Thin-film limits of functionals on A-free vector fields Carolin KreisbeckFilip Rindler 49J4535E9974K35dimension reductionthin filmsPDE constraintsA-quasiconvexityGamma-convergence This paper deals with variational principles on thin films subject to linear PDE constraints represented by a constant-rank operator $\mathcal{A}$. 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 $\mathcal{A}_0$-free vector fields, where the limit operator $\mathcal{A}_0$ is in general not of constant rank. This result extends work by Bouchitt\'e, Fonseca, and Mascarenhas [\emph{J. Convex Anal.} 16 (2009), pp.~351--365] to the setting of $\mathcal{A}$-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 rescaled operators $\mathcal{A}_{\epsilon}$ 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 $\mathcal{A}$. Indiana University Mathematics Journal 2015 text pdf 10.1512/iumj.2015.64.5653 10.1512/iumj.2015.64.5653 en Indiana Univ. Math. J. 64 (2015) 1383 - 1423 state-of-the-art mathematics http://iumj.org/access/