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

TL;DR

This paper proves that for compact symmetric matrix sets, the 1- and infinity-symmetric div-quasiconvex hulls are identical, confirming a recent conjecture, and introduces an $ ext{L}^ ext{infty}$-truncation preserving symmetry and solenoidality.

Contribution

It establishes the equality of symmetric div-quasiconvex hulls for compact sets and introduces a novel $ ext{L}^ ext{infty}$-truncation method that maintains symmetry and solenoidality.

Findings

Proves $K^{(1)}=K^{( ext{infty})}$ for compact symmetric sets.

Constructs an $ ext{L}^ ext{infty}$-truncation preserving symmetry and solenoidality.

Confirms a conjecture from recent work on symmetric div-quasiconvexity.

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, M\"{u}ller 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 $\mathrm{L}^{1}$.