Quantitative homogenization of non-linear elastic periodic composites for small loads

In this talk, I consider periodic homogenization of non-convex integral functionals that are motivated by non-linear elasticity. In this situation long wavelength buckling can occur which mathematically implies that the homogenized integrand is given by an asymptotic multi-cell formula. From this formula it is difficult to deduce qualitative or quantitative properties of the effective energy. Under suitable assumptions, in particular that the integrand has a single, non-degenerate, energy well at the set of rotations, we show that the multi-cell formula reduces to a much simpler single-cell formula in a neighbourhood of the rotations. This allows for a more refined, corrector based, analysis. In particular, for small data, we obtain a quantitative two-scale expansion and uniform Lipschitz estimates for energy minimizer. This is joint work with Stefan Neukamm (Dresden).