Characterization of Generalized Young Measures Generated by $${\mathcal {A}}$$-free Measures

Abstract

We 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.

Shape optimization of light structures and the vanishing mass conjecture

“…• Understanding diffuse concentrations: In [10] a possible approach to the vanishing mass conjecture in terms of (generalized) Young measure generated by the sequence {σ ε µ ε } ε>0 is outlined. However, despite recent advances in the theory of Young measures for A-free sequences [7,32], diffuse concentrations (singular measures converging weakly* to an absolutely continuous one) remain only superficially understood. It is, however, precisely these diffuse concentrations that lie at the core of the general conjecture (as is already observed in [10]).…”

“…Second, the asymptotic concentrations allowed under the PDE constraint − div(σµ) = f are not currently known explicitly and only partial results exist, see, e.g., [7,9,21,32,33] and the references contained therein. It follows from the recent results in [8,20] that σµ must be absolutely continuous with respect to the 1-dimensional Hausdorff measure and det σ(x) = 0 for µ s -almost every x, where µ s is the singular part of µ.…”

$$\displaystyle L^{\infty }$$-truncation of closed differential forms

“…For further reference to classification of Ap-Young measures for p < ∞, let us shortly refer to [2,11,12,19,20].…”

“…If r = N − 1, after identifying N −1 with R N and N with R, the differential operator d becomes the divergence of a vector field which is defined for 2 We have the following product rules for d:…”

An elementary approach to the homological properties of constant-rank operators

“…This homology-type result has proved to be a very useful tool to solve some longstanding questions in the calculus of variations related to the study of oscillations and concentration effects associated to sequences of PDE-constrained maps (see [Arr21,KR19]).…”