MR | Zbl

[17] Y. Amirat, O. Bodard, G. A. Chechkin, A. L. Piatnitski, “Asymptotics of a spectral-sieve problem”, J. Math. Anal. Appl., 435:2 (2016), 1652–1671 | DOI | MR | Zbl

[18] V. A. Marchenko, E. Ya. Khruslov, Homogenization of partial differential equations, Prog. Math. Phys., 46, Birkhäuser Boston, Inc., Boston, MA, 2006, xiv+398 pp. | MR | Zbl

[19] T. Del Vecchio, “The thick Neumann's sieve”, Ann. Mat. Pura Appl. (4), 147 (1987), 363–402 | DOI | MR | Zbl

[20] M. Lobo, O. A. Oleinik, M. E. Pérez, T. A. Shaposhnikova, “On homogenization of solutions of boundary value problems in domains, perforated along manifolds”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25:3-4 (1997), 611–629 | MR | Zbl

[21] T. A. Mel'nyk, “Scheme of investigation of the spectrum of a family of perturbed operators and its application to spectral problems in thick junction”, Nonlinear Oscil., 6:2 (2003), 232–249 | DOI | MR

[22] E. Sánchez-Palencia, “Boundary value problems in domains containing perforated walls”, Nonlinear partial differential equations and their applications, Collège de France seminar, Vol. III (Paris, 1980/1981), Res. Notes in Math., 70, Pitman, Boston, MA–London, 1982, 309–325 | MR | Zbl

[23] D. Onofrei, B. Vernescu, “Asymptotics of a spectral problem associated with the Neumann sieve”, Anal. Appl. (Singap.), 3:1 (2005), 69–87 | DOI | MR | Zbl

[24] T. A. Mel'nyk, “Asymptotic behavior of eigenvalues and eigenfunctions of the Steklov problem in a thick periodic junction”, Nonlinear Oscil., 4:1 (2001), 91–105 | MR | Zbl

[25] W. Stekloff, “Sur les problèmes fondamentaux de la physique mathématique”, Ann. Sci. École Norm. Sup. (3), 19 (1902), 455–490 | DOI | MR | Zbl

[26] V. P. Mikhajlov, Partial differential equations, Moscow, Mir Publishers, 1978, 396 pp. | MR | MR | Zbl | Zbl

[27] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, reprint of the 2nd ed., Springer-Verlag, Berlin, 1998, xiv+517 pp. | DOI | MR | MR | Zbl | Zbl

[28] E. M. Landis, G. P. Panasenko, “A variant of a Phragmén–Lindelöf theorem for elliptic equations with coefficients that are periodic functions of all variables except one”, Topics in modern mathematics, Petrovskii seminar No 5, Contemp. Soviet Math., Consultants Bureau [Plenum], New York, 1985, 133–172 | MR | MR | Zbl | Zbl

[29] I. Pankratova, A. Piatnitski, “Homogenization of convection-diffusion equation in infinite cylinder”, Netw. Heterog. Media, 6:1 (2011), 111–126 | DOI | MR | Zbl

[30] G. A. Chechkin, A. L. Piatnitski, A. S. Shamaev, Homogenization. Methods and applications, Transl. Math. Monogr., 234, Amer. Math. Soc., Providence, RI, 2007, x+234 pp. | MR | Zbl

[31] M. I. Višik, L. A. Lyusternik, “Regular degeneration and boundary layer for linear differential equations with small parameter”, Amer. Math. Soc. Transl. Ser. 2, 20, Amer. Math. Soc., Providence, RI, 1962, 239–364 | DOI | MR | MR | Zbl | Zbl

[32] E. Acerbi, V. Chiadò Piat, G. Dal Maso, D. Percivale, “An extension theorem from connected sets, and homogenization in general periodic domains”, Nonlinear Anal., 18:5 (1992), 481–496 | DOI | MR | Zbl

[33] A. M. Il'in, Matching of asymptotic expansions of solutions of boundary value problems, Transl. Math. Monogr., 102, Amer. Math. Soc., Providence, RI, 1992, x+281 pp. | MR | MR | Zbl | Zbl

[34] V. Maz'ya, S. Nazarov, B. Plamenevskii, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, v. 1, 2, Oper. Theory Adv. Appl., 111, 112, Birkhäuser, Basel, 2000, xxiii+435 pp., xxiii+323 pp. | MR | MR | Zbl

[35] J. M. Lee, G. Uhlmann, “Determining anisotropic real-analytic conductivities by boundary measurements”, Comm. Pure Appl. Math., 42:8 (1989), 1097–1112 | DOI | MR | Zbl