Relaxation of Some Multi { Well Problemskaushik Bhattacharya

Mathematical models of phase transitions in solids lead to the variational problem, minimize R W (Du)dx where W has a multi-well structure: W = 0 on a multi-well set K and W > 0 otherwise. We study this problem in two dimensions in the case of equal determinant, i.e., for K = denotes the (3 2)-matrix formed with the rst two columns of U i. We characterize generalized convex hulls, including the quasiconvex hull, of these sets, prove existence of minimizers and identify conditions for the uniqueness of the minimizing Young measure. Finally, we use the characterization of the quasi-convex hull to proposèapproximate relaxed energies', quasiconvex functions which vanish on the quasiconvex hull of K and grow quadratically away from it.