Characterization of Two-scale Gradient Young Measures and Application to Homogenization

Young (or Parametrized) measures have been introduced in optimal control theory by L. C. Young [39] to study non convex variational problems for which there were no classical solution, and to provide an effective notion of generalized solution for problems in Calculus of Variations. Starting with the works of Tartar [35] on hyperbolic conservation laws, Young measures have been an important tool for studying the asymptotic behavior of solutions of nonlinear partial differential equations (see also DiPerna [17]). A key feature of these measures is their capacity to capture the oscillations of minimizing sequences of non convex variational problems, and many applications arise e.g. in models of elastic crystals (see Chipot & Kinderlehrer [16] and Fonseca [19]), phase transition (see Ball & James [8]), optimal design (see Bonnetier & Conca [11], Maestre & Pedregal [24] and Pedregal [32]). The special properties of Young measures generated by sequences of gradients of Sobolev functions have been studied by Kinderlehrer & Pedregal [21, 22] and are relevant in the applications to nonlinear elasticity. The lack of information on the spatial structure of oscillations presents an obstacle for the application of Young measures to homogenization problems. Two-scale Young measures, which have been introduced by E in [18] to study periodic homogenization of nonlinear transport equations, contain some information on the amount of oscillations and extend Nguetseng’s notion of two-scale convergence (see [29] and Allaire [2]). Other (generalized) multiscale Young measures have been introduced in the works of Alberti & Müller [1] and Ambrosio & Frid [3].