Potentials for $\mathcal{A}$-quasiconvexity

TL;DR

This paper demonstrates that constant rank operators have exact potentials in frequency space, enabling simplified testing of $\mathcal{A}$-quasiconvexity and characterizing $\mathcal{A}$-free Young measures through potential-generated sequences.

Contribution

It establishes the existence of exact potentials for constant rank operators and applies this to test $\mathcal{A}$-quasiconvexity and describe $\mathcal{A}$-free Young measures.

Findings

Existence of exact potentials $\mathbb{B}$ for constant rank operators in frequency space.

$\mathcal{A}$-quasiconvexity can be tested using compactly supported fields.

$\mathcal{A}$-free Young measures are generated by sequences of the form $\mathbb{B}u_j$, up to shifts.

Abstract

We show that each constant rank operator $\mathcal{A}$ admits an exact potential $\mathbb{B}$ in frequency space. We use this fact to show that the notion of $\mathcal{A}$-quasiconvexity can be tested against compactly supported fields. We also show that $\mathcal{A}$-free Young measures are generated by sequences $\mathbb{B}u_j$, modulo shifts by the barycentre.