Thin-film limits of functionals on A-free vector fields

TL;DR

This paper investigates the asymptotic behavior of variational functionals on thin films constrained by linear PDEs, establishing the limit functional and its properties as the film's thickness approaches zero.

Contribution

It extends the analysis of $ ext{A}$-free vector fields to thin film limits, providing explicit formulas and new techniques for the $ ext{Γ}$-convergence analysis.

Findings

The limit functional is an integral constrained to $ ext{A}_0$-free fields.

A new decomposition lemma aids in establishing lower bounds.

Explicit recovery sequences are constructed using Fourier analysis.

Abstract

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 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 Bouchitte, Fonseca and Mascarenhas [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 $\mathcal{A}$.