Reiterated homogenization of integral functionals

We consider the homogenization of sequences of integral functionals defined on media with several length-scales. Our general results connected to the corresponding homogenized functional are used to analyze new types of struc tures and to illustrate the wide range of effective properties achievable t hrough reiteration. In particular, we consider a two-phase structure which is optimal when the integrand is a quadratic form and point out examples wh ere the macroscopic behavior of this structure underlines an effective ener gy density which is lower than that of the best possible multirank laminate . We also present some results connected to a reiterated structure where th e effective property is extremely sensitive of the growth of the integrand.