Some examples of nonlinear homogenization involving nearly degenerate energies

We consider a specific class of nonlinear homogenization problems. The microstructure is a sort of checkerboard polycrystal, and the energy of the basic crystal is degenerate in one direction. We give matching upper and lower bounds for the homogenized energy. The motivation for this problem lies in the recent work of Bhattacharya and Kohn on shape–memory polycrystals. Our results show that a bound proved therein is nearly sharp.