Energies with values in networks and geometric quasiconvexity

Abstract

We study the asymptotic behaviour, as $\varepsilon\to 0^+$, of the $p$-Dirichlet energy of maps $u\colonΩ\subset\mathbb{R}^2 \to \mathbb{R}^2$ 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 $Γ$-limit is an integral functional on the whole of $\rm{W}^{1,p}(Ω, \mathbb{R}^2 )$, 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 $a\otimesν$, $|ν|=1$, 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 $p\ge2$ the constraint survives in the limit as $\det\nabla u=0$, and the $Γ$-limit is the degenerate energy $\int_Ω(|\nabla u_1|+|\nabla u_2|)^p\, \rm{d} x$, 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.