Elliptic problems in long cylinders revisited

Abstract

The asymptotic study of partial differential equations in cylinders becoming unbounded in one or several directions has known important developments in the last years, especially thanks to the works of Michel Chipot and his collaborators, see for example Chipot ($\ell$ goes to plus infinity, Birkäuser Verlag, Basel, 2002, Asymptotic issues for some partial differential equations, Imperial College Press, London, 2016), Chipot and Mardare (J Math Pures Appl 104:921–941, 2015), Chipot and Rougirel (Discrete Contin Dyn Syst Ser B 1(3):319–338, 2001), Chipot et al. (Asympt Anal 85:199–227, 2013), Chipot and Yeressian (C R Acad Sci Paris Ser I 346:21–26, 2008), Guesmia (Nonlinear Anal 70(9):3320–3331, 2009) and Xie (in: Recent advances on elliptic and parabolic issues, Proceedings of the 2004 Swiss-Japanese Seminar, World Scientific, 2006). In this paper, we prove the convergence to the solution of a linear elliptic problem on an infinite cylinder of the solutions of the same problem taken on larger and larger truncations of the cylinder. Following the methods introduced in Chipot and Yeressian (2008) and Chipot and Mardare (2015), we generalize the result of convergence found in Chipot and Yeressian (2008) for the case where the data is not necessarily independent of the coordinate along the axis of the cylinder. We also consider the non-homogenous Dirichlet problem instead of the homogenous one.