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

Behn, Linus; Gmeineder, Franz; Schiffer, Stefan

doi:10.4171/aihpc/66

Geodesic

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On symmetric div-quasiconvex hulls and divsym-free

L^{\infty}

-truncations

Behn, Linus ; Gmeineder, Franz ; Schiffer, Stefan

Annales de l'I.H.P. Analyse non linéaire, Tome 40 (2023) no. 6, pp. 1267-1317

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Résumé

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

Publié le : 2022-12-08

DOI : 10.4171/aihpc/66

Classification : 26B25, 49J45
Keywords: divsym-quasiconvexity, A-quasiconvexity, convex hulls, differential constraints, divergence-free truncation, A-free truncation

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     title = {On symmetric div-quasiconvex hulls and divsym-free $L^\infty$-truncations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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     language = {en},
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Behn, Linus; Gmeineder, Franz; Schiffer, Stefan. On symmetric div-quasiconvex hulls and divsym-free $L^\infty$-truncations. Annales de l'I.H.P. Analyse non linéaire, Tome 40 (2023) no. 6, pp. 1267-1317. doi: 10.4171/aihpc/66

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