Relaxation of some multi-well problems

Mathematical models of phase transitions in solids lead to the variational problem, minimize ∫ Ω W ( Du ) d x , 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 ∪ … ∪SO(2) U k or K = O(2) U 1 ∪ … ∪ O(2) U k for U 1 , … , U k ∈ M 2×2 with det U i = δ in three dimensions when the matrices U i are essentially two-dimensional and also for K = SO(3) Û 1 ∪ … ∪ SO(3) Û k for U 1 , … , U k ∈ M 3×3 with , which arises in the study of thin films. Here, Û i denotes the (3×2) matrix formed with the first 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 quasiconvex hull to propose ‘approximate relaxed energies’, quasiconvex functions which vanish on the quasiconvex hull of K and grow quadratically away from it.