Geometric rigidity principles for 𝒜-free Young measures

Abstract

We study generalized Young measures generated by 𝒜-free measures, 𝒜 a constant-rank operator, in a compactification of the value space adapted to an energy. Given an 𝒜-quasiconvex integrand G of linear growth with coercive recession function G^∞, the unit sphere at infinity is replaced by the indicatrix 𝕊_G={G^∞=1}, which need be neither convex nor symmetric. The two compactifications are equivalent, but in the adapted one the equality cases of the Jensen inequalities characterizing 𝒜-free Young measures become statements of convex geometry about 𝕊_G. We prove the characterization in this setting, with a new streamlined proof of the inhomogenization step based on a dyadic martingale argument. We then show that if 𝒜-free measures μ_joverset*rightharpoonupμ and their G-energies converge, then at |μ^s|-almost every point the concentration-angle measure is supported on the smallest exposed face of co̅(𝕊_G) containing the G-polar of μ^s; in particular, it never charges the set where G^∞ differs from its convex envelope, which, by the automatic convexity of Kirchheim and Kristensen, does not meet the wave cone. The concentration on the singular part of μ is elementary for every such sequence if and only if each point of the wave cone on 𝕊_G is an extreme point of co̅(𝕊_G), as for the Euclidean norm underlying Reshetnyak's continuity theorem; on flat faces of 𝕊_G, rigidity degrades from a point to a face. Finally, we prove a transfer principle for the convergence of energies of BV maps under Dirichlet boundary conditions.

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BibTeX RIS

BibTeX

@misc{scalabrino2026geometric,
  title = {{Geometric rigidity principles for $\mathscrA$-free Young measures}},
  author = {Sergio Scalabrino},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.22560},
  eprint = {2610.22560},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.22560},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Geometric rigidity principles for 𝒜-free Young measures
AU  - Sergio Scalabrino
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.22560
UR  - https://arxiv.org/abs/2610.22560
AB  - We study generalized Young measures generated by 𝒜-free measures, 𝒜 a constant-rank operator, in a compactification of the value space adapted to an energy. Given an 𝒜-quasiconvex integrand G of linear growth with coercive recession function G^∞, the unit sphere at infinity is replaced by the indicatrix 𝕊_G={G^∞=1}, which need be neither convex nor symmetric. The two compactifications are equivalent, but in the adapted one the equality cases of the Jensen inequalities characterizing 𝒜-free Young measures become statements of convex geometry about 𝕊_G. We prove the characterization in this setting, with a new streamlined proof of the inhomogenization step based on a dyadic martingale argument. We then show that if 𝒜-free measures μ_joverset*rightharpoonupμ and their G-energies converge, then at |μ^s|-almost every point the concentration-angle measure is supported on the smallest exposed face of co̅(𝕊_G) containing the G-polar of μ^s; in particular, it never charges the set where G^∞ differs from its convex envelope, which, by the automatic convexity of Kirchheim and Kristensen, does not meet the wave cone. The concentration on the singular part of μ is elementary for every such sequence if and only if each point of the wave cone on 𝕊_G is an extreme point of co̅(𝕊_G), as for the Euclidean norm underlying Reshetnyak's continuity theorem; on flat faces of 𝕊_G, rigidity degrades from a point to a face. Finally, we prove a transfer principle for the convergence of energies of BV maps under Dirichlet boundary conditions.
ER  -

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