Relaxation of some multi-well problems

Mathematical models of phase transitions in solids lead to the variational problem, minimize f(Omega)W(Du) dx, where W has a multi-well structure, i.e . 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 = SO(2)U-1 boo lean OR...boolean OR SO(2)U-k or K = O(2)U-1 boolean OR...boolean OR O(2)U- k for U-1,...,U-k is an element of M-2x2 with det U-i = delta, in three dim ensions when the matrices U, are essentially two-dimensional and also for K = SO(3)(U) over cap (1) boolean OR...boolean OR SO(3)(A) over cap (k) for U-1,...,U-k is an element of M-3X3 with (adj (UiUi)-U-T)(33) = delta (2), w hich arises in the study of thin films. Here, (U) over cap (i) denotes the (3 x 2) matrix formed with the first two columns of U-i. We characterize ge neralized convex hulls, including the quasiconvex hull, of these sets, prov e existence oi minimizers and identify conditions for the uniqueness of the minimizing Young measure. Finally, we use the characterization of the quas iconvex hull to propose 'approximate relaxed energies', quasiconvex functio ns which vanish on the quasiconvex hull of K and grow quadratically away fr om it.