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

TL;DR

This paper investigates the structure of the Rank-1 convex hull of a specific submanifold related to hyperbolic elasticity systems, showing that certain configurations cannot be embedded, which impacts the applicability of convex integration methods.

Contribution

It proves that in a hyperbolic, nonlinear setting, the set does not contain the Tartar square configuration, limiting convex integration approaches for these systems.

Findings

No Tartar square can be embedded into the set in the considered case.

The structure of the Rank-1 convex hull is characterized for hyperbolic elasticity systems.

Implications for the existence of solutions via convex integration are discussed.

Abstract

We address the questions (P1), (P2) asked in Kirchheim-M\"{u}ller-\v{S}ver\'{a}k (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 $\mathcal{K}_1$.