D.I. Borisov. Operator estimates in two-dimensional problems with a frequent alternation in the case of small parts with the Dirichlet condition ... P. 36-55

A two-dimensional boundary value problem is studied for a scalar elliptic second-order equation of the general form with frequent alternation of boundary conditions. The alternation is defined on small, closely spaced parts of the boundary, on which the Dirichlet boundary condition and the nonlinear third boundary condition are set alternately. The distribution and size of these segments are arbitrary. The case is considered when, upon averaging, the Dirichlet boundary condition completely disappears and only the original nonlinear third boundary condition remains. The main result is estimates for the $W_2^1$- and $L_2$-norms of the difference between the solutions of the perturbed and averaged problems, which are uniform in the $L_2$-norm of the right-hand side and characterize the rate of convergence. It is shown that these estimates are exact in the order of smallness.

Keywords: two-dimensional boundary value problem, elliptic equation, frequent alternation, homogenization, operator estimate

Received January 30, 2023

Revised February 16, 2023

Accepted February 20, 2023

Funding Agency: The work is partially supported by the Grant Agency of Czech Republic (grant no. 22-18739S) and by the Ministry of Education of the Russian Federation in the framework of the state task (agreement No.073-03-2023-010 on 26.01.2023).

Denis Ivanovich Borisov, Dr. Phys.-Math. Sci., Prof., Institute of Mathematics of Ufa Federal Research Center of Russian Academy of Sciences, Ufa, 450008 Russia; Bashkir State Pedagogical University named after M. Akmullah, Ufa, the Republic of Bashkortostan, Russia; University of Hradec Králové, Hradec Kralove, 500 03, Czech Republic e-mail: borisovdi@yandex.ru


1.   Marchenko V.A., Khruslov E.Ya. Kraevye zadachi v oblastyakh s melkozernistoi granitsei [Boundary-value problems with fine-grained boundary]. Kiev: Naukova Dumka Publ., 1974, 280 p.

2.   Damlamian Alain, Li Ta-Tsien. Boundary homogenization for elliptic problems. J. Math. Pures Appl. (9), 1987, vol. 66, no. 4, p. 351–361.

3.   Lobo M., Pérez M.E. Asymptotic behaviour of an elastic body with a surface having small stuck regions. Math. Model. Numer. Anal., 1988, vol. 22, no. 4, pp. 609–624. doi: 10.1051/m2an/1988220406091

4.   Lobo M., Pérez M.E. Boundary homogenization of certain elliptic problems for cylindrical bodies. Bull. Sci. Math., Ser. 2, 1992, vol. 116, pp. 399–426.

5.   Chechkin G. A. Averaging of boundary value problems with a singular perturbation of the boundary conditions. Sb. Math., 1994, vol. 79, no. 1, pp. 191–222. doi: 10.1070/SM1994v079n01ABEH003608

6.   Friedman A., Huang Ch. and Yong J. Effective permeability of the boundary of a domain. Comm. Part. Diff. Equat., 1995, vol. 20, no. 1-2, pp. 59–102. doi: 10.1080/03605309508821087

7.   Belyaev A. Yu., Chechkin G. A. Averaging of operators with boundary conditions of fine-scaled structure. Math. Notes., 1999, vol. 65, no. 4, pp. 418–429.

8.   Dávila J. A nonlinear elliptic equation with rapidly oscillating boundary conditions. Asympt. Anal., 2001, vol. 28, no. 3-4, pp. 279–307.

9.   Borisov D., Cardone G. Homogenization of the planar waveguide with frequently alternating boundary conditions. J. Phys. A: Math. Theor., 2009, vol. 42, no. 36, article no. 365205. doi: 10.1088/1751-8113/42/36/365205

10.   Borisov D., Bunoiu R., Cardone G. On a waveguide with frequently alternating boundary conditions: homogenized Neumann condition. Ann. H. Poincaré, 2010, vol. 11, no. 8, pp. 1591–1627. doi: 10.1007/s00023-010-0065-0

11.   Borisov D., Bunoiu R., Cardone G. Waveguide with non-periodically alternating Dirichlet and Robin conditions: homogenization and asymptotics. Zeit. Angew. Math. Phys., 2013, vol. 64, no. 3, pp. 439–472. doi: 10.1007/s00033-012-0264-2

