On correctors to elliptic problems in long cylinders

Abstract

The last years have seen the development of the asymptotic study of partial differential equations in cylinders becoming unbounded in one or several directions, particularly under the impetus of Michel Chipot and his collaborators. In this paper, we aim to improve some results that have already been shown about 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. This aim will be realized by the construction of well-adjusted correctors. Thanks to our main results established in Sect. 2 of this paper, we conclude by an application in a particular case (by taking data that does not depend on the coordinate along the cylinder’s axis) that the convergence results that can be obtained using the methods introduced by Chipot and Yeressian (CR Acad Sci Paris Ser I 346:21–26, 2008) are optimal. The particularity here is that the optimality is taken in the sense of “the largest domain where the convergence takes place” instead of the classical optimality of the speed of convergence itself.