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 \(\mathcal {K}_1\).

我们解决了Kirchheim等人提出的（P1），（P2）问题。（通过矩阵空间中的几何学研究非线性PDE。几何分析和非线性偏微分方程，Springer，柏林，1986年），涉及子流形\（\ mathcal {K} _1 \ subset M ^ { 3 \次2} \）这与DiPerna（Trans Am Math Soc 292（2）：383-420，1985）研究的拉格朗日弹性方程组的二乘二系统的弱解有关，其中熵增加了。对于只有有限多个熵的高阶系统，该系统充当模型问题。等级1凸包在通过凸积分求解的研究中很有趣：等级1凸包必须足够小才能使凸积分成为可能。通常通过嵌入\（\ mathbb {T} _4 \）来显示这种简单性（T广场）入集；参见例如Müller等。（通过凸积分获得两井问题的结果。几何分析和变异演算，国际出版社，剑桥，1996年）以及穆勒和史维拉克（Ann Math（2）157（3）：715–742，2003年） 。我们显示在DiPerna（1985）考虑的严格双曲，真正非线性的情况下，无法将\（\ mathbb {T} _4 \）配置嵌入\（\ mathcal {K} _1 \）中。