Electron. J. Differential Equations, Vol. 2024 (2024), No. 03, pp. 1-29.

Strange non-local operators homogenizing the Poisson equation with dynamical unilateral boundary conditions: asymmetric particles of critical size

Jesus Ildefonso Diaz, Tatiana A. Shaposhnikova, Alexander V. Podolskiy

We study the homogenization of a nonlinear problem given by the Poisson equation, in a domain with arbitrarily shaped perforations (or particles) and with a dynamic unilateral boundary condition (of Signorini type), with a large coefficient, on the boundary of these perforations (or particles). This problem arises in the study of chemical reactions of zero order. The consideration of a possible asymmetry in the perforations (or particles) is fundamental for considering some applications in nanotechnology, where symmetry conditions are too restrictive. It is important also to consider perforations (or particles) constituted by small different parts and then with several connected components. We are specially concerned with the so-called critical case in which the relation between the coefficient in the boundary condition, the period of the basic structure, and the size of the holes (or particles) leads to the appearance of an unexpected new term in the effective homogenized equation. Because of the dynamic nature of the boundary condition this "strange term" becomes now a non-local in time and non-linear operator. We prove a convergence theorem and find several properties of the "strange operator" showing that there is a kind of regularization through the homogenization process.

Submitted November 29, 2023. Published January 4, 2024.
Math Subject Classifications: 35B27, 35K57, 35K91, 35R01, 47B44.
Key Words: Critically scaled homogenization; asymmetric perforated domain; asymmetric particles; unilateral dynamic boundary conditions; strange term; nonlocal monotone operator; Signorini problem.
DOI: 10.58997/ejde.2023.03

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Jesús Ildefonso Díaz
Instituto de Mathematica Interdisciplinar
Universidad Complutense de Madrid, Spain
email: jidiaz@ucm.es
Tatiana A. Shaposhnikova
Lomonosov Moscow State University
Moscow, Russia
email: shaposh.tan@mail.ru
Alexander V. Podolskiy
Lomonosov Moscow State University
Moscow, Russia
email: avpodolskiy@yandex.ru

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