Homogenization of a parabolic problem with alternating boundary conditions.

We consider the homogenization of a parabolic problem in a perforated domain with Robin–Neumann boundary conditions oscillating in time. Boundary conditions alternating in time appear in biological applications, for example in the modeling of ion channels, see [1]. Our approach relies upon a generalization of the unfolding technique, see e.g., [2], to the time-periodic case. To this end we show how the method of periodic unfolding can be applied to classical homogenization problem for a parabolic equation with diffusion and capacity-like coefficients in the diffusion equation oscillating both in space and time, with general independent scales. From an analytical point of view, in the present case such oscillations must compensate the blow up of the boundary measure of the holes.We obtain a macroscopic parabolic problem containing an extra linear term due to the absorption determined by the Robin condition; this term keeps memory of the underlying temporal and spatial microstructures. [1] D. Andreucci, D. Bellaveglia. Permeability of interfaces with alternating pores in parabolic problems. Asymptotic Analysis. 79 (2012), 189–227. [2] D. Cioranescu, A. Damlamian and G. Griso. The periodic unfolding method in homogenization. SIAM Journal on Mathematical Analysis. 40(4) (2008), 1585–1620.