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\(T_5\)-Configurations and non-rigid sets of matrices

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Abstract

In 2003 B. Kirchheim-D. Preiss constructed a Lipschitz map in the plane with 5 incompatible gradients, where incompatibility refers to the condition that no two of the five matrices are rank-one connected. The construction is via the method of convex integration and relies on a detailed understanding of the rank-one geometry resulting from a specific set of five matrices. The full computation of the rank-one convex hull for this specific set was later carried out in 2010 by W. Pompe in Calc. Var. PDE 37(3–4):461–473, (2010) by delicate geometric arguments. For more general sets of matrices a full computation of the rank-one convex hull is clearly out of reach. Therefore, in this short note we revisit the construction and propose a new, in some sense generic method for deciding whether convex integration for a given set of matrices can be carried out, which does not require the full computation of the rank-one convex hull.

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Notes

  1. In some sense this case can be seen as a limiting case from \(\Sigma _t=\{X:\mathrm{det}X=t\}\) with \(t\rightarrow \infty \), see the proof of Lemma 2.5 below.

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Correspondence to László Székelyhidi Jr..

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Communicated by J. Ball.

L. Székelyhidi gratefully acknowledges the support of the ERC Grant Agreement No. 724298.

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Förster, C., Székelyhidi, L. \(T_5\)-Configurations and non-rigid sets of matrices. Calc. Var. 57, 19 (2018). https://doi.org/10.1007/s00526-017-1293-7

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