标题： 对称散度-拟凸包和散度对称自由$\mathrm{L}^\infty$-截断 显示英文标题

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

摘要： 我们证明对于任何非空紧集$K\subset\mathbb{R}_{\mathrm{sym}}^{3\times 3}$，$1$-和$\infty$-对称散度拟凸包$K^{(1)}$和$K^{(\infty)}$是相同的。 这在Conti、Müller和Ortiz（对称散度拟凸性和静态问题的松弛，Arch. Ration. Mech. Anal. 235(2):841-880）的最新工作中肯定地解决了这一猜想。 作为关键创新，我们构造了一个$\mathrm{L}^{\infty}$-截断，它在$\mathrm{L}^{1}$中保持矩阵值映射的对称性和无旋性。 显示英文摘要

摘要： 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 $\mathrm{L}^{1}$.