Weak Convergence of Integrands and the Young Measure Representation

Validity of the Young measure representation is useful in the study of microstructure of ordered solids. Such a Young measure, generated by a minimizing sequence of gradients converging weakly in $L^p $, often needs to be evaluated on functions of the pth power polynomial growth. A sufficient condition for this evaluation is given in terms of the variational principle. The principal result, Theorem 2.1, concerns lower semicontinuity of functionals integrated over arbitrary sets. The question arose in the numerical analysis of equilibrium configurations of crystals with rapidly varying microstructure, whose specific application is treated elsewhere. Several applications are given. Of particular note, Young measure solutions of an evolution problem are found.