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

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 \(\sigma(u)\). 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 \(L^2(0,T)\), independently of the regularity of \(\sigma\).