Homogenization of a heat conduction problem involving tangential operators

We present a model for the heat conduction in a composite having a microscopic structure arranged in a periodic array made by two phases separated by a thermally active membrane. The thermal behavior of the membrane is described by a parabolic equation involving the Laplace–Beltrami operator. Such interface equation furnishes the contact temperature of the two diffusive phases in terms of the jump of the heat fluxes at the interface. We obtain the macroscopic behavior of the material via an homogenization procedure based on the unfolding technique, providing the equation satisfied by the effective temperature. We are also able to prove an error estimate on the rate of convergence of the sequence of approximating solutions to the homogenized solution. These results are part of a joint research with R. Gianni.