Homogenization of Boundary Value Problems in Plane Domains with Frequently Alternating Type of Nonlinear Boundary Conditions: Critical Case
- Authors: Díaz J.I.1,2, Gómez-Castro D.1,3, Podolskiy A.V.4, Shaposhnikova T.A.4
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Affiliations:
- Instituto de Matemática Interdisciplinar
- Depto, Análisis Mat. y Mat. Aplicada
- Departamento de Matemática Aplicada, Escuela Tcnica Superior de Ingeniería—ICAI
- Moscow State University
- Issue: Vol 97, No 3 (2018)
- Pages: 271-276
- Section: Mathematics
- URL: https://journals.rcsi.science/1064-5624/article/view/225511
- DOI: https://doi.org/10.1134/S1064562418030225
- ID: 225511
Cite item
Abstract
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.
About the authors
J. I. Díaz
Instituto de Matemática Interdisciplinar; Depto, Análisis Mat. y Mat. Aplicada
Email: avpodolskiy@yandex.ru
Spain, 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
Spain, Madrid, 28040; Madrid, 28015
A. V. Podolskiy
Moscow State University
Author for correspondence.
Email: avpodolskiy@yandex.ru
Russian Federation, Moscow
T. A. Shaposhnikova
Moscow State University
Email: avpodolskiy@yandex.ru
Russian Federation, Moscow