Abstract

This paper deals with the homogenization of an elliptic boundary value problem in a finite cylindrical domain that consists of two connected components separated by a periodically oscillating interface situated in a band $B$ of positive measure. That is, the amplitude of the oscillating interface is supposed to be fixed, while the period of oscillations is small. On the interface, the flux is assumed to be continuous, and the jump of the solution on the interface is assumed to be proportional to the flux through the interface. Unlike previous works in the literature, here the coefficients are highly oscillating in any directions. For this reason, we need to adapt the periodic unfolding method to our situation, and introduce some related functional spaces. The limit solution is a couple $(u_1,u_2)$, where $u_1$ is defined in one side $Q_1$ and in $B$, and $u_2$ is defined in the other side $Q_2$ and in $B$. We prove that the homogenized problem is a coupled system, where $u_i$ solves a homogenized PDE in $Q_i$, with $i=1,2$, while the two limits solve two coupled differential equations $B$, where only the derivative in one direction appears. We describe also the boundary conditions in each part of the boundaries, and the $L^2$ convergence of the solutions and the fluxes is established. Finally, we prove the convergence of the energies. The main tools when proving these results are a suitable weak compactness result and an accurate study of the limit of the interface integrals on the oscillating boundary. As an illustration of the accuracy of the approximations, a numerical example is provided.

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