We give two characterizations, one for the class of generalized Young measures generated by \({{\,\mathrm{{\mathcal {A}}}\,}}\)-free measures and one for the class generated by \({\mathcal {B}}\)-gradient measures \({\mathcal {B}}u\). Here, \({{\,\mathrm{{\mathcal {A}}}\,}}\) and \({\mathcal {B}}\) are linear homogeneous operators of arbitrary order, which we assume satisfy the constant rank property. The first characterization places the class of generalized \({\mathcal {A}}\)-free Young measures in duality with the class of \({{\,\mathrm{{\mathcal {A}}}\,}}\)-quasiconvex integrands by means of a well-known Hahn–Banach separation property. The second characterization establishes a similar statement for generalized \({\mathcal {B}}\)-gradient Young measures. Concerning applications, we discuss several examples that showcase the failure of \(\mathrm {L}^1\)-compensated compactness when concentration of mass is allowed. These include the failure of \(\mathrm {L}^1\)-estimates for elliptic systems and the lack of rigidity for a version of the two-state problem. As a byproduct of our techniques we also show that, for any bounded open set \(\Omega \), the inclusions

are dense with respect to the area-functional convergence of measures.

我们给出了两个特征，一个是由\({{\,\mathrm{{\mathcal {A}}}\,}}\)生成的广义杨氏测度的类别，另一个是由\( {\mathcal {B}}\) -梯度测量\({\mathcal {B}}u\)。这里，\({{\,\mathrm{{\mathcal {A}}}\,}}\)和\({\mathcal {B}}\)是任意阶的线性齐次算子，我们假设它们满足恒定等级属性。第一个特征将广义\({\mathcal {A}}\) -free Young 测度类与\({{\,\mathrm{{\mathcal {A}}}\,}} \)-quasiconvex 被积函数通过众所周知的 Hahn-Banach 分离特性。第二个特征为广义\({\mathcal {B}}\) -gradient Young 度量建立了类似的陈述。关于应用，我们讨论了几个例子，这些例子展示了当允许质量集中时\(\mathrm {L}^1\) -补偿紧凑性的失败。这些包括对椭圆系统的\(\mathrm {L}^1\) -估计的失败以及对二态问题的一个版本缺乏刚性。作为我们技术的副产品，我们还表明，对于任何有界开集\(\Omega \)，包含

在措施的区域功能收敛方面是密集的。