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

## TL;DR

This paper establishes a necessary condition for solutions of elliptic systems in infinite strips, showing that finite energy solutions must approach specific limits at infinity, extending De Giorgi type conjectures.

## Contribution

It proves that finite energy solutions in infinite strips must converge to zeroes of the potential at infinity, including divergence-free maps on tori, advancing understanding of elliptic systems.

## Key findings

- Solutions have limits at infinity in energy finite cases.
- Convergence occurs in L^2 and almost everywhere.
- Results extend to divergence-free maps on tori.

## Abstract

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