Thin-film limits of functionals on -free vector fields

Abstract: This paper deals with variational principles on thin films with linear PDE constraints represented by a constant-rank operator , and studies the effective behavior, in the sense of -convergence, of integral functionals as the thickness of the domain tends to zero. The limit integral functional turns out to be determined by the -quasiconvex envelope of the original energy density and is constrained to vector fields that satisfy limit PDEs, which in general differ from the ones we started with. While the lower bound follows from a standard Young measure and projection approach together with a new (local) decomposition lemma, the construction of a recovery sequence relies on algebraic considerations in Fourier space. It requires a careful analysis of the limiting behavior of the rescaled operators by a suitable convergence of their symbols, as well as an explicit construction for plane waves inspired by the bending moment formulas common in the theory of elasticity. As an application, the energy of a nonlinear elastic membrane model can be shown to be local, answering a question raised by Bouchitté, Fonseca and Mascarenhas in [J. Convex Anal. 16 (2009), pp. 351-365].