Abstract
We address the homogenization of a boundary value problem posed in perforated media for the p-Laplacian. We consider p = n, that is the n-Laplace operator in a perforated domain of \(\mathbb{R}^{n}\), n ≥ 3, while the flux (associated with the n-Laplacian) on the boundary of the perforations is given by a negative, nonlinear monotonic function of the solution which is multiplied by a parameter which can be very large compared with the periodicity of the structure O(ɛ). A certain non-periodical distribution of the perforations is allowed, while the assumption that their size is much smaller than the periodicity scale ɛ is performed. We consider different relations between the parameters of the problem and, as ɛ → 0, we obtain all the possible homogenized problems. For certain of these relations, in the average constants of the problem, the perimeter of the perforations appears for any shape.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Brillard, A., Gómez, D., Lobo, M., Pérez, E., Shaposhnikova, T.A.: Boundary homogenization in perforated domains for adsorption problems with an advection term. Appl. Anal. 95, 1517–1533 (2016)
Chechkin, G.A., Piatnitski, A.L.: Homogenization of boundary value problem in a locally periodic perforated domain. Appl. Anal. 71, 215–235 (1999)
Cioranescu, D., Donato, P.: Homogénéisation du problème de Neumann non homogène dans des ouverts perforés. Asymptot. Anal. 1, 115–138 (1988)
Cioranescu, D., Murat, F.: Un terme étrange venu d’ailleurs I & II. In: Brezis, H., Lions, J.L. (eds.) Nonlinear Partial Differential Equations and Their Applications, Collège de France Séminar, Volume II & III. Research Notes in Mathematics, vols. 60, 70, pp. 98–138, 154–178. Pitman, London (1982)
Gómez, D., Lobo, M., Pérez, E., Shaposhnikova, T.A., Zubova, M.N.: On critical parameters in homogenization of perforated domains by thin tubes with nonlinear flux and related spectral problems. Math. Methods Appl. Sci. 38, 2606–2629 (2015)
Gómez, D., Lobo, M., Pérez, M.E., Podolskii, A.V., Shaposhnikova, T.A.: Homogenization of a variational inequality for the p-Laplacian in perforated media with nonlinear restrictions for the flux on the boundary of isoperimetric perforations: p equal to the dimension of the space. Dokl. Math. 93, 140–144 (2016)
Gómez, D., Lobo, M., Pérez, E., Podolskii, A.V., Shaposhnikova, T.A.: Average of unilateral problems for the p-Laplace operator in perforated media involving large parameters on the restriction, ESAIM: Control Optim. Calc. Var. doi: 10.1051/cocv/2017026
Goncharenko, M.: The asymptotic behaviour of the third boundary-value problem solutions in domains with fine-grained boundaries. In: Homogenization and Applications to Material Sciences. GAKUTO International Series. Mathematical Sciences and Applications, vol. 9, pp. 203–213. Gakkotosho, Tokyo (1995)
Kaizu, S.: The Poisson equation with semilinear boundary conditions in domains with many tiny holes. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36, 43–86 (1989)
Labani, N., Picard, C.: Homogenization of a nonlinear Dirichlet problem in a periodically perforated domain. In: Recent Advances in Nonlinear Elliptic and Parabolic Problems. Pitman Research Notes in Mathematics Series, vol. 208, pp. 294–305. Longman Sci. Tech., Harlow (1989)
Lions, J.L.: Quelques méthodes de résolution des Problèmes aux Limites non Linéaires. Dunod, Paris (1969)
Marchenko, V.A., Khruslov, E.Ya.: Boundary Value Problems in Domains with a Fine-grained Boundary. Izdat. Naukova Dumka, Kiev (1974) [in Russian]
Oleinik, O.A., Shaposhnikova, T.A.: On homogenization problem for the Laplace operator in partially perforated domains with Neumann’s condition on the boundary of cavities. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl. 6, 133–142 (1995)
Oleinik, O.A., Shaposhnikova, T.A.: On the averaging problem for a partially perforated domain with a mixed boundary condition with a small parameter on the cavity boundary. Differ. Equ. 31, 1086–1098 (1995)
Oleinik, O.A., Shaposhnikova, T.A.: On homogenization of the Poisson equation in partially perforated domains with arbitrary density of cavities and mixed type conditions on their boundary. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl. (9) 7, 129–146 (1996)
Pérez, M.E., Shaposhnikova, T.A., Zubova, M.N.: Homogenization problem in a domain perforated by tiny isoperimetric holes with nonlinear Robin type boundary conditions. Dokl. Math. 90, 489–494 (2014)
Podolskii, A.V.: Homogenization limit for the boundary value problem with the p-Laplace operator and a nonlinear third boundary condition on the boundary of the holes in a perforated domain. Dokl. Math. 82, 942–945 (2010)
Podolskii, A.V.: Solution continuation and homogenization of a boundary value problem for the p-Laplacian in a perforated domain with a nonlinear third boundary condition on the boundary of holes. Dokl. Math. 91, 30–34 (2015)
Podolskii, A.V., Shaposhnikova, T.A.: Homogenization for the p-Laplacian in a n-dimensional domain perforated by fine cavities, with nonlinear restrictions on their booundary, when p = n. Dokl. Math. 92, 464–470 (2015)
Sanchez-Palencia, E.: Boundary value problems in domains containing perforated walls. In: Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, Vol. III. Research Notes in Mathematics, vol. 70, pp. 309–325. Pitman, Boston (1982)
Shaposhnikova, T.A., Podolskii, A.V.: Homogenization limit for the boundary value problem with the p-Laplace operator and a nonlinear third boundary condition on the boundary of the holes in a perforated domain. Funct. Differ. Equ. 19, 351–370 (2012)
Zubova, M.N., Shaposhnikova, T.A.: Homogenization of boundary value problems in perforated domains with the third boundary condition and the resulting change in the character of the nonlinearity in the problem. Differ. Equ. 47, 78–90 (2011)
Acknowledgements
This work has been partially supported by MINECO:MTM2013-44883-P.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Gómez, D., Pérez, E., Podol’skii, A.V., Shaposhnikova, T.A. (2017). On Homogenization of Nonlinear Robin Type Boundary Conditions for the n-Laplacian in n-Dimensional Perforated Domains. In: Constanda, C., Dalla Riva, M., Lamberti, P., Musolino, P. (eds) Integral Methods in Science and Engineering, Volume 1. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-59384-5_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-59384-5_11
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-59383-8
Online ISBN: 978-3-319-59384-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)