AbstractWe give two characterizations, one for the class of generalized Young measures generated by $${{\,\mathrm{{\mathcal {A}}}\,}}$$
A
-free measures and one for the class generated by $${\mathcal {B}}$$
B
-gradient measures $${\mathcal {B}}u$$
B
u
. Here, $${{\,\mathrm{{\mathcal {A}}}\,}}$$
A
and $${\mathcal {B}}$$
B
are linear homogeneous operators of arbitrary order, which we assume satisfy the constant rank property. The first characterization places the class of generalized $${\mathcal {A}}$$
A
-free Young measures in duality with the class of $${{\,\mathrm{{\mathcal {A}}}\,}}$$
A
-quasiconvex integrands by means of a well-known Hahn–Banach separation property. The second characterization establishes a similar statement for generalized $${\mathcal {B}}$$
B
-gradient Young measures. Concerning applications, we discuss several examples that showcase the failure of $$\mathrm {L}^1$$
L
1
-compensated compactness when concentration of mass is allowed. These include the failure of $$\mathrm {L}^1$$
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 $$\Omega $$
Ω
, the inclusions $$\begin{aligned} \mathrm {L}^1(\Omega ) \cap \ker {\mathcal {A}}&\hookrightarrow {\mathcal {M}}(\Omega ) \cap \ker {{\,\mathrm{{\mathcal {A}}}\,}}\,,\\ \{{\mathcal {B}}u\in \mathrm {C}^\infty (\Omega )\}&\hookrightarrow \{{\mathcal {B}}u\in {\mathcal {M}}(\Omega )\} \end{aligned}$$
L
1
(
Ω
)
∩
ker
A
↪
M
(
Ω
)
∩
ker
A
,
{
B
u
∈
C
∞
(
Ω
)
}
↪
{
B
u
∈
M
(
Ω
)
}
are dense with respect to the area-functional convergence of measures.
Characterization of Generalized Young Measures Generated by $${\mathcal {A}}$$-free Measures