Applicability of slicing or blow-up methods to the weakly converging eigendeformation term

Determine whether the slicing method or a blow-up argument can be applied to estimate the weakly converging term depending on the eigendeformation measure gamma when the measure space is equipped only with the flat topology.

Background

The paper establishes the lower bound for the volume contribution of the Gamma-limit by a direct argument. The relevant difficulty is the weak convergence of the eigendeformation measures gamma and the fact that the flat topology is sufficiently weak to obstruct the straightforward use of standard slicing or blow-up techniques.

The authors explicitly state that it is unclear how either of these methods could handle the term involving gamma. Consequently, the paper develops a separate direct estimate rather than resolving the broader methodological question of whether slicing or blow-up arguments can be adapted to this setting.

References

The estimate of the volume term, however, needs to be proved directly, since it is not clear how to apply the slicing method or a blow-up argument to the weakly converging term depending on $\gamma$, as $M$ is merely endowed with the flat topology.

— Eigendamage: an Eigendeformation model for the variational approximation of cohesive fracture in antiplane shear deformations  (2609.29734 - Auer-Volkmann et al., 24 Sep 2026) in Section 5.2, “Estimate from below of the volume term”