Variational integrals modeling solid-to-solid phase transformations often fail to be weakly lower semicontinuous because the energy densities f are not quasiconvex in the sense of Morrey. In this talk we analyse properties of minimizing Young measures generated by minimizing sequences for these variational integrals. We prove that the moments of order q > p exist if the integrand is sufficiently close to the p-Dirichlet energy at infinity. A counterexample related to the one-well problem in two dimensions shows that one cannot expect in general L ∞ estimates, i.e., that the support of the minimizing Young measure is uniformly bounded.