Electron. J. Differential Equations, Vol. 2022 (2022), No. 52, pp. 1-32.

A strange non-local monotone operator arising in the homogenization of a diffusion equation with dynamic nonlinear boundary conditions on particles of critical size and arbitrary shape

Jesus Ildefonso Diaz, Tatiana A. Shaposhnikova, Maria N. Zubova

We characterize the homogenization limit of the solution of a Poisson equation in a bounded domain, either periodically perforated or containing a set of asymmetric periodical small particles and on the boundaries of these particles a nonlinear dynamic boundary condition holds involving a Holder nonlinear σ(u). We consider the case in which the diameter of the perforations (or the diameter of particles) is critical in terms of the period of the structure. As in many other cases concerning critical size, a "strange" nonlinear term arises in the homogenized equation. For this case of asymmetric critical particles we prove that the effective equation is a semilinear elliptic equation in which the time arises as a parameter and the nonlinear expression is given in terms of a nonlocal operator H which is monotone and Lipschitz continuous on L2(0,T), independently of the regularity of σ.

Submitted May 8, 2022. Published July 18, 2022.
Math Subject Classifications: 35B27, 35K57, 35K91, 35R01, 47B44.
Key Words: Critically scaled homogenization; asymmetric particles; dynamic boundary conditions; Holder continuous reactions; strange term; nonlocal monotone operator.

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Jesús Ildefonso Díaz
Instituto de Matematica Interdisciplinar
Universidad Complutense
Madrid, 28040 Spain
email: ji_diaz@ucm.es
Tatiana A. Shaposnikova
Faculty of Mechanics and Mathematics
Moscow State University
Moscow, 119991 Russia
email: shaposh.tan@mail.ru
Maria N. Zubova
Faculty of Mechanics and Mathematics
Moscow State University
Moscow, 119991 Russia
email: zubovnv@mail.ru

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