Energies with values in networks and geometric quasiconvexity
Abstract
We study the asymptotic behaviour, as ε→ 0+, of the p-Dirichlet energy of maps u:Ω⊂ℝ2 → ℝ2 constrained to take values in the rescaled periodic network ε𝒵, where 𝒵 is a standard square network in ℝ2. The oscillations are carried by the constraint in the target, not by the energy density, and their effect depends on p. For 1<p<2 the constraint disappears in the limit: the Γ-limit is an integral functional on the whole of W1,p(Ω, ℝ2 ), whose density is given by an asymptotic cell formula on tilted cells. This density is equal to (|a1|+|a2|)p on rank-one matrices a⊗ν, |ν|=1, and exceeds its convex envelope by a quantity comparable to |det A| |A|p−2, with constants of order (2−p)−1. Consequently, although the original energy density is convex, the limit density is quasiconvex but neither convex nor polyconvex: quasiconvexity is generated by the geometry of the constraint. For p≥2 the constraint survives in the limit as det∇ u=0, and the Γ-limit is the degenerate energy ∫Ω(|∇ u1|+|∇ u2|)p d x, that is, the p-energy of u considered as a map into the normed plane (ℝ2,|·|1), the metric limit of the networks.
Related
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BibTeX
@misc{braides2026energies,
title = {{Energies with values in networks and geometric quasiconvexity}},
author = {Andrea Braides and Jan Kristensen and Sergio Scalabrino},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32444},
eprint = {2610.32444},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32444},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Energies with values in networks and geometric quasiconvexity
AU - Andrea Braides
AU - Jan Kristensen
AU - Sergio Scalabrino
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32444
UR - https://arxiv.org/abs/2610.32444
AB - We study the asymptotic behaviour, as ε→ 0^+, of the p-Dirichlet energy of maps u:Ω⊂ℝ^2 → ℝ^2 constrained to take values in the rescaled periodic network ε𝒵, where 𝒵 is a standard square network in ℝ^2. The oscillations are carried by the constraint in the target, not by the energy density, and their effect depends on p. For 1<p<2 the constraint disappears in the limit: the Γ-limit is an integral functional on the whole of W^1,p(Ω, ℝ^2 ), whose density is given by an asymptotic cell formula on tilted cells. This density is equal to (|a_1|+|a_2|)^p on rank-one matrices a⊗ν, |ν|=1, and exceeds its convex envelope by a quantity comparable to |det A| |A|^p−2, with constants of order (2−p)^−1. Consequently, although the original energy density is convex, the limit density is quasiconvex but neither convex nor polyconvex: quasiconvexity is generated by the geometry of the constraint. For p≥2 the constraint survives in the limit as det∇ u=0, and the Γ-limit is the degenerate energy ∫_Ω(|∇ u_1|+|∇ u_2|)^p d x, that is, the p-energy of u considered as a map into the normed plane (ℝ^2,|·|_1), the metric limit of the networks.
ER -