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Résumé : We consider a vectorial monotone problem posed on a cylinder ${\varOmega}^\varepsilon=\omega\times (0,\varepsilon L)$ of $\mathbb{R}^N$ and we study its limit behavior as $\varepsilon$ goes to $0$ . We show that the limit solution solves an elliptic system posed on $\omega$ . We give also a corrector result as well as an error estimate under some regularity hypotheses.