Correctors for some asymptotic problems

In the theory of anisotropic singular perturbation boundary value problems, the solution $u_\varepsilon$ does not converge, in the $H^1$-norm on the whole domain, towards some $u_0$. In this paper we construct correctors to have good approximations of $u_\varepsilon$ in the $H^1$-norm on the whole domain. Since the anisotropic singular perturbation problems can be connected to the study of the asymptotic behaviour of problems defined in cylindrical domains becoming unbounded in some directions, we transpose our results for such problems.