Upper semicontinuity of the lamination hull

Harris, Terence L.J.

doi:10.1051/cocv/2017033

Harris, Terence L.J.¹

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1503-1510.

Résumé

Let $K \subseteq ℝ^{2 \times 2}$ be a compact set, let $K^{r c}$ be its rank-one convex hull, and let $L (K)$ be its lamination convex hull. It is shown that the mapping $K \mapsto \bar{L (K)}$ is not upper semicontinuous on the diagonal matrices in $ℝ^{2 \times 2}$ , which was a problem left by Kolář. This is followed by an example of a $5$ -point set of $2 \times 2$ symmetric matrices with non-compact lamination hull. Finally, another $5$ -point set $K$ is constructed, which has $L (K)$ connected, compact and strictly smaller than $K^{r c}$ .

MR Zbl

DOI : 10.1051/cocv/2017033

Classification : 49J45, 52A30
Mots clés : Lamination convexity, rank-one convexity

Affiliations des auteurs :

Harris, Terence L.J. ¹

@article{COCV_2018__24_4_1503_0,
     author = {Harris, Terence L.J.},
     title = {Upper semicontinuity of the lamination hull},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1503--1510},
     publisher = {EDP-Sciences},
     volume = {24},
     number = {4},
     year = {2018},
     doi = {10.1051/cocv/2017033},
     mrnumber = {3922434},
     zbl = {1411.26015},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2017033/}
}

TY  - JOUR
AU  - Harris, Terence L.J.
TI  - Upper semicontinuity of the lamination hull
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2018
SP  - 1503
EP  - 1510
VL  - 24
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2017033/
DO  - 10.1051/cocv/2017033
LA  - en
ID  - COCV_2018__24_4_1503_0
ER  -

%0 Journal Article
%A Harris, Terence L.J.
%T Upper semicontinuity of the lamination hull
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2018
%P 1503-1510
%V 24
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2017033/
%R 10.1051/cocv/2017033
%G en
%F COCV_2018__24_4_1503_0

Harris, Terence L.J. Upper semicontinuity of the lamination hull. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1503-1510. doi : 10.1051/cocv/2017033. http://www.numdam.org/articles/10.1051/cocv/2017033/

Bibliographie
Cité par

[1] R.J. Aumann and S. Hart, Bi-convexity and bi-martingales. Israel J. Math. 54 (1986) 159–180 | DOI | MR | Zbl

[2] S.Conti, D. Faraco, F. Maggi and S. Müller, Rank-one convex functions on 2 × 2 symmetric matrices and laminates on rank-three lines. Calc. Var. Partial Differ. Eq. 24 (2005) 479–493 | DOI | MR | Zbl

[3] D. Faraco and L. Székelyhidi Jr. Tartar’s conjecture and localization of the quasiconvex hull in ℝ^2×2. Acta Math. 200 (2008) 279–305 | DOI | MR | Zbl

[4] B. Kirchheim, Rigidity and geometry of microstructures. Habilitation thesis, University of Leipzig (2003)

[5] J. Kolář Non-compact lamination convex hulls. Ann. Inst. Henri Poincaré Anal. Non Linéaire 20 (2003) 391–403 | DOI | Numdam | MR | Zbl

[6] C.F. Kreiner and J. Zimmer, Topology and geometry of nontrivial rank-one convex hulls for two-by-two matrices. ESAIM COCV 12 (2006) 253–270 | DOI | Numdam | MR | Zbl

[7] L. Székelyhidi Jr. Rank-one convex hulls in ℝ^2×2. Calc. Var. Partial Differ. Eq. 22 (2005) 253–281 | DOI | MR | Zbl

[8] V. Šverák On Tartar’s conjecture. Ann. Inst. Henri Poincaré Anal. Non Linéaire 10 (1993) 405–412 | DOI | Numdam | MR | Zbl

[9] K. Zhang, On the stability of quasiconvex hulls. Preprint, Max-Plank Inst. for Mathematics in the Sciences, Leipzig (1998), Vol. 33

Cité par Sources :