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On the Rank-1 convex hull of a set arising from a hyperbolic system of Lagrangian elasticity

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Abstract

We address the questions (P1), (P2) asked in Kirchheim et al. (Studying nonlinear PDE by geometry in matrix space. Geometric analysis and nonlinear partial differential equations, Springer, Berlin, 1986) concerning the structure of the Rank-1 convex hull of a submanifold \(\mathcal {K}_1\subset M^{3\times 2}\) that is related to weak solutions of the two by two system of Lagrangian equations of elasticity studied by DiPerna (Trans Am Math Soc 292(2):383–420, 1985) with one entropy augmented. This system serves as a model problem for higher order systems for which there are only finitely many entropies. The Rank-1 convex hull is of interest in the study of solutions via convex integration: the Rank-1 convex hull needs to be sufficiently non-trivial for convex integration to be possible. Such non-triviality is typically shown by embedding a \(\mathbb {T}_4\) (Tartar square) into the set; see for example Müller et al. (Attainment results for the two-well problem by convex integration. Geometric analysis and the calculus of variations, Int. Press, Cambridge, 1996) and Müller and Šverák (Ann Math (2) 157(3):715–742, 2003). We show that in the strictly hyperbolic, genuinely nonlinear case considered by DiPerna (1985), no \(\mathbb {T}_4\) configuration can be embedded into \(\mathcal {K}_1\).

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Notes

  1. Contrast this with the well known result of Evans [14] that minimizers do have partial regularity.

  2. Indeed as noted in [31], \(\mathbb {T}_4\) configurations played an important role in [35] and seem to have been discovered independently by a number of authors.

  3. This goal and this approach has been introduced to us by Šverák [39].

  4. Note that a differential inclusion into set \(\mathcal {K}_1\) gives a solution to (3) with the inequality replaced by an equality.

  5. For the general case in \(M^{m\times n}\) the construction is the same, simply slightly harder to visualize.

  6. Here we are stating a more restrictive version of their theorem to avoid some technicalities.

  7. It is likely that the sharp results of [17] could also be used to generate explicit examples in \(M^{2\times 2}\).

  8. The two by two system (2) has infinitely many entropies, and it is known from [13] that the method of compensated compactness works even for the system adjoined by two appropriate entropies. It seems to the authors of this paper that for two by two systems augmented by infinitely many entropies there is little hope to counterexamples of uniqueness and regularity by differential inclusions and convex integration.

  9. On a somewhat related well known result, it is known that the differential inclusion into any finite set of four matrices without Rank-1 connections has no convex integration solutions [6], however there exists a set of five matrices without Rank-1 connections that admits convex integration solutions of the corresponding differential inclusion; see [23, Chapter 4, Section 3].

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Acknowledgements

The first author would like to thank V. Šverák for many very helpful discussions during a visit to Minnesota in summer of 2018. The idea to study entropy solutions of systems of conservation laws via differential inclusions and convex integration is from him. Also a number of key ideas used in this paper (in particular Lemmas 10 and 16) are from Šverák [40]. The first author also gratefully acknowledges the support of the Simons foundation, collaboration Grant #426900. Both authors warmly thank the anonymous referee for very careful reading of the paper and for pointing out a number of improvements.

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  1. Mathematics Department, University of Cincinnati, 2600 Clifton Ave., Cincinnati, OH, 45221, USA

    Andrew Lorent

  2. Department of Mathematics, University of Arizona, 617 N. Santa Rita Ave., Tucson, AZ, 85721, USA

    Guanying Peng

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Lorent, A., Peng, G. On the Rank-1 convex hull of a set arising from a hyperbolic system of Lagrangian elasticity. Calc. Var. 59, 156 (2020). https://doi.org/10.1007/s00526-020-01805-6

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