We would like to study the asymptotic behaviour of problems set in cylinders. Let $\Omega_{\ell} = (-\ell,\ell) \times(-1,1)$. The simplest case that we can have in mind is to study


\begin{displaymath}\left\{ \begin{array}{lll} - \partial^2_{x_1} u_{\ell} - \par... ...= 0 \ \ {\rm on}\ \ \partial \Omega_{\ell}, \end{array}\right. \end{displaymath}

when $ \ell \rightarrow \infty$ and show in particular that the solution converges toward the solution of the problem in lower dimension


\begin{displaymath}\left\{ \begin{array}{lll} - \partial^2_{x_2} u_{\infty} = f(... ...fty} = 0 \ \ {\rm on}\ \ \partial (-1,1). \end{array} \right. \end{displaymath}