Thin-Film Limits of Functionals on A-free Vector Fields

This paper deals with variational principles on thin films subject to linear PDE constraints represented by a constant-rank operator A. We study the effective behavior of integral functionals as the 'thickness of the domain tends to zero, investigating both upper and lower bounds for the Gamma-limit. Under certain conditions, we show that the limit is an integral functional, and we give an explicit formula. The limit functional turns out to be constrained to A(0)-free vector fields, where the limit operator A is in general not of constant rank. This result extends work by Bouchitte, Fonseca, and Mascarenhas [J Convex Anal. 16 (2009), pp. 351-365] to the setting of A-free vector fields. While the lower bound follows from a Young measure approach together with a new decomposition lemma, the construction of a recovery sequence relies on algebraic considerations in Fourier space. This part of the argument requires a careful analysis of the limiting behavior of the resealed operators A(epsilon) by a suitable convergence of their symbols, as well as an explicit construction for plane waves inspired by the bending moment formulas in the theory of (linear) elasticity. We also give a few applications to common operators A.