Homogenization of an alternating Robin-Neumann boundary condition via time-periodic unfolding
In this paper the authors deal with a problem of homogenization of a parabolic equation, where the coefficients are oscillating, in space and in time, in a domain with holes. Two type of behaviors are considered respect to the suitable balance between the space and time oscillations. Let \(\tau\) and \(\varepsilon\) be the time and space period of oscillations respectively, the authors consider the fast oscillations in time when \(\tau\theta_\tau / \varepsilon \leq C\) and the case of slow oscillation in time when \(\tau\theta_\tau / \varepsilon \) goes to \(+\infty\), where \(\theta_\tau\) is relate on the degeneracy of the estimate of the time derivative, whose \(L^p\)-norm is assumed to behave in the limit as the parameter \(\theta_\tau\). The Robin-Neumann oscillating boundary conditions allow relevant estimates for homogenization process. In the limit, using unfolding time-periodic techniques, a macroscopic parabolic problem is achieved, which contain an extra linear term due to the absoption determined by the Robin condition.
scientific article; zbMATH DE number 6695865
scientific article