Submitted Paper

Inserted: 23 oct 2026
Last Updated: 23 oct 2026

Year: 2026

Abstract:

We study the asymptotic behaviour, as $\varepsilon\to 0^+$, of the $p$-Dirichlet energy of maps $u\colon\Omega\subset\mathbb R^2 \to \mathbb R^2$ constrained to take values in the rescaled periodic network $\varepsilon\mathcal Z$, where $\mathcal Z$ is a standard square network in $\mathbb R^2$. 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 $\Gamma$-limit is an integral functional on the whole of $W^{1,p}(\Omega , \mathbb R^2 )$, 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 $a\otimes\nu$, $
\nu
=1$, and exceeds its convex envelope by a quantity comparable to $
{\rm 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 $p\ge2$ the constraint survives in the limit as $\det\nabla u=0$, and the $\Gamma$-limit is the degenerate energy $\int_\Omega(
\nabla u_1
+
\nabla u_2
)^p d x$, that is, the $p$-energy of $u$ considered as a map into the normed plane $(\mathbb R^2,
\cdot
_1)$, the metric limit of the networks.

Keywords: Homogenization, quasiconvexity, networks, manifold-valued energies

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