On symmetric div-quasiconvex hulls and divsym-free ${\mathrm{L}}^{\infty }$-truncations

Abstract

We establish that for any non-empty, compact set $K\subset {\mathbb{R}}_{\mathrm{sym}}^{3\times 3}$ the $1$- and $\infty$-symmetric div-quasiconvex hulls $K^{(1)}$ and $K^{(\infty )}$ coincide. This settles a conjecture in a recent work of Conti, Müller and Ortiz (Symmetric Div-Quasiconvexity and the Relaxation of Static Problems. Arch. Ration. Mech. Anal. 235(2):841-880) in the affirmative. As a key novelty, we construct an ${\mathrm{L}}^{\infty }$-truncation that preserves both symmetry and solenoidality of matrix-valued maps in ${\mathrm{L}}^1$.

35 pages, 1 figure