Sharp alpha-dependence of the disclination energy lower bound

Prove a lower bound of order \(|\alpha|h^2\log(1/h)\) for an appropriate two-dimensional elastic-sheet model with a disclination, thereby establishing the sharp dependence on the disclination parameter \(\alpha\) and transferring the result to the corresponding fully nonlinear three-dimensional model.

Background

For thin elastic sheets with disclinations, the paper proves a three-dimensional lower bound of order α2h2log⁡(1/h)\alpha^2h^2\log(1/h) and an upper bound of order ∣α∣h2log⁡(1/h)|\alpha|h^2\log(1/h). Thus, the scaling in the thickness hh is sharp, but the dependence on the defect parameter α\alpha is not.

The authors explain that a lower bound of order ∣α∣h2log⁡(1/h)|\alpha|h^2\log(1/h) for a suitable two-dimensional model would, through their dimension-reduction comparison theorem, imply the same lower bound for the full three-dimensional model. Establishing this bound requires extending existing pure-bending arguments to models that also permit stretching.

References

We believe the upper bound to be tight. Indeed, a lower bound of $ |\alpha| h2 \log{1/h}$ was obtained for a reduced 2D model with a geometrically-linearized bending , in a von-Karm\n an model and in a pure bending model . To prove its tightness, it would be sufficient to obtain an $ |\alpha| h2 \log{1/h}$ lower bound for a 2D model, since, as shown in #1{thm:3D2D}, such a bound would also apply to the 3D setting.

— Energy scaling laws for thin elastic sheets with topological defects  (2608.23134 - Kupferman et al., 24 Aug 2026) in Section 1, paragraph “Disclinations” in the discussion following Theorem 1