Homogenization of Boundary Value Problems in Plane Domains with Frequently Alternating Type of Nonlinear Boundary Conditions: Critical Case
- 作者: Díaz J.1,2, Gómez-Castro D.1,3, Podolskiy A.4, Shaposhnikova T.4
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隶属关系:
- Instituto de Matemática Interdisciplinar
- Depto, Análisis Mat. y Mat. Aplicada
- Departamento de Matemática Aplicada, Escuela Tcnica Superior de Ingeniería—ICAI
- Moscow State University
- 期: 卷 97, 编号 3 (2018)
- 页面: 271-276
- 栏目: Mathematics
- URL: https://journals.rcsi.science/1064-5624/article/view/225511
- DOI: https://doi.org/10.1134/S1064562418030225
- ID: 225511
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详细
In the present paper we consider a boundary homogenization problem for the Poisson’s equation in a bounded domain and with a part of the boundary conditions of highly oscillating type (alternating between homogeneous Neumman condition and a nonlinear Robin type condition involving a small parameter). Our main goal in this paper is to investigate the asymptotic behavior as ε → 0 of the solution to such a problem in the case when the length of the boundary part, on which the Robin condition is specified, and the coefficient, contained in this condition, take so-called critical values. We show that in this case the character of the nonlinearity changes in the limit problem. The boundary homogenization problems were investigate for example in [1, 2, 4]. For the first time the effect of the nonlinearity character change via homogenization was noted for the first time in [5]. In that paper an effective model was constructed for the boundary value problem for the Poisson’s equation in the bounded domain that is perforated by the balls of critical radius, when the space dimension equals to 3. In the last decade a lot of works appeared, e.g., [6–10], in which this effect was studied for different geometries of perforated domains and for different differential operators. We note that in [6–10] only perforations by balls were considered. In papers [11, 12] the case of domains perforated by an arbitrary shape sets in the critical case was studied.
作者简介
J. Díaz
Instituto de Matemática Interdisciplinar; Depto, Análisis Mat. y Mat. Aplicada
Email: avpodolskiy@yandex.ru
西班牙, Madrid, 28040; Madrid, 28040
D. Gómez-Castro
Instituto de Matemática Interdisciplinar; Departamento de Matemática Aplicada, Escuela Tcnica Superior de Ingeniería—ICAI
Email: avpodolskiy@yandex.ru
西班牙, Madrid, 28040; Madrid, 28015
A. Podolskiy
Moscow State University
编辑信件的主要联系方式.
Email: avpodolskiy@yandex.ru
俄罗斯联邦, Moscow
T. Shaposhnikova
Moscow State University
Email: avpodolskiy@yandex.ru
俄罗斯联邦, Moscow