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Given a bounded Lipschitz domain $\omega\subset\mathbb{R}^{d-1}$ and a lowersemicontinuous function $W:\mathbb{R}^N\to\mathbb{R}_+\cup\{+\infty\}$ thatvanishes on a finite set and that is bounded from below by a positive constantat infinity, we show that every map $u:\mathbb{R}\times\omega\to\mathbb{R}^N$with \[ \int_{\mathbb{R}\times\omega}\big(\lvert\nablau\rvert^2+W(u)\big)\mathop{}\mathopen{}\mathrm{d}x_1\mathop{}\mathopen{}\mathrm{d}x'<+\infty\] has a limit $u^\pm\in\{W=0\}$ as$x_1\to\pm\infty$. The convergence holds in $L^2(\omega)$ and almost everywherein $\omega$. We also prove a similar result for more general potentials $W$ inthe case where the considered maps $u$ are divergence-free in$\mathbb{R}\times\omega$ with $\omega$ being the $(d-1)$-torus and $N=d$.

A necessary condition in a De Giorgi type conjecture for elliptic systems in infinite strips