Transmission problems in a thin layer set in the framework of the Hölder spaces: Resolution and impedance concept
Abstract
We consider a family of singular transmission problems depending on some small positive parameter δ set in the juxtaposition of two rectangular domains. They are written in the form of abstract elliptic equations and the study, here, is given in the Hölder spaces completing in this way the work in Lp cases given in [6]. In this first part, we present a new approach for the resolution of these problems by using the concept of impedance operator. This method is different of the one performing a rescaling in the thin layer, see [6]. It leads to obtain direct and simplified problems. We use the Dunford calculus and some techniques similar to that in [10], [7] and [4], to prove existence, uniqueness, results and some specific estimates on the impedance operator. This study will allow us, in a forthcoming work, to obtain respectively optimal regularities and the limit problem when δ → 0.