Homogenization of line tension energies (Q6637154)

The authors consider the scaled energy functional: \(E_{\sigma }(\mu )=\int_{\gamma \cap \Omega }\sigma \psi (\frac{b}{\sigma },t)d\mathcal{H}^{1} \), if \(\mu =b\otimes t\mathcal{H}^{1}\llcorner \gamma \in \mathcal{M} _{df}^{1}(\Omega ;\sigma \mathcal{B}\times \mathbb{S}^{2})\), \(E_{\sigma }(\mu )=+\infty \), otherwise, where \(\Omega \subset \mathbb{R}^{3}\) is an open, bounded, uniformly Lipschitz, and simply connected set, \(\gamma \subset \mathbb{R}^{3}\) a 1-rectifiable set, \(t\in \mathbb{S}^{2}\) its tangent, \(\psi :\mathbb{Z}^{N}\times \mathbb{S}^{2}\rightarrow \lbrack 0,+\infty )\) a \(\mathcal{H}^{1}\)-elliptic function that satisfies \(\psi (b,t)\geq \frac{1}{c}\left\vert b\right\vert \), for all \(b\in \mathbb{Z}^{N}\) and \(t\in \mathbb{S}^{2}\), \(b\) a vector that belongs to a discrete lattice \( \mathcal{B}\) in \(\mathbb{R}^{N}\), with \(N\geq 2\), \(\mu =b\otimes t\mathcal{H} ^{1}\llcorner \gamma \) a divergence-free matrix-valued measure, and \( \mathcal{M}_{df}^{1}(\Omega ;\sigma \mathcal{B}\times \mathbb{S}^{2})\) the set of such measures. The purpose of the paper is to describe the \(\Gamma \)-convergence with respect to the weak\(^{\ast }\) topology of measures of \( E_{\sigma }\), as \(\sigma \rightarrow 0\). The limit energy is proved to be: \( E_{0}(\mu )=\int_{\Omega }g(\frac{d\mu (x)}{d\left\vert \mu \right\vert } )d\left\vert \mu \right\vert \), if \(\mu \in \lbrack \mathcal{M}(\Omega )]^{N\times 3}\), \(div(\mu )=0\), \(E_{0}(\mu )=+\infty \), otherwise, where \(g: \mathbb{R}^{N\times 3}\rightarrow \lbrack 0,+\infty )\) is the convex 1-homogeneous function defined as the convex envelope of \(g_{\infty }\) defined as: \(g_{\infty }(A)=\psi _{\infty }(b,t)\), if \(A=b\otimes t\), \(b\in \mathbb{R}^{N}\), \(t\in \mathbb{S}^{2}\), and \(g_{\infty }(A)=+\infty \), otherwise, \(\psi _{\infty }\) being the recession function of \(\psi \). The proof of the \( \Gamma \)-liminf is quite straightforward, based on the subadditivity property of \(\psi \), and on Reshetnyak's theorem to prove that \(E_{0}\) is weakly lower semicontinuous. For the proof of the \(\Gamma \)-limsup, the authors first prove that any divergence-free measure can be approximated strictly with measures that are absolutely continuous with respect to the Lebesgue measure, piecewise constant and divergence-free in dimension \(n\geq 2\), if \(\Omega \subset \mathbb{R}^{n}\) is simply connected. They also prove that piecewise constant measures in \([\mathcal{M}(\Omega )]_{df}^{N\times 3}\) can be approximated with measures in \(\mathcal{M}_{df}^{1}(\Omega ;Q\times \mathbb{S}^{2})\). As a consequence of this main result, they prove that if \( \Omega \subset \mathbb{R}^{3}\) is an open, bounded, uniformly Lipschitz, and simply connected set, then for any \(\mu \in \mathcal{M}_{df}(\Omega ;\mathbb{ R}^{N\times 3})\) there exists a sequence of polyhedral measures \(\mu _{n}\in \mathcal{M}_{df}^{1}(\Omega ;\sigma \mathbb{Z}^{N}\times \mathbb{S}^{2})\) strictly converging to \(\mu \) with respect to the Schatten 1-norm.