We study the homogenization problem for the Poisson equation in a domain perforated along a manifold with the dynamic Signorini boundary condition on the boundary of perforations (particles), with a parameter ε−γ. We focus on the case where the perforations (particles) are of an arbitrary shape and all the parameters of the problem take the critical value. The main result is the derivation and justification of a homogenized model containing a new nonlocal nonlinear operator, called the strange operator.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. I. Díaz, A. V. Podolskiy, and T. A. Shaposhnikova, “Unexpected regionally negative solutions of the homogenization of Poisson equation with dynamic unilateral boundary conditions: critical symmetric particles,” Rev. R. Acad. Cienc. Exactas. Fis. Nat., Ser. A Mat., RACSAM 118, No. 1, Paper 9 (2024).
A. V. Podolskiy and T. A. Shaposhnikova, “Homogenization of a parabolic equation in a perforated domain with unilateral dynamic boundary conditions. Critical case” [in Russian], Sovrem. Mat., Fundam. Napravlenia 68, No. 4, 671–685 (2022).
W. Jager, M. Neuss-Radu, and T. A. Shaposhnikova, “Homogenization of a variational inequality for the Laplace operator with nonlinear restriction for the flux on the interior boundary of a perforated domain,” Nonlinear Anal., Real World Appl. 15, 367–380 (2014).
D. Gómez, M. Lobo, M. E. Pérez, A. V. Podolskiy, and T. A. Shaposhnikova, “Unilateral problems for the p-Laplace operator in perforated media involving large parameters,” ESAIM Control Optim. Calc. Var. 24, No. 3, 921–964 (2018).
J. I. Díaz, D. Gómez-Castro, and T. A. Shaposhnikova, Nonlinear Reaction-Diffusion Processes for Nanocomposites. Anomalous Improved Homogenization, De Gruyter, Berlin (2021).
J. I. Díaz, D. Gómez-Castro, A. V. Podolskiy, and T. A. Shaposhnikova, “Homogenization of a net of periodic critically scaled boundary obstacles related to reverse osmosis ‘nanocomposite’ membranes,” Adv. Nonlinear Anal. 9, 193–227 (2020).
C. Conca, F. Murat, and C. Timofte, “A generalized strange term Signorini’s type problems,” M2AN, Math. Model. Numer. Anal. 57, No. 3, 773–805 (2003).
A. Yu. Vorob’ev and T. A. Shaposhnikova, “Homogenization of a nonhomogeneous Signorini problem for the Poisson equation in a periodically perforated domain,” Differ. Equ. 39, No. 3, 387–396 (2003).
S. E. Pastukhova, “Homogenization of a mixed problem with Signorini condition for an elliptic operator in a perforated domain,” Sb. Math. 192, No. 2, 245–260 (2001).
J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux limites non linéaires [in French], Dunod, Paris (1968).
J. I. Díaz, T. A. Shaposhnikova, and M. N. Zubova, “A strange non-local monotone operatot arising in the homogenization of a diffusion equation with dynamic nonlinear boundary conditions on particles of critical size and arbitrary shape,” Electron. J. Differ. Equ 2022, Paper No. 52 (2022).
M. N. Zubova and T. A. Shaposhnikova, “Appearance of a nonlocal monotone operator in transmission conditions when homogenizing the Poisson equation in a domain perforated along an (n − 1)-dimensional manifold by sets of arbitrary shape and critical size with nonlinear dynamic boundary conditions on the boundary of perforations,” J. Math. Sci. 255, No. 4, 423–443 (2021).
O. A. Oleinik and T. A. Shaposhnikova, “On homogenization problem for the Laplace operatot in partially perforated domains with Neumann’s condition on the boundary od cavities,” Rend. Lincei, Mat. Appl. 6, No. 3, 133–142 (1995).
A. V. Podolskiy and T. A. Shaposhnikova, “Homogenization of the boundary value problem for the Laplace operator in domain perforated along (n − 1)–dimensional manifold with nonlinear Robin type boundary condition on the boundary of arbitrary shaped holes: Critical case,” Dokl. Math. 96, No. 3, 601–606 (2017).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Problemy Matematicheskogo Analiza 126, 2024, pp. 65-86.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Podolskiy, A.V., Shaposhnikova, T.A. Strange Operator in Homogenization of the Diffusion Equation in a Domain Perforated Along of a Manifold with Dynamic Signorini Condition on Perforation Boundary. Critical Case. J Math Sci 279, 525–549 (2024). https://doi.org/10.1007/s10958-024-07030-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-024-07030-2