12.   Sharapov T.F. On the resolvent of multidimensional operators with frequently changing boundary conditions in the case of the homogenized Dirichlet condition. Sb. Math., 2014, vol. 205, no. 10, pp. 1492–1527. doi: 10.1070/SM2014v205n10ABEH004427

13.   Sharapov T.F. On resolvent of multi-dimensional operators with frequent alternation of boundary conditions: critical case. Ufa Math. J., 2016, vol. 8, no. 2, pp. 65–94. doi: 10.13108/2016-8-2-65

14.   Borisov D.I., Konyrkulzhaeva M.N. Operator $L_2$-estimates for two-dimensional problems with rapidly alternating boundary conditions. J. Math. Sci., 2022, vol. 267, no. 3, pp. 319–337. doi: 10.1007/s10958-022-06136-9

15.   Kato T. Perturbation Theory for Linear Operators. Heidelberg: Springer, 1966, 592 p. doi: 10.1007/978-3-662-12678-3 Translated to Russian under the title Teoriya vozmushchenii lineinykh operatorov, Moscow, Mir Publ., 1972, 740 p.

16.   Borisov D.I. Operator estimates for planar domains with irregularly curved boundary. The Dirichlet and Neumann сondition. J. Math. Sci., 2022, vol. 264, no. 5, pp. 562–580. doi: 10.1007/s10958-022-06017-1

17.   Borisov D.I., Norm resolvent convergence of elliptic operators in domains with thin spikes. J. Math. Sci., 2022, vol. 261, no. 3, pp. 366–392. doi: 10.1007/s10958-022-05756-5

18.   Vainberg M.M. Variational method and method of monotone operators in the theory of nonlinear equations. NY: Halsted Press Book; London: John Wiley & Sons, 1973, 356 p. ISBN-10: 0685138119. Original Russian text published in Vainberg M.M. Variatsionnyi metod i metod monotonnykh operatorov v teorii nelineinykh uravnenii, Moscow, Nauka Publ., 1972, 416 p.

19.   Dubinskii Yu.A. Nonlinear elliptic and parabolic equations. J. Math. Sci., 1979, vol. 12, no. 5, pp. 475–554. doi: 10.1007/BF01089137

20.   Borisov D.I., Kříž J. Operator estimates for non-periodically perforated domains with Dirichlet and nonlinear Robin conditions: vanishing limit. Anal. Math. Phys., 2023, vol. 13, article id. 5. doi: 10.1007/s13324-022-00765-8

21.   Senik N.N. Homogenization for non-self-adjoint periodic elliptic operators on an infinite cylinder. SIAM J. Math. Anal., 2017, vol. 49, no. 2, pp. 874–898. doi: https://doi.org/10.1137/15M1049981

22.   Senik N.N. Homogenization for locally periodic elliptic operators. J. Math. Anal. Appl., 2022, vol. 505, no. 2, article no. 125581. doi: 10.1016/j.jmaa.2021.125581

23.   Pastukhova S.E. Homogenization estimates for singularly perturbed operators. J. Math. Sci., 2020, vol. 251, no. 5, pp. 724–747. doi: 10.1007/s10958-020-05125-0

24.   Pastukhova S.E. $L_2$-approximation of resolvents in homogenization of higher order elliptic operators. J. Math. Sci., 2020, Vol. 251, no. 6, pp. 902–925. doi: 10.1007/s10958-020-05135-y

25.   Pastukhova S.E. Approximation of resolvents in homogenization of fourth-order elliptic operators. Sb. Math., 2021, vol. 212, no. 1, pp. 111–134. doi: 10.1070/SM9413

26.   Borisov D.I. Asymptotics and estimates for the eigenelements of the Laplacian with frequently alternating nonperiodic boundary conditions. Izv. Math., 2003, vol. 67, no. 6, pp. 1101–1148. doi: 10.1070/IM2003v067n06ABEH000459

Cite this article as: D.I. Borisov. Operator estimates in two-dimensional problems with a frequent alternation in the case of small parts with the Dirichlet condition. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 1, pp. 36–55; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2023, Vol. 321, Suppl. 1, pp. S33–S52.