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 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 Muller et al. (Attainment results for the two-well problem by convex integration. Geometric analysis and the calculus of variations, Int. Press, Cambridge, 1996) and Muller and Sverak (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 $$\mathcal {K}_1$$ .