Доклады Академии наукДоклады Академии наук0869-5652The Russian Academy of Sciences1282810.31857/S0869-56524853263-268Research ArticleHomogenization of the boundary value problem for the poisson equation with rapidly oscillating nonlinear boundary conditions: space dimension n ≥ 3, critical casePodolskiyA. V.shaposh.tan@mail.ruShaposhnikovaT. A.shaposh.tan@mail.ruMoscow State University2105201948532632682205201922052019Copyright © 2019, Russian academy of sciences2019<p>The homogenization of the Poisson equation in a bounded domain with rapidly oscillating boundary conditions specied on a part of the domain boundary is studied. A Neumann boundary condition alternates with an -periodically distributed nonlinear Robin condition involving the coefficient <sup></sup>, where R. The diameter of the boundary portions with a nonlinear Robin condition is of order O(), 1. A critical relation between the parameters and is considered</p>homogenizationPoisson’s equationboundary value problemoscillating boundary conditionусреднениеуравнение Пуассонакраевая задачаосциллирующее краевое условие.[Диаз Ж.И., Гомез-Кастро Д., Подольский А.В., Шапошникова Т.А. // ДАН. 2018. Т. 480. № 6. С. 644-649.][Zubova M. N., Shaposhnikova T. A. // Math. Notes. 2007. V. 82. Iss. 4. P. 481-491.][Damlamian A., Li Ta-Tsien // J. Math. Pures Appl. (9). 1987. V. 66. № 4. P. 351-361.][Chechkin G. A. // Sb. Math. 1994. V. 79. № 1. P. 191-222.][Borisov D. I. // Asymptot. Anal. 2003. V. 35. № 1. P. 1-26.][Oleinik O. A., Shaposhnikova T. A. // Rend. Mat. Acc. Lincei. Ser. 9. 1995. V. 6. P. 133-142.][Oleinik O. A., Shaposhnikova T. A. // Rend. Mat. Acc. Lincei. Ser. 9. 1996. V. 7. P. 129-146.][Kaizu S. // J. Fac. Sci. Univ. Tokyo Sect. IA Math. 1989. V. 36. P. 43-86.][Jager W., Neuss-Radu M., Shaposhnikova T. A. // Nonlin. Anal.: Real World Appl. 2014. V. 15. P. 367-380.][Gomez D., Lobo M., Perez E., Podol’skii A.V., Shaposhnikova T. A. // ESAIM Control Optim. Calc. Var. 2017. DOI: 10.1051/cocv/2017026.][Диаз Ж.И., Гомез-Кастро Д., Подольский А.В., Шапошникова T. A. // ДАН. 2016. Т. 469. № 2. С. 148-153.]