Nondegenerate gradient Young measures and dimension reduction for nonlinear membranes
Abstract
We consider the rigorous derivation of membrane theories from three-dimensional nonlinear elasticity. A salient feature of our framework is the inclusion of an orientation-preservation constraint together with energy growth that penalizes vanishing local volume. We embed the 3D variational problem into a space of parametrized measures, and obtain a membrane Γ-limit defined on a class of nondegenerate gradient Young measures. This formulation offers a twofold advantage: first, it uniquely identifies a non-relaxed membrane energy density capable of capturing the fine oscillations (microstructure) of minimizing sequences; second, the resulting membrane density preserves the unbounded energy growth near degenerate configurations. We further show that the classical relaxed membrane energy of Le Dret and Raoult is recovered as the minimum of our functional over all measures with a prescribed barycenter. The main difficulty lies in generating such measures by maps whose gradients have rank two almost everywhere, which we address through piecewise isometric (origami-like) constructions.
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BibTeX
@misc{nair2026nondegenerate,
title = {{Nondegenerate gradient Young measures and dimension reduction for nonlinear membranes}},
author = {Gokul G. Nair and David Padilla-Garza and Marco Picchi Scardaoni},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30411},
eprint = {2609.30411},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30411},
}
RIS
TY - UNPB
TI - Nondegenerate gradient Young measures and dimension reduction for nonlinear membranes
AU - Gokul G. Nair
AU - David Padilla-Garza
AU - Marco Picchi Scardaoni
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30411
UR - https://arxiv.org/abs/2609.30411
AB - We consider the rigorous derivation of membrane theories from three-dimensional nonlinear elasticity. A salient feature of our framework is the inclusion of an orientation-preservation constraint together with energy growth that penalizes vanishing local volume. We embed the 3D variational problem into a space of parametrized measures, and obtain a membrane Γ-limit defined on a class of nondegenerate gradient Young measures. This formulation offers a twofold advantage: first, it uniquely identifies a non-relaxed membrane energy density capable of capturing the fine oscillations (microstructure) of minimizing sequences; second, the resulting membrane density preserves the unbounded energy growth near degenerate configurations. We further show that the classical relaxed membrane energy of Le Dret and Raoult is recovered as the minimum of our functional over all measures with a prescribed barycenter. The main difficulty lies in generating such measures by maps whose gradients have rank two almost everywhere, which we address through piecewise isometric (origami-like) constructions.
ER -