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

Abstract

We address the questions (P1), (P2) asked in Kirchheim-Müller-Šverák (2003) 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 (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. We show that in the strictly hyperbolic, genuinely nonlinear case considered by DiPerna (1985), no ${\mathbb{T}}_4$ configuration can be embedded into ${\mathcal{K}}_1$.

This version is the revised preprint correcting an error we made in the 2d case of our original proof. This version contains the rearrangement of the order of lemmas and different arguments required to fix this error. We warmly thank Sam Krupa and Laszlo Szekelyhidi for pointing out the error

References in corpus (1)

• Dissipative continuous Euler flows

Cited by in corpus (1)

• Theory of shifts, shocks, and the intimate connections to $L^2$-type a posteriori error analysis of numerical schemes for hyperbolic problems