Aperiodical Isoperimetric Planar Homogenization with Critical Diameter: Universal Non-local Strange Term for a Dynamical Unilateral Boundary Condition

Abstract

We study the asymptotic behavior of the solution to the diffusion equation in a planar domain, perforated by tiny sets of different shapes with a constant perimeter and a uniformly bounded diameter, when the diameter of a basic cell, \(\varepsilon \), goes to 0. This makes the structure of the heterogeneous domain aperiodical. On the boundary of the removed sets (or the exterior to a set of particles, as it arises in chemical engineering), we consider the dynamic unilateral Signorini boundary condition containing a large-growth parameter \(\beta (\varepsilon )\). We derive and justify the homogenized model when the problem’s parameters take the “critical values”. In that case, the homogenized problem is universal (in the sense that it does not depend on the shape of the perforations or particles) and contains a “strange term” given by a non-linear, non-local in time, monotone operator H that is defined as the solution to an obstacle problem for an ODE operator. The solution of the limit problem can take negative values even if, for any \(\varepsilon \), in the original problem, the solution is non-negative on the boundary of the perforations or particles.