On symmetric div-quasiconvex hulls and divsym-free L∞-truncations

Abstract

We establish that for any non-empty, compact set $K\subset\mathbb{R}_{\operatorname{sym}}^{3\times 3}$ the $1$- and $\infty$-symmetric div-quasiconvex hulls $\smash{K^{(1)}}$ and $\smash{K^{(\infty)}}$ coincide. This settles a conjecture in a recent work of Conti,Müller & Ortiz [Arch. Ration. Mech. Anal. 235 (2020)] in the affirmative. As a key novelty, we construct an $\operatorname{L}^{\infty}$-truncation that preserves both symmetry and solenoidality of matrix-valued maps in $\operatorname{L}^{1}$. For comparison, we moreover give a construction of $\mathscr{A}$-free truncations in the regime $1<p<\infty$ which, however, does not apply to the case $p=1$.