Homogenization of the diffusion equation with nonlinear flux condition on the interior boundary of a perforated domain — the influence of the scaling on the nonlinearity in the effective sink-source term

Abstract

In this paper, we study the asymptotic behavior of solutions ue of the initial boundary value problem for parabolic equations in domains \( {\Omega_\varepsilon } \subset {\mathbb{R}^n} \), n ≥ 3, perforated periodically by balls with radius of critical size eα, α = n/(n − 2), and distributed with period e. On the boundary of the balls a nonlinear third boundary condition is imposed. The weak convergence of the solutions ue to the solution of an effective equation is given. Furthermore, an improved approximation for the gradient of the microscopic solutions is constructed, and a corrector result with respect to the energy norm is proved.