Exponential rate of convergence for elliptic problems in star-shaped cylindrical domains

Abstract

The aim of this paper is to use a technique introduced by Bruyère in order to prove an exponential rate of convergence for solutions to linear elliptic problems in generalized cylinders of the type $\ell {\omega }_1\times {\omega }_2$ , as ℓ goes to infinity and ${\omega }_1$ and ${\omega }_2$ being two fixed bounded domains. This method allows to avoid the use of the iteration technique introduced by Chipot and Yeressian. However, the method of Bruyère only works in the case where ${\omega }_1$ is a ball centered at the origin. We will prove here that this method can be adapted to the case where ${\omega }_1$ is a bounded domain, star-shaped with respect to an open ball centered at the origin. This approach is possible thanks to a remarkable property of such domains, that will be proved in the second section of this paper.