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

  • Linus Behn

    Universität Bielefeld, Germany
  • Franz Gmeineder

    Universität Konstanz, Germany
  • Stefan Schiffer

    Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Germany
On symmetric div-quasiconvex hulls and divsym-free L∞-truncations cover
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Abstract

We establish that for any non-empty, compact set KRsym3×3K\subset\mathbb{R}_{\operatorname{sym}}^{3\times 3} the 11- and \infty-symmetric div-quasiconvex hulls K(1)\smash{K^{(1)}} and K()\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 L\operatorname{L}^{\infty}-truncation that preserves both symmetry and solenoidality of matrix-valued maps in L1\operatorname{L}^{1}. For comparison, we moreover give a construction of A\mathscr{A}-free truncations in the regime 1<p<1<p<\infty which, however, does not apply to the case p=1p=1.

Cite this article

Linus Behn, Franz Gmeineder, Stefan Schiffer, On symmetric div-quasiconvex hulls and divsym-free LL^\infty-truncations. Ann. Inst. H. Poincaré Anal. Non Linéaire (2022),

DOI 10.4171/AIHPC/66