Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 52, pp. 1-32.

Title: 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
       
Authors: Jesus Ildefonso Diaz (Univ. Complutense, Madrid, Spain)
         Tatiana  A. Shaposhnikova (Moscow State Univ., Moscow, Russia)
         Maria N. Zubova (Moscow State Univ., Moscow, Russia)

Abstract: 
 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 L<sup>2</sup>(0,T), independently of
the regularity of &sigma;.  

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.