On the local structure of rank-one convex hulls

Székelyhidi, László, Jr.

doi:10.1090/S0002-9939-05-08299-7

On the local structure of rank-one convex hulls
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Proc. Amer. Math. Soc. 134 (2006), 1963-1972 Request permission

Abstract:

In this note we prove that if $K$ is a compact set of $m\times n$ matrices containing an isolated point $X$ with no rank-one connection into the convex hull of $K\setminus \{X\}$, then the rank-one convex hull separates as \[ K^{rc}=\bigl (K\setminus \{X\}\bigr )^{rc}\cup \{X\}. \] This is an extension of a result of P. Pedregal, which holds for $2\times 2$ matrices.

References

Kari Astala, Daniel Faraco, and László Székelyhidi, Jr., Convex integration and the $L^p$ theory of elliptic equations, Preprint, MPI-MIS, 2004.
Sergio Conti, Francesco Maggi, and Daniel Faraco, A new approach to counterexamples to $L^1$ estimates: Korn’s inequality, geometric rigidity and regularity for gradients of separately convex functions, Preprint, MPI-MIS, 2003.
Bernd Kirchheim, Rigidity and Geometry of microstructures, Habilitation thesis, University of Leipzig, 2003.
Bernd Kirchheim, Stefan Müller, and Vladimír Šverák, Studying nonlinear pde by geometry in matrix space, Geometric analysis and nonlinear partial differential equations, Springer, Berlin, 2003, pp. 347–395. MR 2008346
J. Matoušek, On directional convexity, Discrete Comput. Geom. 25 (2001), no. 3, 389–403. MR 1815439, DOI 10.1007/s004540010069
J. Matoušek and P. Plecháč, On functional separately convex hulls, Discrete Comput. Geom. 19 (1998), no. 1, 105–130. MR 1486640, DOI 10.1007/PL00009331
S. Müller and V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity, Ann. of Math. (2) 157 (2003), no. 3, 715–742. MR 1983780, DOI 10.4007/annals.2003.157.715
Pablo Pedregal, Laminates and microstructure, European J. Appl. Math. 4 (1993), no. 2, 121–149. MR 1228114, DOI 10.1017/S0956792500001030
László Székelyhidi Jr., The regularity of critical points of polyconvex functionals, Arch. Ration. Mech. Anal. 172 (2004), no. 1, 133–152. MR 2048569, DOI 10.1007/s00205-003-0300-7
László Székelyhidi Jr., Rank-one convex hulls in $\Bbb R^{2\times 2}$, Calc. Var. Partial Differential Equations 22 (2005), no. 3, 253–281. MR 2118899, DOI 10.1007/s00526-004-0272-y