Abstract
In previous works, the homogenization of the problem with p-Laplace diffusion and nonlinear reaction in the boundary of periodically distributed particles in n-dimensional domains has been studied in the cases where \(p \le n\). The main trait of the cases \(p \le n\) is the existence of a critical size of the particles, for which the nonlinearity arising of the limit problem does not coincide with the non linear term of the microscopic reaction. The main result of this paper proves that in the case \(p > n\) there exists no critical size.
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References
Cioranescu, D., Murat, F.: A strange term coming from nowhere. In: Cherkaev, A., Kohn, R. (eds.) Topics in Mathematical Modelling of Composite Materials, pp. 45–94. Springer Science+Business Media, LLC, New York (1997)
Díaz, J.I.: Nonlinear Partial Differential Equations and Free Boundaries. Pitman, London (1985)
Díaz, J.I., Gómez-Castro, D., Podol’skii, A.V., Shaposhnikova, T.A.: Homogenization of the p-Laplace operator with nonlinear boundary condition on critical size particles: identifying the strange terms for some non smooth and multivalued operators. Doklady Math. 94(1), 387–392 (2016)
Díaz, J.I., Gómez-Castro, D., Podol’skii, A.V., Shaposhnikova, T.A.: Homogenization of variational inequalities of Signorini type for the \(p\)-Laplacian in perforated domains when \(p \in (1, 2)\). Doklady Math. (2017). (To appear)
Díaz, J.I., Gómez-Castro, D., Podol’skiy, A.V., Shaposhnikova, T.A.: Characterizing the strange term in critical size homogenization: quasilinear equations with a nonlinear boundary condition involving a general maximal monotone graph (2017). (To appear)
Goncharenko, M.V.: Asymptotic behavior of the third boundary-value problem in domains with fine-grained boundaries. In: Damlamian, A. (ed.) Proceedings of the Conference “Homogenization and Applications to Material Sciences” (Nice. 1995), volume GAKUTO of GAKUTO, pp. 203–213. Gakkötosho, Tokyo (1997)
Oleinik, O.A., Shaposhnikova, T.A.: On homogeneization problems for the Laplace operator in partially perforated domains with Neumann’s condition on the boundary of cavities. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 6(3), 133–142 (1995)
Podol’skii, 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. Doklady Math. 91(1), 30–34 (2015)
Podol’skiy, A.V., Shaposhnikova, T.A.: Homogenization for the p-Laplacian in an n-dimensional domain perforated by very thin cavities with a nonlinear boundary condition on their boundary in the case p = n. Doklady Math. 92(1), 464–470 (2015)
Shaposhnikova, T.A., Podolskiy, 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(3–4), 1–20 (2012)
Simon, J.: Journées d’Analyse Non Linéaire: Proceedings, Besançon, France, June 1977. Chapter Regularite, pp. 205–227. Springer, Berlin (1978)
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(1), 78–90 (2011)
Zubova, M.N., Shaposhnikova, T.A.: Averaging of boundary-value problems for the Laplace operator in perforated domains with a nonlinear boundary condition of the third type on the boundary of cavities. J. Math. Sci. 190(1), 181–193 (2013)
Acknowledgements
The research of the first two authors was partially supported by the project ref. MTM2014-57113-P of the DGISPI (Spain) and as members of the Research Group MOMAT (Ref. 910480) of the UCM . The research of D. Gómez-Castro was supported by a FPU Grant from the Ministerio de Educación, Cultura y Deporte (Spain).
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Díaz, J.I., Gómez-Castro, D., Podolskii, A.V. et al. Non existence of critical scales in the homogenization of the problem with p-Laplace diffusion and nonlinear reaction in the boundary of periodically distributed particles in n-dimensional domains when \(p > n\) . RACSAM 112, 331–340 (2018). https://doi.org/10.1007/s13398-017-0381-z
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DOI: https://doi.org/10.1007/s13398-017-0381-z