On the Rank-1 convex hull of a set arising from a hyperbolic system of Lagrangian elasticity
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Calculus of Variations
On the Rank-1 convex hull of a set arising from a hyperbolic system of Lagrangian elasticity Andrew Lorent1 · Guanying Peng2 Received: 29 September 2019 / Accepted: 30 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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 K1 ⊂ M 3×2 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 T4 (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 T4 configuration can be embedded into K1 . Mathematics Subject Classification 35L65 · 35L40
1 Introduction There has recently been a lot of progress on a number of outstanding problems in PDE by reformulating the PDE as a differential inclusion. In [29] counterexamples to partial regularity of weak solutions to elliptic systems that arise as the critical point of a strongly quasiconvex 1 Contrast this with the well known result of Evans [14] that minimizers do have partial regularity.
Communicated by J.Ball.
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Andrew Lorent [email protected] Guanying Peng [email protected]
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Mathematics Department, University of Cincinnati, 2600 Clifton Ave., Cincinnati, OH 45221, USA
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Department of Mathematics, University of Arizona, 617 N. Santa Rita Ave., Tucson, AZ 85721, USA 0123456789().: V,-vol
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functional were provided.1 This was later extended to polyconvex functionals in [41] and parabolic systems in [28]. Prior to this Scheffer [35] provided counterexamples to related regularity problems. In [8], De Lellis and Székelyhidi reproved (and considerably strengthened) the well known result of Scheffer [36] on weak solutions to the Euler equation with compact support in space and time, with a much shorter and simpler proof via reformulation as a differential inclusion. Previously Shnirelman [37] provided a somewhat simpler proof by a different method. The advance provided by [8] opened an approach to Onsager’s conjecture which was subsequently studied intensively by a number of authors [3,9,10,20,21] with a final solution being prov
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