Operator Estimates in Two-Dimensional Problems with a Frequent Alternation in the Case of Small Parts with the Dirichlet Condition
Tóm tắt
A two-dimensional boundary value problem is studied for a general 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 Robin boundary condition are set alternately. The distribution and size of these segments are arbitrary. The case is considered when, upon homogenization, the Dirichlet boundary condition completely disappears and only the original nonlinear Robin 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 homogenized 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 order sharp.
Tài liệu tham khảo
V. A. Marchenko and E. Ya. Khruslov, Boundary Value Problems with Fine-Grained Boundary (Naukova Dumka, Kyiv, 1974) [in Russian].
A. Damlamian and L. Ta-Tsien “Boundary homogenization for elliptic problems,” J. Math. Pures Appl. (9), 66 (4), 351–361 (1987).
M. Lobo and M. E. Pérez, “Asymptotic behaviour of an elastic body with a surface having small stuck regions,” Math. Model. Numer. Anal. 22 (4), 609–624 (1988). https://doi.org/10.1051/m2an/1988220406091
M. Lobo and M. E. Pérez, “Boundary homogenization of certain elliptic problems for cylindrical bodies,” Bull. Sci. Math., Ser. 2, 116, 399–426 (1992).
G. A. Chechkin, “Averaging of boundary value problems with a singular perturbation of the boundary conditions,” Sb. Math. 79 (1), 191–222 (1994). https://doi.org/10.1070/SM1994v079n01ABEH003608
A. Friedman, Ch. Huang, and J. Yong, “Effective permeability of the boundary of a domain,” Comm. Part. Diff. Equat. 20 (1–2), 59–102 (1995). https://doi.org/10.1080/03605309508821087
A. Yu. Belyaev and G. A. Chechkin, “Averaging of operators with boundary conditions of fine-scaled structure,” Math. Notes 65 (4), 418–429 (1999).
J. Dávila, “A nonlinear elliptic equation with rapidly oscillating boundary conditions,” Asympt. Anal. 28 (3–4), 279–307 (2001).
D. Borisov and G. Cardone, “Homogenization of the planar waveguide with frequently alternating boundary conditions,” J. Phys. A: Math. Theor. 42 (36), 365205 (2009). https://doi.org/10.1088/1751-8113/42/36/365205
D. Borisov, R. Bunoiu, and G. Cardone, “On a waveguide with frequently alternating boundary conditions: Homogenized Neumann condition,” Ann. H. Poincaré 11 (8), 1591–1627 (2010). https://doi.org/10.1007/s00023-010-0065-0
D. Borisov, R. Bunoiu, and G. Cardone, “Waveguide with non-periodically alternating Dirichlet and Robin conditions: Homogenization and asymptotics,” Zeit. Angew. Math. Phys. 64 (3), 439–472 (2013). https://doi.org/10.1007/s00033-012-0264-2
T. F. Sharapov, “On the resolvent of multidimensional operators with frequently changing boundary conditions in the case of the homogenized Dirichlet condition,” Sb. Math. 205 (10), 1492–1527 (2014). https://doi.org/10.1070/SM2014v205n10ABEH004427
T. F. Sharapov, “On resolvent of multi-dimensional operators with frequent alternation of boundary conditions: Critical case,” Ufa Math. J. 8 (2), 65–94 (2016). https://doi.org/10.13108/2016-8-2-65
D. I. Borisov and M. N. Konyrkulzhaeva, “Operator \(L_{2}\)-estimates for two-dimensional problems with rapidly alternating boundary conditions,” J. Math. Sci. 267 (3), 319–337 (2022). https://doi.org/10.1007/s10958-022-06136-9
T. Kato, Perturbation Theory for Linear Operators (Springer, New York, 1966; Mir, Moscow, 1972).
D. I. Borisov, “Operator estimates for planar domains with irregularly curved boundary. The Dirichlet and Neumann condition,” J. Math. Sci. 264 (5), 562–580 (2022). https://doi.org/10.1007/s10958-022-06017-1
D. I. Borisov, “Norm resolvent convergence of elliptic operators in domains with thin spikes,” J. Math. Sci. 261 (3), 366–392 (2022). https://doi.org/10.1007/s10958-022-05756-5
M. M. Vainberg, Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations (Nauka, Moscow, 1972; Wiley, New York, 1974).
Yu. A. Dubinskii, “Nonlinear elliptic and parabolic equations,” J. Math. Sci. 12 (5), 475–554 (1979). https://doi.org/10.1007/BF01089137
D. I. Borisov and J. Kříž, “Operator estimates for non-periodically perforated domains with Dirichlet and nonlinear Robin conditions: Vanishing limit,” Anal. Math. Phys. 13, 5 (2023). https://doi.org/10.1007/s13324-022-00765-8
N. N. Senik, “Homogenization for non-self-adjoint periodic elliptic operators on an infinite cylinder,” SIAM J. Math. Anal. 49 (2), 874–898 (2017). https://doi.org/10.1137/15M1049981
N. N. Senik, “Homogenization for locally periodic elliptic operators,” J. Math. Anal. Appl. 505 (2), 125581 (2021). https://doi.org/10.1016/j.jmaa.2021.125581
S. E. Pastukhova, “Homogenization estimates for singularly perturbed operators,” J. Math. Sci. 251 (5), 724–747 (2020). https://doi.org/10.1007/s10958-020-05125-0
S. E. Pastukhova, “\(L_{2}\)-approximation of resolvents in homogenization of higher order elliptic operators,” J. Math. Sci. 251 (6), 902–925 (2020). https://doi.org/10.1007/s10958-020-05135-y
S. E. Pastukhova, “Approximation of resolvents in homogenization of fourth-order elliptic operators,” Sb. Math. 212 (1), 111–134 (2021). https://doi.org/10.4213/im459
D. I. Borisov, “Asymptotics and estimates for the eigenelements of the Laplacian with frequently alternating nonperiodic boundary conditions,” Izv. Math. 67 (6), 1101–1148 (2003). https://doi.org/10.1070/IM2003v067n06ABEH000459