Transmission problems in a thin layer set in the framework of the Hِlder spaces: Resolution and impedance concept

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 [A. Favini, R. Labbas, K. Lemrabet, S. Maingot, Study of the limit of transmission problems in a thin layer by the sum theory of linear operators, Rev. Mat. Complut. 18 (1) (2005) 143–176]. 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 [A. Favini, R. Labbas, K. Lemrabet, S. Maingot, Study of the limit of transmission problems in a thin layer by the sum theory of linear operators, Rev. Mat. Complut. 18 (1) (2005) 143–176]. It leads to obtain direct and simplified problems. We use the Dunford calculus and some techniques similar to that in [R. Labbas, Problèmes aux limites pour une équation différentielle abstraite de type elliptique, Thèse dʹétat, Université de Nice, 1987; A. Favini, R. Labbas, S. Maingot, H. Tanabe, A. Yagi, Unified study of elliptic problems in Hölder spaces, C. R. Math. Acad. Sci. Paris 134 (2005); G. Dore, A. Favini, R. Labbas, K. Lemrabet, S. Maingot, A transmission problem in a thin layer, Part I, Sharp estimates, in press], 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 .